Download Unit 8 Statistics and Probability: Probability Models

Unit 8 Statistics and Probability:
Probability Models
Introduction
In this unit, students will understand probability as a way to measure the likelihood of events.
Students will evaluate probabilities based on theoretical models and on doing experiments,
including simple and compound events. Students will use simulation to estimate probabilities.
Random number generators. When simulating experiments, students will use random number
generators, available online or on some graphing calculators. Here are some useful Web sites
for online random number generators.
You can find a random integer generator at www.random.org. It allows you to input
● the number of random integers you want it to generate, up to 10,000 numbers;
● the values that the integers need to be between, up to +/−1,000,000,000;
● the number of columns you want to format it in.
For example, if you want to simulate choosing 20 samples of 5 numbers from 0 to 9, you would
tell it to generate 100 random integers between 0 and 9, and format them in 5 columns.
The Web site also has a coin flipper, dice roller, and playing-card shuffler. The limitation with the
dice roller, however, is that it gives you only one result at a time. A Web site that will allow you
to simulate many dice rolls at a time is www.roll-dice-online.com.
If your classroom does not have access to online random number generators, we have provided
some randomly generated numbers as Blackline Masters, and we explain how to use spinners
to generate random numbers.
Fraction notation. We show fractions in two ways in our lesson plans:
Stacked:
1
2
Not stacked: 1/2
If you show your students the non-stacked form, remember to introduce it as new notation.
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability
I-1
SP7-1
Events and Outcomes
Pages 192–194
Standards: 7.SP.C.5, 7.SP.C.7a
Goals:
Students will describe events as impossible, unlikely, even, likely, or certain.
Students will compare events by deciding which event is more likely or less likely.
Vocabulary: certain, even chance, event, experiment, impossible, likely, line segment,
outcome, probability line, unlikely
Materials:
a bottle cap, such as from a medicine bottle
Introduce experiments and outcomes. Ask students if they have ever done a science
experiment. Have volunteers give examples. Tell students that any time they do something that
has different possible results, and they know what all the possible results are, they are doing a
probability experiment. SAY: Rolling a die is an experiment because there are six possible
results: rolling a 1, 2, 3, 4, 5, or 6. ASK: Is tossing a coin an experiment? (yes) What are the
possible results? (heads or tails) Tell students that the different results of an experiment are
called outcomes. ASK: What are the possible outcomes when two teams play soccer? (win,
lose, or tie)
Hold up a bottle cap and ASK: What are the possible outcomes when I toss this lid in the air?
(landing with the flat side down, flat side up, or on its side) Then toss it several times until you
see all three possibilities come up.
Exercises: What are the possible outcomes when you spin the spinner?
a)
b)
c)
d)
1
2
3
4
3
1
6
5
7
7
8
Answers: a) 1, 2, 3, 4; b) 1, 3, 5; c) 6, 7, 8; d) 7
Introduce events. Tell students that an event is any set of outcomes. For example, when rolling
a die, the event “rolling an odd number” has the set of outcomes 1, 3, and 5.
Exercises:
1. Mandy spins the spinner. What outcomes make up the event of Mandy spinning …
a) a multiple of 3
b) an even number
6 1
c) a number greater than 4
d) a factor of 6
5
2
e) a prime number
f) a multiple of 4
4 3
g) a number greater than 6
Answers: a) 3, 6; b) 2, 4, 6; c) 5, 6; d) 1, 2, 3, 6; e) 2, 3, 5; f) 4; g) no possible outcomes
I-2
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability 2. How many outcomes make up the event of spinning …
a) green (G)
b) an odd number
c) a 5
1
G B
B
7
G
B R
3
3
3
5 5
d) a multiple of 4
1
5
1
5
5 2
2
6
3
5 7
Answers: a) 2, b) 6, c) 3, d) 0
Refer students to the spinner in Exercise 2, part a), above. SAY: Spinning green might seem
like it is a single outcome because there is one color, but in fact two regions have green. There
are six outcomes because there are six regions on the spinner, and you can define events by
combinations of outcomes. Spinning green is a combination of two outcomes, so it is an event.
ASK: How many outcomes are blue? (3) How many outcomes are a primary color—red, yellow,
or blue? (4) SAY: You can describe an event however you want, so there can be many events.
But in this spinner, there are exactly six outcomes.
Introduce events that are impossible or certain. Tell students that an event is impossible if
there are no possible outcomes that produce it. Refer students to Exercise 2 above, and
ASK: Which event is impossible? (spinning a multiple of 4 in part d)) Tell students that an event
is certain if all the possible outcomes produce it. ASK: Which event in is certain? (spinning an
odd number in part b)) SAY: Anything that is not impossible and not certain is in between.
Exercises: John rolls a regular die. Is the event certain, impossible, or in between?
a) rolling a 5
b) rolling an even number
c) rolling a number less than 7
d) rolling a 0
Answers: a) in between, b) in between, c) certain, d) impossible
Comparing likelihoods. Draw on the board:
1
2
ASK: If we use this spinner to play a game in which you win if it lands on 1, and I win if it lands
on 2, is it fair? (no) Why not? (because it should land on 2 more often) Point out that it might not
land on 2 more often, but it should land on 2 more often in most games. SAY: So, landing on 2
is more likely than landing on 1.
Exercises: Which outcome is more likely, spinning white or spinning gray?
a)
b)
c)
Answers: a) gray, b) white, c) gray
SAY: Landing on a bigger region is more likely to happen than landing on a smaller region.
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability
I-3
Exercises: Which is outcome more likely? If they are equally likely, write “neither.”
1 2
9
8
7 6
3
4
5
a) 1 or 2
b) 3 or 6
c) 4 or 5
d) 5 or 9
Bonus: an even number or an odd number
Answers: a) neither, b) 6, c) 5, d) neither, Bonus: an odd number
Showing likelihood on a probability line. Tell students that how likely any event is to happen
is somewhere between impossible and certain. SAY: So you can show how likely something is
to happen by drawing a line segment with “Impossible” at one end and “Certain” at the other
end. Draw on the board:
Impossible
Certain
SAY: This line segment is called a probability line because it shows us whether something will
probably happen or probably not happen. Write on the board:
A. You will see someone you don’t know on your way to school.
B. You will see a kangaroo on your way to school.
C. You will roll a 7 on a regular die.
D. The sun will rise tomorrow in the east.
E. It will rain in the next hour.
F. It will snow in the next hour.
Have volunteers tell you where they would place the event on the line. Students’ answers will
vary as to how likely E and F are, depending on the current weather in your location. Answers to
A, B, C, and D should be approximately as shown below:
CB
Impossible
AD
Certain
Tell students that A is very close to certain, so it is very likely, and B is very close to impossible,
so it is very unlikely. SAY: An event that is more likely to happen than not to happen is at least
somewhat likely, but could be very likely. An event that is more likely not to happen than to
happen is at least somewhat unlikely, but could be very unlikely.
I-4
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability (MP.6) Exercises: Describe the event in terms of how likely or unlikely it is.
a) going to sleep before midnight tonight
b) going outside at some point tomorrow
c) rolling a 6 the next ten times you roll a die
d) rolling a 6 the next time you roll a die
Sample answers: a) very likely, b) very likely, c) very unlikely, d) somewhat unlikely
Now add a marking for “Even” on the probability line to show that an even chance is halfway
between impossible and certain, as shown below:
CB
AD
Impossible
Even
Certain
SAY: Halfway between impossible and certain, I put “Even.” An even chance means that there
is an equal chance of the event happening and of the event not happening. An example of an
event that has an even chance of happening is getting tails when tossing a coin. You may need
to tell students not to confuse “even chance” with “even number,” which are two completely
different concepts. In fact, an even number could have an even chance of happening.
(MP.4, MP.7) Exercises: Jen rolls a regular die. Is there an even chance of Jen rolling …
a) a 3
b) a number greater than 3
c) an even number
d) a prime number
e) a number greater than 4
f) a factor of 20
Answers: a) no, b) yes, c) yes, d) yes, e) no, f) no
Refer students to the exercises they just did. SAY: Parts b), c), and d) all have an even chance
of happening because they consist of 3 of the 6 possible outcomes, which is half. Guide
volunteers to predict where each event above would be on the probability line. ASK: Which two
events are less likely than even? (parts a) and e)) SAY: You can tell they are less likely because
less than 3 of the possible outcomes give the event. ASK: Which is more likely, rolling a 3 or
rolling a number greater than 4? (rolling a number greater than 4) SAY: Two of the possible
outcomes are greater than 4 and only one possible outcomes is 3, so rolling a number greater
than 4 is more likely than rolling a 3. ASK: How many possible outcomes will give a factor of 20?
(four—rolling 1, 2, 4, or 5) So where should it go on the probability line? (see sample answers
below; approximate positions are fine)
CB
Impossible
a)
e)
b), c), d)
f)
Even
AD
Certain
When outcomes are equally likely, the event with more outcomes is more likely.
SAY: When all the outcomes are equally likely, the way they are when you roll a die, the event
with more outcomes is more likely. So rolling a factor of 20 is more likely than rolling a prime
number because there are 4 factors of 20 on a die, but only 3 prime numbers.
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability
I-5
Exercises: a) How many outcomes make up the event of spinning …
8 1
i) an even number
ii) an odd number
2
7
iii) a 7
iv) a 2
7
7
9 4
v) a number less than 3
vi) a number greater than 6
b) Which event is more likely?
i) spinning an odd number or an even number
ii) spinning a 7 or a 2
iii) spinning a number greater than 6 or a number less than 3
Bonus: Order the events from part a) from least likely to most likely.
Answers: a) i) 3, ii) 5, iii) 3, iv) 1, v) 2, vi) 5; b) i) odd, ii) 7, iii) a number greater than 6;
Bonus: from part a), iv) < v) < iii) = i) < vi) = ii)
The importance of outcomes being equally likely to compare the likelihood of events.
Draw on the board:
ASK: Which is more likely, spinning white or spinning gray? (gray) How do you know? (more of
the spinner is gray) SAY: The same number of parts are gray and white—there is one of each—
but the gray part is bigger, so spinning gray is more likely.
Now split the white region into two smaller regions, as shown below:
(MP.3) Tell students that now there are two white parts and only one gray part. ASK: Can we
say that spinning white is more likely than spinning gray? (no) PROMPT: Did that change which
color the spinner is more likely to land on? (no) Why not? (you didn’t change how much of the
spinner is gray or white by dividing the white part into smaller parts) SAY: If you want to count
outcomes to decide which event is more likely, you have to make sure that all the outcomes are
equally likely.
Exercises: Which event is more likely, spinning white or spinning gray?
a)
b)
c)
Answers: a) gray, b) white, c) white
I-6
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability Extensions
(MP.1) 1. Draw a spinner with 10 equal regions, all marked A, B, or C, so that …
a) spinning A and B are equally likely, and spinning C is the most likely.
b) spinning A is more likely than spinning B, and spinning B is more likely than spinning C.
c) spinning A and B are equally likely and spinning C is the least likely, but not impossible.
Sample answers: a) A, B, C, C, C, C, C, C, C, C; b) A, A, A, A, A, B, B, B, C, C
Answer: c) A, A, A, A, B, B, B, B, C, C
(MP.1) 2. Draw a spinner with 10 equal regions, all marked A, B, C, or D, so that …
a) spinning A and B are equally likely, spinning C and D are equally likely, and spinning A is
more likely than spinning C.
b) spinning A, B, and C are equally likely, and spinning D is most likely, but less than even.
c) spinning A, B, and C are equally likely, and spinning D is least likely.
Sample answer: a) A, A, A, B, B, B, C, C, D, D
Answers: b) A, A, B, B, C, C, D, D, D, D; c) A, A, A, B, B, B, C, C, C, D
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability
I-7
SP7-2
Probability
Pages 195–197
Standards: 7.SP.C.5, 7.SP.C.7a
Goals:
Students will understand that probability is a measure of how likely an event is to happen, with
probability 0 meaning impossible and probability 1 meaning certain.
Students will evaluate probabilities of given events when all outcomes are equally likely.
Students will express probabilities as a fraction, a decimal, or a percent.
Prior Knowledge Required:
Can describe events as likely or unlikely
Can compare likelihoods of different events when all outcomes are equally likely
Is familiar with measurements for different attributes
Can convert between fractions, decimals, and percents
Vocabulary: certain, even chance, event, experiment, impossible, likely, outcome, probability
Materials:
two pencils of different lengths
(MP.1) Measuring likelihood. Show students two pencils of different lengths. Ask students
how they could determine which pencil is longer. Then present two measurements that cannot
be compared directly, such as the length of a ruler and the circumference of a cup. (Students
might suggest using a measuring tape to compare them indirectly.) Ask students how they could
compare the weight of two objects, such as a book and a cup, or the temperature in two different
places. Point out that in all cases, students tried to attach a number to the characteristic or
quantity and to compare the numbers. They used different tools—a measuring tape, a scale, or
a thermometer—to get a number, or a measurement. Explain that probability is the branch of
mathematics that studies the likelihood of events and expresses this likelihood in numbers. The
measure of the likelihood of an event is called probability.
Probability assigns numbers between 0 and 1 to likelihoods of events. Tell students that
an event that is impossible has probability 0 and an event that is certain has probability 1, and
any event has a probability that is between 0 and 1. SAY: Halfway between 0 and 1 is 1/2, or
50%, so an event that has an even chance of occurring—neither more nor less likely to occur
than not to occur—would have probability 1/2, or 50%.
Exercises: What is the probability of spinning red?
a)
b)
c)
G
B
R
B
R
R
Answers: a) 0, b) 1/2, c) 1
I-8
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability (MP.8) For each spinner in the exercises above, ASK: What is the fraction of the spinner that is
red? (0, 1/2, 1) How does this compare to your answers above? (they are the same) Have
students use the same pattern to complete the following exercises.
(MP.7) Exercises: What is the probability of spinning red?
a)
b)
c)
Y
B
R
G
B
G
R
R
d)
R
G
R
R
Answers: a) 0, b) 1/3, c) 2/3, d) 1
SAY: When all outcomes are equally likely, the probability of spinning red is the fraction of
outcomes that are red. This is always a fraction between 0 and 1. When one third of the spinner
is red, then the probability of spinning red is 1/3.
In the following exercises, some students might need the added structure shown below:
______ red outcome(s)
______ outcomes in all
The probability of red is ________
(MP.7) Exercises: What is the probability of spinning red?
a)
b)
c)
R
G
R
B
G
Y
R R
R
R
R Y
Answers: a) 1/2, b) 1/4, c) 5/6
SAY: All the probabilities are fractions between 0 and 1. That’s because the number of
outcomes that are red cannot be less than 0 and cannot be more than the total number of
outcomes. So the fraction of outcomes that are red cannot be less than 0 or more than 1.
A formula for finding probability. SAY: When all outcomes are equally likely, the probability of
any event is the fraction of outcomes that make up that event. Write on the board:
Probability =
# of outcomes when the event happens
# of outcomes in total
SAY: The spinner with the most red has the greatest likelihood of spinning red—it also has the
highest fraction of red outcomes, so it has the greatest probability of spinning red. That’s good
because it tells us that the way we defined probability is a good measure of how likely an event
is to happen. Since probabilities are numbers between 0 and 1, they can be compared.
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability
I-9
(MP.2) Exercises: Is spinning red or blue more likely? Explain how you know.
1
1
and the probability of spinning blue is .
a) The probability of spinning red is
3
4
2
3
and the probability of spinning blue is .
b) The probability of spinning red is
5
8
Answers: a) red, because 1/3 > 1/4; b) red, because 2/5 > 3/8 (or 16/40 > 15/40)
Attaching numbers to probability lines. SAY: When an event is impossible, its probability
is 0. When an event is certain, its probability is 1. So you can put 0 and 1 at the ends of a
probability line. Draw on the board:
1
2
1
Even
Certain
0
Impossible
SAY: I put “Even” at the halfway point because it is neither more likely to happen nor more likely
not to happen. I put “1/2” at the halfway point because it is halfway between 0 and 1. So an
event with an even chance of happening has probability 1/2.
Exercises:
(MP.1, MP.4) 1. Copy the probability line and show where you would put the probability of …
A. spinning red
B. spinning blue
C. spinning yellow
R
Y
Y
R R
Y Y
B
Y
Y
Answers:
0
B
A
C
1
(MP.7) 2. A 6-sided die has 2 sides colored red, 1 side colored blue, and 3 sides colored yellow.
What is the probability that the die lands with …
a) a red side face up
b) a yellow side face up
c) a blue side face up
d) a green side face up
e) a primary color side face up
Answers: a) 2/6 or 1/3, b) 3/6 or 1/2, c) 1/6, d) 0 or 0/6, e) 6/6 or 1
SAY: A class has 11 girls and 14 boys. I want to pick a student at random from that class. That
means all students are equally likely to be chosen. ASK: What is the fraction of students who
are girls? (11/25) What is the probability that the student is a girl? (11/25) Point out that the two
answers are the same because that’s the formula for how to find the probability.
I-10
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability Writing probabilities as decimals and percents. SAY: Because a probability is just a number
between 0 and 1, it can be written as a fraction, a decimal, or percent. Write on the board:
11
25
SAY: The first step to writing the fraction as a decimal is to write it as a fraction with denominator
equal to a power of 10. ASK: What is the smallest power of 10 we can use? (100) Write on the board:
11
=
25 100
ASK: What is the numerator when the denominator is 100? (44) ASK: How would you write
44 hundredths as a decimal? (0.44) And as a percent? (44%) SAY: A percent is just a
hundredth, so if the decimal is already in hundredths, you can immediately write it as a percent.
But sometimes you have to change it to hundredths. Write on the board:
4
5
Ask volunteers to change it to a decimal (0.8), then to a percent (80%).
Exercises:
(MP.7) 1. Write the probability of spinning gray as a fraction, a decimal, and a percent.
a)
b)
c)
Answers: a) 1/2, 0.5, 50%; b) 7/10, 0.7, 70%; c) 3/4, 0.75, 75%
(MP.4) 2. Estimate the probability of spinning gray. Write your answer to the nearest 10%.
a)
b)
c)
d)
Sample answers: a) 40%, b) 20%, c) 60%, d) 10%
(MP.4) Probabilities as decimals and percents in real life. Tell students that probabilities are
often written as decimals or percents in real life. Write on the board:
a) The probability that it will rain.
b) The probability that a given baseball batter will get a hit.
c) The probability of a flood occurring.
d) The probability that a given hockey goalie will make a save on the next shot.
e) The probability that a given basketball player will sink the next free throw.
Have volunteers tell you whether each probability is usually seen as a decimal or as a percent.
(a) percent, b) decimal, c) percent, d) percent, e) percent)
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability
I-11
Exercises: Write the probability as a fraction in lowest terms.
a) The probability of rain is 60%.
b) The probability of the next free throw being successful is 75%.
c) The probability of a given baseball player getting a hit is 0.350, meaning the player gets a hit
350 times out of every 1,000 chances.
Answers: a) 3/5, b) 3/4, c) 7/20
Making equal outcomes from unequal outcomes. Draw on the board:
SAY: Two of the five parts are gray. ASK: Is the probability of spinning gray equal to 2/5? (no)
How do you know? PROMPT: What fraction of the spinner is gray? (1/4) So what is the real
probability of spinning gray? (1/4) SAY: If you want the actual probability, you need to divide the
spinner into equal outcomes. Demonstrate this on the board, as shown below:
SAY: Now there are still 2 gray parts but there are 8 parts altogether, so the probability of
spinning gray is 2/8. ASK: Is that the same as we got before? (yes, 2/8 = 1/4)
(MP.1) Exercises: Divide the spinner to make all outcomes equally likely. Then find the
probability of spinning gray.
a)
b)
c)
Bonus:
Answers: a) 2/4 = 1/2, b) 1/6, c) 3/10, Bonus: 11/20
Extensions
1. Find the probability of the event.
a) A factor of 24 is chosen from the numbers 1 to 24.
b) A multiple of 3 is chosen from all the composite numbers (numbers with more than two
factors) from 1 to 20.
c) A multiple of 3 is chosen from the factors of 48.
d) A word is formed from writing the letters in ART in random order.
e) A word is formed from writing the letters from OPST in random order.
Answers: a) 1/3; b) 5/11; c) 1/2; d) 1/2, the three words are ART, RAT, and TAR;
e) 1/4, the six words are OPTS, POST, POTS, SPOT, STOP, TOPS
I-12
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability (MP.1) 2. Billy has 13 coins with a total value of 25¢. If Billy loses one coin, what is the
probability that it will be a nickel?
Answer: The coins are 3 nickels and 10 pennies, so the probability that a nickel is lost is 3/13.
(MP.3) 3. a) Tess randomly picks a letter of the alphabet from a bag. What is the probability that
she picks a letter from A to M?
b) A teacher randomly picks a student from the class. Is the probability that the student’s name
starts with a letter from A to M equal to 1/2? Hint: Is each letter equally likely?
Answers: a) 1/2; b) no, because the letters are not all equally likely to start a name
(MP.3) 4. Sam randomly picks a marble from a bag. The probability of picking a red marble
2
is . What is the probability of not picking red? Explain.
5
Answer: 3/5, because if 2 out of every 5 marbles are red, then 3 out of every 5 marbles are
not red.
(MP.1) 5. a) Use the spinner to determine the probability of spinning red …
i) by adding fractions
ii) by breaking the spinner into smaller parts
B
R
G
Y
R
b) Is your answer to part a) more than half or less than half? Why does that make sense?
c) Do the Bonus in the previous exercises by adding fractions instead of dividing the spinner into
equal outcomes. Make sure you get the same answer.
d) How is adding fractions similar to dividing the spinner into equal outcomes?
Answers: a) i) 1/6 + 1/4 = 5/12, ii) break the pieces so that each piece has size 1/12, 5 pieces
are red so the probability of red is 5/12; b) less than half—this makes sense because half the
right side of the spinner is red, but less than half of the left side is red, so less than half of the
spinner overall is red; c) 1/4 + 3/10 = 5/20 + 6/20 = 11/20; d) the common denominator of the
fractions you are adding is the number of parts you divide the spinner into
(MP.3) 6. Is it possible for a spinner to have probability
2
5
of spinning red and probability
of
5
8
spinning blue? Explain.
Answer: No, because 2/5 + 5/8 = 41/40, which is greater than 1. The blue and red regions
would have to make up more than the whole spinner.
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability
I-13
SP7-3
Expectation
Pages 198–199
Standards: 7.SP.C.6
Goals:
Students will determine the expected number of times an event will happen when an experiment
is repeated a given number of times.
Vocabulary: certain, even, event, experiment, impossible, likely, outcome, probability,
probability line, unlikely
Materials:
transparency of BLM Spinner (p. I-55)
BLM Spinner (p. I-55)
a paper clip and a pencil for each student
Expected number of a given event when repeating an experiment. Project the transparency
of BLM Spinner. ASK: If I spin this spinner 20 times, how many times would you expect it to
spin gray? (take all guesses) Then do 20 trials and record the number of gray outcomes. To
spin the spinner, place the tip of a pencil at the center of the spinner and a paper clip around the
pencil point. Tell students that if the spinner lands between a white region and a gray region,
you will not count that result.
Provide each student with BLM Spinner, a pencil, and a paper clip. Have students spin the
spinner 20 times and keep a tally of how many times they spin white and how many times they
spin gray. Students should not count any spins that land exactly between two regions. Have
several students record their results on the board. Then ask the class how many got gray
exactly one time, two times, and three times, and record the results. Continue tallying the results
until all students have answered. ASK: What appears to be the expected number of grays, now
that you have seen everyone’s results? (5)
SAY: The expected number of grays seems to be 5. This makes sense because 1/4 of the
spinner is gray and 1/4 of 20 is 5. Write on the board:
1
of 20 = 20 ÷ 4 = 5
4
SAY: To show one fourth of 20 dots, you can divide the dots into 4 equal groups and take one of
the groups. Since you can use 20 ÷ 4 to find how many dots are in each group when you have 4
equal groups, you can use 20 ÷ 4 to find one fourth of 20.
(MP.2) Exercises: How many times would you expect the spinner to land on gray if you spin it …
a) 40 times
b) 100 times
c) 32 times
d) 1,000 times
Answers: a) 10, b) 25, b) 8, d) 250
I-14
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability Using proportions to find expectation. SAY: One out of every four spins should land on gray,
so you can also solve the question by using a proportion. Write on the board:
1
=
4 20
Have a volunteer solve the proportion (5) and point out how both the numerator and
denominator are multiplied by 5. SAY: So if 1 out of every 4 spins land on gray, then 5 out of the
20 spins should land on gray.
(MP.1) Exercise: Solve the previous exercises again, this time using proportions. Make sure
you get the same answers as before.
The expected value doesn’t have to be likely to occur. Refer students to the example above.
Point out that 5 isn’t expected in the sense that most people got gray exactly 5 times. That didn’t
happen at all. But 5 is expected in the sense that the class average was probably very close to
5. If you added up all the number of grays and all the spins altogether, the number of gray spins
will be very close to one quarter of all the spins. You might wish to do this to demonstrate the
point that even with a large number of spins, you won’t get exactly a quarter being gray, but you
will get very close to a quarter.
The expected number of times when the probability has a numerator greater than 1.
ASK: If the spinner is expected to land on gray 5 times out of 20, how many times out of the 20
times is it expected to land on white? (15) How did you get that? (students might subtract
20 − 5, or students might multiply by 3) If both solutions do not come up, ask for other possible
ways of seeing it. Once you have elicited both solutions, SAY: You would expect that 3 out of
every 4 spins will land on white. Write on the board:
1
3
of 20 = 3 ´ of 20
4
4
=3×5
3
of 20 is 3 of 4 equal groups
4
= 15
SAY: An event that happens 3/4 of the time should happen three times as often as an event that
happens only one quarter of the time.
Exercises: Megan spins the spinner 30 times. How many times should she expect the spinner
to land on red (R)?
a)
b)
c)
d)
R B
R B
R B
R B
O
G
R
Y
R
Y
R
R
Y
G
R
R
Answers: a) 6, b) 12, c) 18, d) 24
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability
I-15
Using proportions to find expectation when the probability has a numerator greater than 1.
SAY: Again, you can use proportions to solve these problems. Write on the board:
3
=
4 20
Have a volunteer solve the proportion (15) and point out that both the numerator and
denominator were multiplied by 5. SAY: If 3 out of every 4 spins should land on white, then you
can expect 15 out of the 20 spins to land on white.
(MP.1) Exercises: Use proportions to solve the previous exercises. Make sure you get the
same answer.
NOTE: Later in the unit, students will solve the same type of problem using the experimental
probability instead of the theoretical probability.
(MP.4) Exercises: Pretend that three cereal companies offer prizes.
Company A: 1 out of every 3 boxes wins $2
Company B: 1 out of every 4 boxes wins $3
Company C: 1 out of every 6 boxes wins $5
a) If you buy a pack of 24 boxes of each type of cereal, how much money do you expect to win
from each company?
b) Which company would you buy the 24 boxes from if you want better prizes?
Answers: a) $16 from Company A, $18 from Company B, $20 from Company C; b) Company C
Deciding whether given results are likely or unlikely. Then, referring back to BLM Spinner,
write two more results on the board:
White: 6
Gray: 14
White: 14
Gray: 6
(MP.3) ASK: Which of these results do you think would be more likely? (White: 14, Gray: 6)
Why? (because there is more chance to land on white than on gray)
(MP.4) Exercises:
1. If you spin the spinner 25 times …
a) how many times would you expect to spin gray?
b) which of the charts shows a result you would be most likely to get?
A.
Gray
White
B.
Gray
White
C.
Gray
|||| ||||
|||| |||
|||| ||||
|||| ||||
|||
||||
|
|||| ||
White
|||| ||||
|||| ||||
||
c) Which result would surprise you the most? Why?
Answers: a) 20; b) Chart A; c) Chart C because the spinner is mostly gray, but the results in
Chart C show many more white than gray
I-16
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability (MP.4) 2. Matt spins a spinner 300 times and gets red 96 times. Which spinner did he likely
use? Explain.
A.
B
B
R B
G Y
B.
G
Y
C.
R
D.
R
B
R
Y
B
R
B
G
Y
Answer: Spinner C because about 1/3 of the spins were red.
Extensions
(MP.4) 1. The pie chart shows the fraction of students who walk, ride the bus, bike, or
skateboard to school. If you surveyed 200 students from the school, about how many would you
expect to have biked to school?
walk
bus
bike
skateboard
Answer: 25
(MP.4) 2. The chart shows the survival rate, or how many birds will survive, under two different
environmental protection programs. If a program could be implemented in only one
forest, which one would you choose? Explain.
Forest A
Forest B
Number of Endangered Birds
5,000
15,000
Survival Rate
80%
2 in 5
Answer: The program in Forest B because it will save 6,000 birds and the program in Forest A
will save only 4,000 birds.
3. (MP.4) a) A batting average of .427 means a baseball player had 427 hits in 1,000 times at
bat. How many hits, on average, would each player likely get in 60 times at bat?
i) Player A has a .200 batting average
ii) Player B has a .250 batting average
iii) Placer C has a .400 batting average
iv) Player D has a .350 batting average
(MP.1) b) Check that your answers make sense. Are the answers in the order you expect based
on the decimals? Is Player B’s number more than Player A’s number by the same amount that
Player C’s number is more than Player D’s? Is the difference between Player D and Player B’s
numbers twice that amount?
Answers: a) i) 12, ii) 15, iii) 24, iv) 21; b) yes, 12 < 15 < 21< 24 with differences of 3 between
Player B and Player A, 6 between Player B and Player D, and 3 between Player D and Player C
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability
I-17
4. These nets are folded together to make a die and each die is rolled 300 times.
Match each die net to the correct statement.
A.
B.
C.
D.
4
1
2
3
4
5
3
4
6
2
3
1
3
5
2
5
4
3
5
2
5
6
1
4
a) I would expect to roll 4 about 50 times.
b) I would expect to roll an even number about 150 times.
c) I would expect to roll 3 about 150 times.
d) I would expect to roll 1 about the same number of times as rolling 5.
Answers: a) D, b) A, c) C, d) B
(MP.3) 5. If there are 6,000,000,000 people in the world, how many would you expect to be born
on February 29th? Explain your reasoning and your assumptions.
Answer: February 29th occurs approximately once every four years, and so about 1/1,461 of
the time. So you would expect about 1/1,461 × 6,000,000,000 ≈ 4,106,776 people to have been
born on a February 29th. This assumes that all birthdays are equally likely.
6. Read the following problem:
In what year did women in the United States gain the right to vote?
A. 1916
B. 1917
C. 1918
D. 1919
E. 1920
a) If you guess randomly, what is the probability of answering correctly?
b) On a test of 30 similar questions, how many questions would you expect to guess correctly?
Incorrectly? Assume you answer each question blindly.
c) You get 4 points for each correct answer and −1 for each incorrect answer. What do you
expect your final score to be?
Answers: a) 1/5 or 20%, b) 6, c) 4 × 6 + (−1) × 24 = 0
I-18
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability SP7-4
Tree Diagrams
Pages 200–201
Standards: 7.SP.C.8b
Goals:
Students will draw tree diagrams to represent the total number of outcomes from consecutive or
simultaneous experiments.
Prior Knowledge Required:
Can list the outcomes from a single experiment
Vocabulary: experiment, outcome, tree diagram
Introduce tree diagrams. Explain to students that mathematicians often use tree diagrams
when they have to make choices and they want to keep track of all the possible combinations.
For example, Katie has 3 pairs of mittens: a white pair, a blue pair, and a green pair. She also
has 2 hats: a white hat and a blue hat. Tell students that you want an organized way to keep
track of her options, and one way of doing that is to use a tree diagram. SAY: Let’s start with
choosing the hat. Draw on the board:
white hat
blue hat
SAY: Now let’s choose the mittens. If Katie chooses the white hat, she has three options for
mittens. She could pick the white mittens, the blue mittens, or the green mittens. ASK: What if
she chooses the blue hat—what options does she have for the mittens? (the white mittens, the
blue mittens, or the green mittens) Continue drawing on the board:
white hat
white
blue
blue hat
green
white
blue
green
SAY: We can also write a short form by just writing the initial letters of the colors: W for white, B
for blue, and G for green. Draw on the board:
W
W
B
B
G
W
B
G
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability
I-19
SAY: This is called a tree diagram because it looks like an upside down tree with branches.
Leave this tree diagram on the board for later reference.
Exercises:
a) A restaurant offers three main courses: chicken (C), fish (F), or tofu (T). It also offers four
desserts: chocolate cake (C), chocolate ice cream (I), a fruit plate (F), or pie (P). Draw a tree
diagram to show all the possible outcomes for one meal.
b) A camp offers two options in the morning: drama (D) or visual arts (V). It also offers five
options in the afternoon: swimming (S), tennis (T), baseball (B), karate (K), or football (F). Draw
a tree diagram to show all the possible outcomes for one day.
c) Draw a tree diagram to show the possible outcomes if you were to roll a die, then toss a coin.
Answers:
a)
C
C
I
F
F
P
C
I
T
F
P
C
I
F
P
b)
D
S
T
V
B
K
F
S
T
B
K
F
c)
1
H
2
T
H
3
T
H
4
T
H
5
T
H
6
T
H
T
Refer students to the example on the board. Point out how the tree diagram shows all the
outcomes. SAY: You can write each outcome underneath its path. Do so for the first path (WW)
and have volunteers dictate the remaining paths. (WB, WG, BW, BB, BG) Then have students
highlight, in part a) from the exercises above, the path that shows tofu and cake.
Tree diagrams can have more than two levels. SAY: All of the tree diagrams above have two
levels of branches, but you can have three levels of branches, too.
I-20
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability Exercise: Draw a tree diagram to show the possible outcomes of tossing a coin three times.
Answer:
H
T
H
H
T
T
H
H
T
H
T
T
H
T
Counting outcomes. Refer students to part a) from the first set of exercises in this lesson.
ASK: How can you see from the tree diagram how many outcomes there are in total? (the total
is the number of outcomes written on the bottom row) SAY: The bottom row of the tree has 4
branches from the first main course, 4 branches from the second main course, and 4 branches
from the third main course. So the total number of options is 4 + 4 + 4 = 12, or 3 × 4 = 12.
SAY: When you have two or more experiments done in a row, the rule to get the total number
of outcomes is to multiply the number of outcomes from each experiment. That allows you to
find the number of outcomes even when it is very large, because now you don’t need a tree
diagram.
(MP.7) Exercises: How many outcomes are there when you …
a) toss two coins
R
B
b) toss a coin, then spin the spinner
c) spin the spinner twice
G
d) toss two coins, then spin the spinner
Answer: a) 4, b) 6, c) 9, d) 12
Demonstrate part a) below to review the definition of a power.
(MP.7) Bonus: a) Write the number of outcomes for the given event as a power.
i) roll three regular dice
ii) toss a coin eight times
iii) toss eight coins
iv) toss 100 coins
v) roll eight regular dice
b) Write the number of outcomes as a product of powers when you roll three dice, then toss
five coins.
Answers: a) 63, b) 28, c) 28, d) 2100, e) 68, f) 63 × 25
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability
I-21
Extensions
(MP.1) 1. Kim is going to a summer camp with different activities. The camp has a rule that if
you pick an indoor activity in the morning, you must pick an outdoor activity in the afternoon.
Outdoor activities: tennis (T), swimming (S), or football (F)
Indoor activities: cards (C), or board games (B)
a) Draw a tree diagram to show all of Kim’s options.
b) How many outcomes are there altogether?
c) How many outcomes have outdoor activities in the morning and in the afternoon?
d) If Kim picks randomly, would you describe the chances of doing outdoor activities in both the
morning and the afternoon as likely, unlikely, or about even? Explain.
Answers:
a)
T
S
F
C
B
T S F C B T S F C B T S F C B T S F
T S F
b) 21, since there are 21 outcomes in the bottom row of the tree diagram; c) 9, by reading
from the tree diagram: TT, TS, TF, ST, SS, SF, FT, FS, FF; d) 9 is close to half of 21, so
about even.
(MP.7) 2. Han performs an experiment by rolling a die with 20 faces and a die with 10 faces,
once each. He draws a tree diagram to see all the possible outcomes. How many paths will his
tree diagram have?
Answer: 200
3. In one pocket, Alexa has a $1 bill, a $5 bill, and a $10 bill. In the other pocket, she has a
$5 bill, a $10 bill, and a $20 bill. She randomly pulls out one bill from each pocket. What is the
probability that she pulls out at least $20 in total?
Answer: The probability is 4/9. The following tree diagram shows the 9 outcomes with the 4 that
have at least $20 circled:
1
5
I-22
10
5
20
5
10
10
20
5
10
20
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability (MP.4) 4. The color of a flower is determined by its genes. A red rose, for example, has two
R genes, a white rose has two r genes, and a pink rose has one R gene and one r gene. If two
pink roses are crossbred, the “child” rose can be red, white, or pink. The possible results are:
Gene from parent 1
Gene from parent 2
R
Resulting gene
RR
Color
red
R
r
Rr
pink
R
rR
pink
r
r
rr
white
a) When two pink roses are crossbred, what is the probability the resulting rose will be pink?
b) When a pink rose is crossbred with a red rose, what is the probability that the resulting rose
will be red? Pink? White?
c) When a pink rose is crossbred with a white rose, what is the probability that the resulting rose
will be pink? Red? White?
d) What happens when a red rose is crossbred with a white rose?
Answers: a) 2/4 or 1/2, b) red = 1/2, pink = 1/2, white = 0; c) pink = 1/2, red = 0, white = 1/2;
d) the result is always a pink rose
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability
I-23
SP7-5
Charts and Organized Lists
Pages 202–204
Standards: 7.SP.C.8a, 7.SP.C.8b
Goals:
Students will list all the outcomes from two or more consecutive or simultaneous experiments by
drawing charts and making organized lists.
Students will use the list of all the outcomes to evaluate probabilities of given events.
Prior Knowledge Required:
Can draw tree diagrams to list outcomes for consecutive experiments
Can evaluate probabilities when all outcomes are equally likely
Vocabulary: chart, event, experiment, fair, likely, organized list, outcome, probability,
tree diagram
Review drawing tree diagrams.
Exercise: Draw a tree diagram showing all the outcomes for rolling two 4-sided dice with sides
labeled 1, 2, 3, and 4.
Answer:
1
1
2
2
3
4
1
2
3
3
4
1
2
4
3
4
1
2
3
4
Tree diagrams are hard to draw when there are many outcomes for each experiment.
When students finish, SAY: Let’s try to draw a tree diagram to show all the outcomes for rolling
two dice with 12 sides. Demonstrate on the board how it becomes cramped very easily, even
with the large space on the board. SAY: Drawing it in your notebook would be impossible. In this
class, we’re going to learn a way that makes it easy to list all the 144—that’s 12 × 12—outcomes.
Using charts to list outcomes. SAY: Let’s start by seeing how to draw a chart to list all the
outcomes of tossing two coins, or tossing a coin twice. Each coin has two outcomes, so draw a
chart with 2 rows and 2 columns. Draw on the board:
H
T
H
T
I-24
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability SAY: Each box represents an outcome, so there are four outcomes. If you toss the same coin
twice, the chart shows the outcomes of both tosses. If you toss two different coins at the same
time, the chart shows the outcomes of both coins. Point to the top-left corner box and SAY: This
box represents the outcome of heads on both coins. Write “HH” in the box. Point to the other
boxes, one at a time, and ASK: What does this box represent? (the top-right and bottom-left
boxes both represent a head and a tail; the bottom-right box represents two tails) Fill in the rest
of the chart, as shown below:
H
T
H
HH
HT
T
TH
TT
SAY: You don’t need to write the outcomes in the boxes to show all the outcomes, though.
Exercise: Use the chart format to show all the outcomes of rolling two dice with …
a) 4 sides, labeled 1, 2, 3, and 4
b) 12 sides, labeled 1 to 12
Selected answer:
a)
1 2 3 4
1
2
3
4
(MP.8) Using charts to show that the number of outcomes of two experiments is the
product of the number of outcomes for each experiment. Refer students back to the
previous exercises and ASK: How many outcomes does rolling the two 4-sided dice have? (16)
Is that the same answer you got when you did the tree diagram? (yes) How many outcomes
does rolling the two 12-sided dice have? (144) How did you get that? (multiplied 12 by 12)
SAY: The chart being a 12 by 12 array makes it clear that you multiply the number of outcomes
of one die by the number of outcomes of the second die.
Using charts to find probabilities. Tell students that you want to find the probability that, when
you roll the two 4-sided dice, the numbers you get are one apart. Draw on the board:
1
2
3
4
1
2
3
4
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability
I-25
SAY: It might be helpful to write the outcomes in the boxes. Write on the board:
1
2
3
4
1
1,1
1,2
1,3
1,4
2
2,1
2,2
2,3
2,4
3
3,1
3,2
3,3
3,4
4
4,1
4,2
4,3
4,4
Have a volunteer shade all the outcomes in which the numbers are one apart. (see answers
below)
1
2
3
4
1
1,1
1,2
1,3
1,4
2
2,1
2,2
2,3
2,4
3
3,1
3,2
3,3
3,4
4
4,1
4,2
4,3
4,4
ASK: How many outcomes are shaded? (6) How many outcomes are there altogether? (16) Are
all the outcomes equally likely? (yes) SAY: Because all the outcomes are equally likely, the
probability is the fraction of outcomes that have the property we are looking for. So the
probability of the two numbers rolled being one apart is 6/16 or 3/8.
For the following exercises, students should draw the chart in pen. But they should do the
shading in pencil for each question, then erase the shading before going on to the next
question. Some students might find it helpful to write the outcomes in the chart. If so, they
should also write the entries in pen, so that they have the entries for every question.
(MP.1) Exercises: Draw a chart showing all the outcomes of rolling two regular 6-sided dice.
Then find the probability of the event when …
a) the numbers are equal
b) the numbers add to 7
c) the numbers add to 4
d) you roll two 4s
e) the numbers add to 8
f) you roll at least one 4
g) the numbers are one apart
h) the numbers add to an odd number
i) the numbers add to an even number
Answers: a) 6/36 or 1/6, b) 6/36 or 1/6, c) 3/36 or 1/12, d) 1/36, e) 5/36, f) 11/36, g) 10/36 or
5/18, h) 18/36 or 1/2, i) 18/36 or 1/2
I-26
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability When students finish the exercises above, ASK: Is the probability of rolling two 4s greater than
or less than the probability of rolling numbers that add to 8? (less) Why does that make sense?
(because the two 4s is only one way of rolling a total of 8—there are other ways, too)
(MP.1, MP.3) Exercises: Find as many examples as you can in the previous exercises in which
one probability has to be less than the other probability because it is part of it. Then verify
whether that is the case.
Sample answers: d) < f), b) < h), g) < h), c) < i), e) < i), a) < i)
Selected solutions: b) < h) because 7 is odd; g) < h) because two consecutive numbers
always add to an odd number
Making organized lists. Tell students that there are many ways to list outcomes. SAY: So far, we
have seen tree diagrams and charts. You can also make an organized list. Write on the board:
Roll a die, then toss a coin.
SAY: Let’s imagine there’s a game that involves rolling a die, then tossing a coin. ASK: What
are the outcomes for rolling a die? (1, 2, 3, 4, 5, and 6) What are the outcomes for tossing a
coin? (heads and tails) SAY: To start making an organized list to find all the outcomes, write
each outcome for rolling a die twice, because the second process—tossing a coin—has 2
outcomes. Then, beside each outcome from rolling the die, write the two outcomes from tossing
the coin. Demonstrate on the board, as shown below:
Die
1
1
2
2
3
3
4
4
5
5
6
6
Coin
Die
1
1
2
2
3
3
4
4
5
5
6
6
Coin
H
T
H
T
H
T
H
T
H
T
H
T
Keep this on the board for later use.
Exercises: Make an organized list to show all the outcomes. Use the spinner shown.
a) toss a coin, then spin the spinner
R
B
b) spin the spinner, then toss a coin
c) roll a 4-sided die, then spin the spinner
G
d) spin the spinner twice
Bonus: Yu and Marco play Rock, Paper, Scissors. Hint: Yu winning is an event, not an outcome.
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability
I-27
Answers:
a) Coin Spinner
H
R
H
B
H
G
T
R
T
B
T
G
Bonus:
b) Spinner Coin
R
H
R
T
B
H
B
T
G
H
G
T
c)
Die
1
1
1
2
2
2
3
3
3
4
4
4
Spinner
R
B
G
R
B
G
R
B
G
R
B
G
d) Spinner Spinner
R
R
B
R
G
R
B
R
B
B
B
G
R
G
B
G
G
G
Yu
Marco
Rock
Rock
Rock
Paper
Rock
Scissors
Paper
Rock
Paper
Paper
Paper
Scissors
Scissors
Rock
Scissors
Paper
Scissors Scissors
Using organized lists to find probability. Refer students back to the example on the board
that involves rolling a regular die, then tossing a regular coin. SAY: I want the probability of
rolling an odd number and tossing heads. So I’m going to list all the outcomes in a row, then
underline all the outcomes where that happens. Write on the board:
1H
1T
2H
2T
3H
3T
4H
4T
5H
5T
6H
6T
Pointing to each outcome, have students signal thumbs up if you should underline it, and
thumbs down if you should not. The final picture should look like this:
1H
1T
2H
2T
3H
3T
4H
4T
5H
5T
6H
6T
SAY: 3 of the 12 outcomes have what I need, and all outcomes are equally likely, so the
probability is 3/12 or 1/4. Write on the board:
3
1
=
12 4
I-28
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability (MP.1) Exercises: Yu spins the spinner, then rolls a 6-sided die. Find the probability that …
a) the sum is odd
b) the sum is 7
1
2
c) the product is even
3
d) the product is odd
e) the product is 6
Answers: The list of outcomes is:
1,1
1,2
1,3
1,4
1,5
1,6
2,1
2,2
2,3
2,4
2,5
2,6
3,1
3,2
3,3
3,4
3,5
3,6
a) 9/18 or 1/2, b) 3/18 or 1/6, c) 12/18 or 2/3, d) 6/18 or 1/3, e) 3/18 or 1/6
Bonus: In Rock, Paper, Scissors, what is the probability of a draw?
Answer: 3/9 or 1/3
Fair individual games. SAY: A game is fair if the probability of winning is the same as the
probability of losing. Draw on the board:
R
B
B
R
R
R
B
B
SAY: Tony plays a game in which he spins both spinners. He wins if he spins blue both times,
and loses if he spins red both times. The game is a draw if he spins one red and one blue. Have
students predict if the game is fair, or if he will win or lose more often. Then SAY: To find out,
let’s start by listing the outcomes. There is one red and two blue outcomes on the first spinner,
so we have to list them both in our list. ASK: How many times do we have to list the red from the
first spinner? (5 times) Why? (because the second spinner has 5 outcomes) Show the first table
below on the board:
1st B
2nd B
First
Spinner
R
R
R
R
R
B
B
B
B
B
B
B
B
B
B
Second
Spinner
First
Second
Spinner Spinner
R
R
R
R
R
R
R
B
R
B
B
R
B
R
B
R
B
B
B
B
B
R
B
R
R
B
B
B
B
B
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability
I-29
SAY: There are 3 red outcomes and 2 blue outcomes on the second spinner, so we have to
write those beside each outcome on the first spinner. Do so as shown in the second table,
above, then ASK: How many outcomes have two red? (3) How many outcomes have two blue?
(4) Is it a fair game? (no) Does Tony have an advantage or a disadvantage? (an advantage)
(MP.3, MP.4, MP.6) Exercises: Decide if the game is fair. Explain how you know.
a) Blanca tosses two coins. She wins if she tosses heads both times, and loses if she tosses
tails both times. The game is a draw if she tosses one head and one tail.
b) Amit spins the spinner shown below, then tosses a regular die. He wins if he spins red and
rolls a factor of 20. Otherwise, he loses.
R
R
R
B
Answers: a) The outcomes are: HH, HT, TH, TT, and the probability of Blanca winning and the
probability of her losing are both 1/4, so the game is fair; b) The outcomes are:
Spinner R R R R R R R R R R R R B B B B B B R R R R R R
1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6
Die
The game is fair because Amit wins on 12 of the 24 outcomes and loses on the other 12, and
the probability of his winning and his losing are both 1/2
Fair contests between two people. SAY: I would like to play a game with you. The rules of the
game are simple. I will spin a spinner. If I get red, I win; if I get blue, the class wins. Ask
students if they agree to play by these rules. Now show them the spinner.
R
B
ASK: Do you still want to play? Why isn’t the game fair? (you have a greater probability of
winning) SAY: A fair contest between two people is when both people have an equal probability
of winning.
Exercises: Player 1 must get red to win and Player 2 must get blue to win. Is the game fair?
a)
b)
c)
d)
R B
G Y
R R
Y B
B
G
B
G
R
B
B
B
B
B
B
R
Y
Y
R
B
R G
R G
R R
R R
Bonus: Create rules for a game that would be fair between two players. Use the given spinner.
B
R
Y
Answers: a) yes, b) no, c) no, d) yes
Sample answers: Bonus: Player 1 wins if she spins B, Player 2 wins if he spins Y; or Player 1
wins if she spins red, Player 2 wins if he spins blue or yellow
I-30
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability Discuss students’ answers to the Bonus exercise above. Then draw on the board:
1
2
3
1
2
4
3
Have a volunteer write on the board the organized list of outcomes for spinning both spinners. If
they do not put commas between the entries, point out how the outcome can look like a 2-digit
number. SAY: To avoid confusion, add commas between the first spinner’s result and the
second spinner’s result. The picture should look like this:
Outcome: 1,1
1,2
1,3
1,4
2,1
2,2
2,3
2,4
3,1
3,2
3,3
3,4
SAY: Cathy and Ethan play a game in which Cathy gets a point if the results of the spinners add
to 5, and Ethan gets a point if the results of the spinners multiply to 5. I want to know who is
more likely to get a point in the next round. Ask a volunteer to write the sums above each
outcome and another volunteer to write the products below each outcome. The final picture
should look like this:
Add:
2
Outcome: 1,1
Multiply:
1
3
1,2
2
4
1,3
3
5
1,4
4
3
2,1
2
4
2,2
4
5
2,3
6
6
2,4
8
4
3,1
3
5
3,2
6
6
3,3
9
7
3,4
12
ASK: What is the probability that the two rolls add to 5? (3/12 = 1/4) What is the probability that
the two rolls multiply to 5? (0) Is the game fair? (no)
(MP.1) Exercise: Make up a fair game that Ava and Carlos could play using the spinners
shown above.
Bonus: Make up as many such games as you can.
Sample answers:
Ava gets a point if the results …
add to 2
add to 6
add to 2 or 3
are equal
add to 4
Carlos gets a point if the results …
multiply to 2
multiply to 6
multiply to an odd number
add to 4
add to 5
Extensions
(MP.1) 1. As the number of sides on a pair of dice (with sides labeled consecutively with
numbers 1 through the number of sides) increases, does the probability of the given event
increase, decrease, or neither? Start with a 4-sided die, then move to 5-sided and 6-sided dice.
a) the numbers are one apart
b) the numbers add to an even number
c) the numbers are not equal.
Answers: a) 6/16 = 3/8, 8/25, 10/36 = 5/18, decreases; b) 8/16 = 1/2, 13/25, 18/36 = 1/2,
neither; c) 12/16 = 3/4, 20/25 = 4/5, 30/36 = 5/6, increases
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability
I-31
(MP.8) 2. a) Tell students to imagine that they are rolling two 10-sided dice. On Die A, 5 faces
are red, 4 faces are blue, and 1 face is yellow. On Die B, 6 faces are yellow, 2 faces are red,
and 2 faces are blue. Have students draw a chart showing all the possible outcomes, then
determine the probability of …
i) both dice showing yellow
ii) both dice showing red
iii) both dice showing blue
iv) Die A showing red and Die B showing yellow
v) Die A showing blue and Die B showing red
b) In part a), how can you get the probability of both outcomes occurring from the probability of
each outcome?
Answers:
a) i) 6/100 = 3/50, ii) 10/100 = 1/10, iii) 8/100 = 2/25, iv) 30/100 = 3/10, v) 8/100 = 2/25;
b) multiply together the probabilities of each outcome to get the probability of both outcomes
occurring; for example, the probability of two red faces is the number of outcomes with two red
faces divided by the total number of outcomes. Since there are 5 red faces on Die A and 2 on
Die B, the number of outcomes with two red faces is 5 × 2. The total number of outcomes is
5´2
5
2
= ´ . The chart of all possible outcomes, with Die A
10 × 10, so the probability is
10 ´10 10 10
along the top and Die B along the side:
R
R R
R R
B B
B
B Y
Y
Y
Y
Y
Y
Y
R
R
B
B
I-32
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability SP7-6
More Tree Diagrams, Charts, and Organized Lists
Pages 205–206
Standards: 7.SP.C.8a, 7.SP.C.8b
Goals:
Students will decide whether to use tree diagrams or charts when listing all the outcomes of two
or more independent experiments.
Students will construct organized lists from tree diagrams and charts.
Prior Knowledge Required:
Can use tree diagrams, charts, and organized lists as ways of listing all the outcomes from
simultaneous or consecutive experiments
Can evaluate probabilities from the list of outcomes organized as a chart or organized list
Vocabulary: chart, event, experiment, likely, organized list, outcome, probability, tree diagram,
unlikely
Making organized lists from tree diagrams. Remind students that in the last two lessons,
they learned three ways to record all the possible outcomes from two or more experiments.
Have volunteers name the three ways. (tree diagrams, charts, organized lists) Tell students that
in this lesson, they will see how to draw organized lists from tree diagrams and charts. Draw on
the board:
R
R
P
P
S
R
S
P
S
R
P
S
Tell students that these are the outcomes of two people playing Rock, Paper, Scissors. The two
experiments are the two players picking rock, paper, or scissors, and they are done
simultaneously rather than one after the other. SAY: It’s the difference between tossing two
coins together or tossing one coin, then tossing it again. You get all the same outcomes.
SAY: You can make an organized list from the tree diagram by listing all the outcomes at the
bottom of the tree from left to right. Point to the first one and demonstrate how the path leading
to it is RR. Point to the second one and have a volunteer tell you the outcome it represents by
showing the path that leads to it. (RP) Write the volunteer’s answer on the board. Repeat for the
third one. (RS) Ask a volunteer to finish writing the organized list on the board, as shown below:
RR
RP
RS
PR
PP
PS
SR
SP
SS
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability
I-33
(MP.1) Exercises: Make an organized list from the tree diagram.
a)
A
1
B
2
3
1
C
2
3
1
D
2
3
1
2
3
b)
H
T
H
H
T
T
H
H
T
H
T
T
H
T
H
T H
T H
T H
T
H
T H
T H
T H
T
Answers: a) A1, A2, A3, B1, B2, B3, C1, C2, C3, D1, D2, D3; b) HHHH, HHHT, HHTH, HHTT,
HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT, TTHH, TTHT, TTTH, TTTT
Making organized lists from charts. Tell students that they can make an organized list from a
chart, too. Use the Rock, Paper, Scissors example above to demonstrate, but in a chart format
as shown in the diagram below.
R
P
S
R
P
S
Explain that Player 1’s options are written beside the chart and Player 2’s options are written on
top of the chart. SAY: Each box in the chart represents an outcome, and if you read the chart
row by row, you will make an organized list. Demonstrate using the first row, then have a
volunteer finish the organized list, as shown below:
RR, RP, RS, PR, PP, PS, SR, SP, SS
In the following exercises, some students might find it helpful to write the outcome represented
by the box inside each box before making the organized list.
I-34
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability (MP.1) Exercises: Write an organized list for the chart.
a) roll two 4-sided dice
b) toss a coin, then roll a 6-sided die
1
2
3
4
1
2
3
4
5
6
1
H
2
T
3
4
Answers: a) 1,1; 1,2; 1,3; 1,4; 2,1; 2,2; 2,3; 2,4; 3,1; 3,2; 3,3; 3,4; 4,1; 4,2; 4,3; 4,4;
b) H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6
Choosing between tree diagrams and charts to list outcomes for more than two
experiments. Tell students that since they now know it is easy to make an organized list from a
tree diagram or a chart, they will now learn how to choose whether it is easiest to draw a tree
diagram or a chart. SAY: The best way to get a feel for how to decide is to try it both ways at
first. That’s like anything in life—the only way to know if you prefer oranges or apples is to try
them both.
Tell students that, in the following exercises, they might find that one of the ways becomes
difficult very quickly. If so, they can abandon that method and try the other method.
(MP.5) Exercises: Draw a tree diagram and a chart to show the outcomes of the experiment.
Which way is easiest? Which is hardest?
a) toss 2 coins
b) toss 3 coins
c) toss 4 coins
Answers: part a) can be done both ways equally easily; a tree diagram is easiest for parts b)
and c) because you can’t draw a chart going in 3 different directions, or 4 directions
a)
H
T
H
H
T
T
H
T
H
T
b)
c)
H
T
H
H
T
T
H
H
H
T
H
T
T
H
T
H
T
H
H
T
T
T H
H
T H
H
T
H
T H T H T H
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability
T
T
H
T H
T
T H
T
I-35
When all students have finished at least up to part b), ASK: Which way was easiest for part b)?
(using a tree diagram) Why? (you can’t draw the chart going up out of the page; a chart has only
two directions, and you would need a third direction for the third coin toss) Which way was
easiest for part c)? (a tree diagram) Why? (you can draw a chart for listing the results of two
experiments only, not more) To summarize, SAY: If you want to list the outcomes for doing
more than two experiments, a chart won’t work, so tree diagrams are better.
NOTE: Some students might find clever ways to make a chart for three coin tosses, but they
would likely involve a combination of a chart with an organized list, or drawing two charts to
show both layers. Tell students that these are good solutions, but ask them if the tree diagram is
still easier.
Choosing between tree diagrams and charts for two experiments. SAY: But now let’s look
at cases of doing two experiments to see if we can decide when to use charts and when to use
tree diagrams.
Again, tell students that, in the following exercises, they may find one of the ways becomes
difficult very quickly, and they can then abandon it.
(MP.5) Exercises: Draw a tree diagram and a chart to show the outcomes of the experiment.
Which way is easiest? Which is hardest?
A. roll two 4-sided dice
B. roll two 6-sided dice
C. roll two 20-sided dice
Answers: A chart is easiest for all three of them, but more noticeable for C. A tree diagram
showing 400 outcomes would not fit on the page; even the 36 outcomes for B would be tight.
Extensions
(MP.4) 1. In a dice game, Sharon rolls 5 dice, as shown below:
Right now, she has 3 of a kind. She is allowed one more roll to try to improve her score. If she
rolls the dice that show 1 and 4, what is the probability that she will get the following outcomes?
a) 4 of a kind (just one more die showing 6)
b) a full house (the two other dice are both the same number, but not 6)
c) 5 of a kind (both dice rolled show 6)
Answers: a) 10/36 = 5/18, b) 5/36, c) 1/36
Selected solution: a)
1 2 3 4 5 6
1
2
3
4
5
6
I-36
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability (MP.1, MP.4) 2. Make a dice game with the following rules.
On each turn, a player rolls two dice to score points. The player has the option of
rolling both dice again to try to improve on this score (and risk lowering it), or the
player can choose to keep the original score.
Determine the probability of each kind of roll, then award greater points for events with
lower probability.
Roll
Probability
Points
A pair
Two numbers 1 apart
At least 1 six
Sum is 7
Sum is a multiple of 5
Sum is prime
Answers:
Roll
6/36 = 1/6
Points
(Sample answers)
5
10/36 = 5/18
3
11/36
2
6/36 = 1/6
5
7/36
4
15/36 = 5/12
1
Probability
A pair
Two numbers 1 apart
At least 1 six
Sum is 7
Sum is a multiple of 5
Sum is prime
(MP.1) 3. Ross spins three spinners. He makes a chart by writing an organized list for the
outcomes of the first two spinners. Find the probability that the sum of all three numbers Ross
gets is a multiple of 5.
1
2
3
4
5
6
7
8
9
Answer: The probability is 4/24 = 1/6
1,3 1,4 1,5 2,3 2,4 2,5
6
7
8
9
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability
I-37
SP7-7
Empirical Probability
Pages 207–209
Standards: 7.SP.C.7, 7.SP.C.7b
Goals:
Students will use empirical probability to estimate theoretical probability for simple experiments.
Students will understand that repeating more trials results in a closer estimate for theoretical
probability than repeating fewer trials.
Prior Knowledge Required:
Knows that the expected number of occurrences of a particular outcome is the product of the
probability of that outcome and the number of trials
Can recognize equivalent ratios
Can recognize proportional quantities
Vocabulary: empirical probability, equivalent ratios, event, experiment, likely, outcome,
probability, probability line, ratio table, theoretical probability
Materials:
BLM Game Board (p. I-56)
a pair of dice or an online dice roller per student
chapter books (see Extension 2)
access to an electronic version of the encoded message for each student (see Extension 3)
access to an online Caesarean shift cipher encoder for each student (see Extension 3)
Review expectation. Draw on the board:
Number of
Spins
20
40
Expected Number
of Red Spins
R
B
G
Y
100
ASK: What fraction of the spins do you expect to land on red? (1/4) If you spin the spinner
20 times, how many times would you expect it to land on red? (5) SAY: 1/4 of 20 is 5. Repeat
for 40 spins (10) and 100 spins (25), completing the table as you do so.
I-38
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability Exercises: Write the number of times you expect to get red in each case.
a)
Spins Expected Red
b)
Spins Expected Red
R B
20
20
O
Y
40
40
G
100
100
B
R
R
G
Y
c)
Spins Expected Red
R R
20
40
G R
100
Answers: a) 4, 8, 20; b) 8, 16, 40; c) 15, 30, 75
Have volunteers write their answers on the board so that the whole class has the answers.
Expected number of outcomes of a given type is proportional to the number of times the
experiment is repeated. Remind students that two quantities are proportional if the T-table
comparing their values is a ratio table. SAY: That means that each row is a multiple of the
first row.
(MP.7) Exercises: Look at the tables from the previous exercises. Is the total number of spins
proportional to the number of expected red spins?
Answers: a) yes, b) yes, c) yes
SAY: The fraction of spins that you expect to be red is always the same, no matter how many
spins you do. So, if you multiply the number of spins by 10, for example, then you multiply the
number of expected red spins by 10, too.
Using empirical probability to estimate theoretical probability. SAY: If you don’t know what
a particular spinner looks like, you can use what actually happens to estimate it. Tell students
that you spun a spinner 10 times, and got red 7 times. But you are not going to show students
what the spinner looks like. ASK: How many times would you expect the spinner to land on red
if I spin it 20 times? (14) Write on the board:
# of red spins
7
=
=
total # of spins 10 20
SAY: The theoretical probability is what should happen. The empirical probability is what does
happen. For example, when I toss a coin, the theoretical probability of getting heads is 50% or
one half. But if I toss a coin 100 times and get heads 46 times, then the empirical probability
of getting heads is 46%, or just less than one half. If you don’t know what the true theoretical
probability is, you can estimate it using repeated experiments to find the empirical probability.
SAY: We are doing exactly what we did when we had the theoretical probability. But now we
only have the empirical probability, so we are making the assumption that we are at least close
to the theoretical probability.
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability
I-39
Exercises: Use the example on the board. How many times would you expect the spinner to
land on red after …
a) 30 spins?
b) 50 spins?
c) 100 spins?
Answers: a) 21, b) 35, c) 70
Assessing confidence. ASK: How confident are you that the real probability is 7/10? Why?
(not very confident, because you didn’t spin it very many times) SAY: We don’t really know that
it wasn’t just luck to get red 7 times out of 10. Maybe the spinner is half red and half green, and
when we spin it 100 times, it will land on red only about half the time. Or, maybe the spinner is
90% red and it was only luck that we didn’t get red more often. But our best estimate is that it
will spin red 70 times out of 100, if it spins red 7 times out of 10.
Exercises: How many times would you expect the spinner to land on red in 1,000 spins? How
confident are you that the true answer is close to your estimate?
a) spinner A lands on red 15 times out of 20 spins
b) spinner B lands on red 9 times out of 10 spins
c) spinner C lands on red 30 times out of 80 spins
Answers: a) 750, not very confident; b) 900, not very confident; c) 375, a little more confident
Estimating with contexts other than spinners. Tell students to pretend that you have a bag
of 200 candies. SAY: Some of them are red and some of them are white, but you don’t know
how many you have of each. Suppose I pull out 20 candies and 3 of them are red. ASK: How
many red candies would you expect are in the bag altogether? (30) SAY: You can use a
proportion to solve this problem. Write on the board:
# of red candies
3
=
=
total # of candies 20 200
Point to 3/20 and SAY: This is the experimental probability that we know. Point to the unfinished
fraction on the right and SAY: This is the actual probability of a red candy, but we don’t know it,
so we are trying to estimate it. You could count all 200, but it is easier and faster to just count 20.
(MP.4) Word problems practice.
Exercises:
1. Use a proportion to solve the problem.
a) You pull out 25 candies from a bag and 8 of them are red. If the bag has 100 candies, how
many would you estimate are red?
b) A soccer player scores 3 goals in 10 games. How many goals would you expect the player to
score in 30 games?
c) A hockey team wins 14 of their first 20 games. How many games would you expect the team
to win in 80 games?
Answers: a) (number of red candies)/(number of candies) = 8/25 = ?/100, so estimate 32 red
candies; b) 9 goals; c) 56 wins
I-40
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability 2. Find the empirical probability and use it to estimate the actual probability.
a) 31 out of 100 randomly selected students said they would vote for Candidate A in an election.
How many of the 800 students in the school would you estimate will vote for Candidate A?
b) A baseball player got 47 hits in her first 200 times at bat. How many hits would you expect
the player to get in 1,000 times at bat?
Answers: a) 248, b) 235
The importance of equally likely outcomes when estimating probabilities. SAY: 20 out of
25 students in Liz’s class said they would vote for her if she runs for school elections.
ASK: What percent is that? (20/25 = 80/100 = 80%) SAY: Liz thinks that 80% of the whole
school will vote for her. ASK: Is she right? (no) Why not? What assumption did she make? (she
assumed that everyone in the school is equally as likely to vote for her as her own classmates)
(MP.2) Exercises: Kyle is in Grade 5 in a school that goes from Kindergarten to Grade 5. Can
you use the empirical probability for Kyle’s Grade 5 class to get a good estimate for the
theoretical probability in the whole school? Why or why not?
a) Kyle asks the students in his class if their name starts with a letter from A to M.
b) Kyle asks the students in his class if they walk to school on their own.
c) Kyle asks the students in his class if they prefer chapter books or picture books.
Answers: a) yes, people don’t often change their names as they get older; b) no, people get
more likely to walk to school on their own as they get older; c) no, people become more likely to
prefer chapter books as they get older
A larger sample is expected to be closer to the actual probability. Tell students that a
hockey team won 4 of its first 5 games, and 32 of its first 50 games. Tell students that you want
to know how many games you should expect the team to win if it played 100 games. Write on
the board:
4
=
5 100
32
=
50 100
Have volunteers fill in the answers. ASK: Why did we get two different answers? (because the
team won a greater fraction of their first 5 games than it did of its first 50 games) Tell students
that there could be lots of reasons for this. Maybe the team played against easier teams at first.
Maybe some players got injured. Maybe the team just had a lucky streak at the beginning.
ASK: Which is more likely, that the team was lucky for 5 games or that it was unlucky for 50
games? (that it was lucky for 5 games) Why? (because 50 games is a lot longer than 5 games,
so the team’s true ability has more time to show up) Tell students that they hit on a fundamental
point in probability and statistics. The larger the part of the population they use, the more
confident they can be that it is closer to the results for the whole population. SAY: The more
trials you perform, the more confident you can be that you are getting close to the true
theoretical probability.
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability
I-41
Activity
Provide each student with one game board from BLM Game Board, and a pair of dice (or an
online dice roller). Play the game as follows: Eleven runners are entered in a race. Each runner
gets a number from 2 to 12. Runners move one place forward every time their number is the
total number rolled on a pair of dice. Which number wins the race? Have students predict the
answer, then roll the dice 20 times and move the runners accordingly. Were their predictions
correct?
After performing the experiment, have students make a chart to show all possible outcomes and
to find the theoretical probabilities of each result. Discuss the results of the experiment as a
class. What was the most common number to win in the class? (most likely 7) Was 7 always the
winner? (most likely not) Explain why the actual winner wasn’t always the expected winner.
(empirical probability is not the same as theoretical probability)
(end of activity)
Finding the total population from a proportion. Write on the board:
# of red marbles
2 10
= =
total # of marbles 5
?
SAY: Suppose I know that 2/5 of the marbles in a bag are red, and I also know that 10 red
marbles are in the bag. ASK: How many marbles are in the bag? (25) SAY: Since the numerator
is multiplied by 5, so is the denominator.
Exercises: How many marbles are in the bag?
2
of the marbles are red. There are 60 red marbles in the bag.
3
3
b)
of the marbles are blue. There are 60 blue marbles in the bag.
5
a)
Answers: a) 90, b) 100
Estimating the fraction that has a given property to estimate the total number.
SAY: Scientists want to estimate the number of fish in a lake. They pick 100 fish out of the lake,
tag them, and put them back. If they can figure out what fraction of the fish are tagged, then
they can figure out how many fish are in the lake in total. Suppose, the following week, they pick
out 100 fish and 20 of them are tagged. ASK: What fraction of their sample is tagged? (20/100
or 1/5) Write on the board:
# of tagged fish 1 100
= =
total # of fish
5
?
SAY: I am estimating from my sample that one fifth of the fish are tagged. But I also know that I
tagged 100 fish. ASK: What should I estimate is the number of fish in the lake? (500)
I-42
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability Exercises: A large jar is half full of marbles. Sindi wins a prize if she guesses the correct
number in the jar, within 50 marbles. She pulls out 30 marbles and puts a sticker on all of them.
She then puts them all back, shakes the jar to mix them up, and pulls out another 30 of them.
Sindi finds that 2 of them have a sticker.
a) Estimate the fraction of marbles that have a sticker.
b) Estimate how many marbles there are altogether.
Answers: a) 2/30 or 1/15; b) 1/15 = 30/?, so estimate 450 marbles in the jar
Extensions
1. Give each student two dice of different colors, such as red and blue, or mark one die in some
way. Students roll the dice and write the result of the red die as the numerator of a fraction, and
the result of the blue die as the denominator of the fraction. Do this 20 times and record the
fractions. Students can then determine their experimental probability of getting …
a) a proper fraction (less than 1)
b) an improper fraction (greater than or equal to 1)
c) a fraction that is in lowest terms
Students then combine their results with at least 9 other students to make a total of 200
fractions. Now what is the experimental probability of each? What is the true theoretical
probability of each? Which experimental results do they expect to be closer to what the actual
theoretical probability is—the results with 20 trials or with 200 trials? Why?
Answer: The theoretical probabilities are a) 15/36 or 5/12, b) 21/36 or 7/12, c) 23/36.
Repeating the experiment 200 times should get closer to the actual probabilities for most
students, but some students might have gotten very close to the actual probabilities even with
20 trials.
2. Take any chapter book and determine the experimental probability that the first word in a
chapter is “The.” To get a result closer to the actual theoretical probability, combine results with
classmates who chose a different book.
NOTE: For Extension 3, part e), you will need to electronically copy the encoded message from
the PDF version of this lesson on our Web site and make it available to students.
(MP.4) 3. Using probability to decode messages. The ancient Roman dictator Julius Caesar
developed a way to encode messages known as the Caesar cipher. His method was to shift the
alphabet and replace each letter by its shifted letter. A shift of 3 letters would be:
ABCDEFGHI J K LMNOPQRSTUVWXYZ
DEFGH I JKLMNOPQRS TUVWXYZABC
Look at the following message: “What’s gone with that boy, I wonder?” Using the shift of 3
letters, the message would begin: “Zkdw’v jrqh…”
a) Finish encoding the message.
b) Tally the occurrence of each letter in the original message and in the encoded text.
c) Which letters occur most often in the original message? Which letters occur most often in the
encoded message? How are these letters related?
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability
I-43
d) In the English language, the letter occurring most often is almost always “e,” especially for
large samples of the language. Use a Caesarean shift cipher encoder (search for one online) to
decode and encode messages. If you encode a message with a shift of 3, you will need to
decode the message using a shift of 26 − 3 = 23 to get back your original message. What shift
would you use to decode a message that was encoded using a shift of 7? A shift of 16? A shift
of 24?
e) To decode the following message, first determine which letter occurs most often by using an
online letter frequency counter.
“ZUS!”
Tu gtyckx.
“ZUS!”
Tu gtyckx.
“Cngz’y mutk cozn zngz hue, O cutjkx? Eua ZUS!”
Tu gtyckx.
Znk urj rgje varrkj nkx yvkizgirky juct gtj ruuqkj ubkx znks ghuaz znk xuus; znkt ynk vaz
znks av gtj ruuqkj uaz atjkx znks. Ynk ykrjus ux tkbkx ruuqkj znxuamn znks lux yu ysgrr
g znotm gy g hue; znke ckxk nkx yzgzk vgox, znk vxojk ul nkx nkgxz, gtj ckxk haorz lux
“yzerk,” tuz ykxboik—ynk iuarj ngbk ykkt znxuamn g vgox ul yzubk-rojy payz gy ckrr. Ynk
ruuqkj vkxvrkdkj lux g susktz, gtj znkt ygoj, tuz lokxikre, haz yzorr ruaj ktuamn lux znk
laxtozaxk zu nkgx: “Ckrr, O rge ol O mkz nurj ul eua O’rr—“ Ynk joj tuz lotoyn, lux he
znoy zosk ynk cgy hktjotm juct gtj vatinotm atjkx znk hkj cozn znk hxuus, gtj yu ynk tkkjkj
hxkgzn zu vatizagzk znk vatinky cozn. Ynk xkyaxxkizkj tuznotm haz znk igz. “O tkbkx joj
ykk znk hkgz ul zngz hue!”
Which letter do you think represents the letter “e”? Why? What shift do you think was used?
What shift do you think will decode the message? Check your answer using an online
Caesarean shift cipher encoder.
Answers:
a) Zkdw’v jrqh zlwk wkdw erb, L zrqghu?
b) The original has 2 a’s, 1 b, 1 d, 2 e’s, 1 g, 3 h’s, 2 i’s, 2 n’s, 3 o’s, 1 r, 1 s, 4 t’s, 3 w’s, 1 y; The
encoded message has 1 b, 2 d’s, 1 e, 1 g, 2 h’s, 1 j, 3 k’s, 2 l’s, 2 q’s, 3 r’s, 1 u, 1 v, 4 w’s, 3 z’s
c) The letter “t” occurs most often in the original; the letter “w” occurs most often in the encoded
message. The letter “t” encodes as “w.”
d) To decode a message encoded using shift 7, use shift 19; to decode a message encoded
using shift 16, use shift 10; to decode a message encoded using shift 24, use shift 2.
e) Since “k” occurs most often, it probably represents “e.” That would mean the shift used is the
6-letter shift, and we should decode it using the 20-letter shift. The decoded message is the
beginning of The Adventures of Tom Sawyer by Mark Twain.
I-44
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability SP7-8
Simulating Real-World Problems
Pages 210–212
Standards: 7.SP.C.7, 7.SP.C.8c
Goals:
Students will use random number generators to model probabilities of simple events.
Students will understand that more trials produce more reliable estimates of the theoretical
probability.
Prior Knowledge Required:
Can find the probability of simple events
Can convert between fractions and percents
Can write a fraction from a part-to-part ratio
Vocabulary: empirical probability, event, experiment, likely, model, outcome, probability,
probability line, random number generator, theoretical probability
Materials:
small, folded pieces of paper each labeled with a number from 1 to 5
hat or bag
a sample from BLM 10, 50, and 100 Trials (pp. I-57–62) for each student, or access to an
online random number generator and word processor for each student
BLM 100 Random Numbers from 0 to 9 (p. I-63)
Modeling probabilities from spinners with a random number generator. Tell students that
mathematicians often represent real-world events mathematically, and that this is called
modeling a real-world event. Draw on the board (and keep it on the board for later use):
R B
R
B
B
ASK: What is the probability of spinning red on this spinner? (2/5) SAY: You can model spinning
the spinner mathematically by picking numbers from 1 to 5 because the spinner has 5
outcomes. Draw on the board:
1
2
Red
3
4
5
Blue
SAY: If you pick a number from 1 to 5 at random, you will get either 1 or 2 with the same
probability that you will get red if you spin the spinner. This is because two of the five equally
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability
I-45
likely outcomes are red on the spinner and two of the five numbers from 1 to 5 are 1 or 2. You
might as well use the first two numbers to represent the two red outcomes. It might not seem
any easier to pick a random number than to spin a spinner, but if you use a computer to pick the
numbers, it is easier to generate many random numbers than to spin a spinner many times, so
computers can be very useful.
Exercises: Which numbers from 1 to 8 would you use to represent landing on red?
a)
b)
c)
R
O
B B
R
R
G
Y
Y O
B B
G
R
Y O
R
R
R R
R
R
R O
Answers: a) 1; b) 1, 2, 3; c) 1, 2, 3, 4, 5, 6, 7
SAY: If you write the probability as a fraction, then the denominator tells you how many
numbers to pick from, and the numerator tells you how many numbers you can count as a
success. You might as well choose numbers from 1 to the denominator, and use numbers from
1 to the numerator to represent the event.
Exercises: Write the numbers you would randomly generate to model spinning the spinner.
Then circle the numbers you would use to represent landing on red.
a)
b)
c)
R
G
R
R
G
G
R
B
G
B
R
R
B R
R O
B
R
Answers: a) 1, 2, 3, 4, circle 1 to 3; b) 1, 2, 3, 4, 5, 6, circle 1; c) 1, 2, 3, 4, 5, 6, 7, 8, circle 1 to 5.
SAY: You might wonder why we want to simulate a probability that we already know. There are
two reasons. One is that you can simulate probabilities that you know to help simulate
probabilities of other events that you don’t know. Point to the example on the board (the spinner
with 2 red regions and 3 blue regions). SAY: I know the probability of red is 2/5, but suppose I
play a game in which I spin the spinner 5 times, and I win if I get exactly two reds. ASK: What is
the probability of winning the game? SAY: One way to find out is to simulate playing the game
several times, and seeing how many of the times you will win. This is what we will be learning
next class, and you will see then how powerful this method is.
Tell students that another reason to simulate probabilities that we already know is to see how
close our simulations are to the real probabilities, so then we can know how much we should
trust the simulation. Maybe some Web sites will be better than others, for example, so if we are
using one in particular, we want to know how good it is.
(MP.4) Modeling probabilities from the real world with a random number generator.
SAY: Suppose there is a 60% chance of rain. Write on the board:
60% =
I-46
60
100
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability SAY: If I randomly pick a number from 1 to 100, the probability that it will be between 1 and 60
is the same probability that it will rain. ASK: But do I have to pick a number from 1 to 100, or can
I use a smaller range? PROMPT: Can I use a smaller denominator to write the fraction 60/100?
(yes, 60/100 = 3/5) SAY: If I randomly pick a number from 1 to 5, the probability that it will be 1,
2, or 3 is the same probability that it will rain. Show this on the board:
1
2
Rain
3
4
5
No rain
(MP.4) Exercises: Write the numbers you would randomly generate. Then circle the numbers
that would represent each event.
a) A baseball player gets a hit 40% of the time at bat.
b) The probability of snow is 30%.
c) There is a 75% chance of an earthquake each year.
d) The probability of getting heads when you toss a regular coin.
e) The probability of rolling a number greater than 4 when you roll a regular die.
Answers: a) generate 1 to 5, circle 1 and 2; b) generate 1 to 10, circle 1 to 3; c) generate 1 to 4,
circle 1 to 3; d) generate 1 and 2, circle 1; e) generate 1 to 3, circle 1, since the probability that
we are simulating is 1/3
Refer students back to the example on the board that shows a 60% chance of rain. SAY: Let’s
check how close simulating the probability is to the real probability. Put 5 small folded pieces of
paper labeled 1 to 5 in a hat or bag. Have a volunteer pick a number from the hat. Record the
result on the board. Repeat with four other students. Have a volunteer circle the 1s, 2s, and 3s.
ASK: What is our simulated experimental probability of rain? (answers will vary, depending on
the class outcome)
More trials produce probabilities closer to the true probability. If your class got an
experimental probability of 60%, repeat again until the probability is off. Point out that when the
probability is off, it is off by at least 20% because you picked only 5 numbers. SAY: If I pick
more numbers, I think I might get closer to the true probability, but picking a lot of numbers from
a hat is a lot of work. Tell students that there is something on the Internet called a random
number generator that will do the work for them.
Tell students that you want to experiment with different numbers of trials. Give each student a
sample from BLM 10, 50, and 100 Trials and have them count the occurrences of 1, 2, and 3—
make sure some students get a sample of size 10, some get a sample of size 50, and some get
a sample of size 100. Alternatively, have students use an online random number generator to
simulate a trial by following the steps below:
Generate 10, 50, or 100 numbers from 1 to 5.
Copy the numbers to a word processor.
Count the occurrences of 1, 2, and 3 (using Ctrl+F usually works).
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability
I-47
Have students find the simulated experimental probability of rain for their trials. When students
finish, ask several volunteers to say what their experimental probability was, and the number of
repeated trials they performed. Record the answers in a chart on the board with headings as
shown below:
10 experiments
50 experiments
100 experiments
When several volunteers have provided their answer, point to the value that is farthest from
60% in the “10 experiments” column and ASK: Did anyone get farther from 60% than that?
Repeat for 50 experiments, and 100 experiments, recording all the values that are farthest away
from 60%. Point out that the farthest away value generally got closer to 60% as the number of
trials became larger, so you can trust your answer as being fairly reliable if you use more
experiments. SAY: Some small trials probably got exactly 60%, but when small trials are off,
they can be off by a lot.
Students will need BLM 100 Random Numbers from 0 to 9 to do the exercises below.
Exercises:
a) A random number generator performed ten trials of the experiment “pick a number from 0 to 9,”
with the following results:
2
6
9
7
6
6
1
1
6
4
Write the experimental probability of each number as a percent.
b) A random number generator performed 100 trials of the same experiment. The results are on
BLM 100 Random Numbers from 0 to 9. Write the experimental probability of each number as a
percent.
c) Use a random number generator to pick a number from 0 to 9 one thousand times. Copy and
paste them to a separate file and do a search for each number. Be sure to use a word
processing program in which such a search will tell you how many occurrences there were.
NOTE: Choosing numbers 0 to 9 instead of 1 to 10 was done so that the number of occurrences
of 1 would not interfere with the occurrences of 10.
If you do not have access to a computer, you can use the following results instead:
0
1
2
3
4
5
6
7
8
9
Number
Number of
90 109 109 101 96 106 103 101 97
88
Occurrences
Write the experimental probability of each number as a percent.
Bonus: Generate a list of 10,000 numbers, or use the data below, and repeat the exercise.
0
1
2
3
4
5
6
7
8
9
Number
Number of
1,076 1,005 1,041 972 963 971 1,018 1,015 1,006 933
Occurrences
I-48
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability (MP.1) Check that the numbers in part c) actually add to 1,000, and that the numbers in the
Bonus question add to 10,000, or that their percents add to 100%.
Answers: The probabilities of each number from 0 to 9 are: a) 0%, 20%, 10%, 0%, 10%, 0%,
40%, 10%, 0%, 10%; b) 10%, 15%, 7%, 9%, 13%, 9%, 8%, 9%, 9%, 11%; c) using the chart:
9%, 10.9%, 10.9%, 10.1%, 9.6%, 10.6%, 10.3%, 10.1%, 9.7%, 8.8%; Bonus: 10.76%, 10.05%,
10.41%, 9.72%, 9.63%, 9.71%, 10.18%, 10.15%, 10.06%, 9.33%
When students finish the exercises above, emphasize that while some small samples got some
of the probabilities exactly right, they also got some probabilities very wrong. The larger
samples didn’t get any probabilities exactly right, but they weren’t off by much.
Using a random number generator to model probabilities expressed as a ratio. Remind
students that they can write ratios as fractions. Write on the board:
girls : boys = 2 : 3
SAY: In a classroom, the ratio of girls to boys is 2 to 3. ASK: What fraction of the students are
girls? (2/5) If you were to pick a student at random from the class, what is the probability that the
student would be a girl? (2/5) If you were to model this probability using a random number
generator, how many numbers would you need to pick from? (5 numbers) Write on the board:
1
2
3
4
5
ASK: If you choose a number from 1 to 5, which numbers would represent the student being a
girl? (1 and 2) Show this on the board:
1
2
Girl
3
4
5
Boy
(MP.4) Exercises: A student is randomly chosen from a class. How would you use a random
number generator to represent the probability of choosing a boy if the ratio of boys to girls is …
a) 2 : 1
b) 1 : 5
c) 3 : 4
d) 5 : 2
Answers: a) pick a number from 1 to 3, and let 1 and 2 represent the student being a boy;
b) pick a number from 1 to 6, and let 1 represent the student being a boy; c) pick a number from
1 to 7, and let 1, 2, and 3 represent the student being a boy; d) pick a number from 1 to 7, and
let 1, 2, 3, 4, and 5 represent the student being a boy
Extensions
1. How would you use a random number generator to simulate the probability of …
a) getting heads and an even number when tossing a die and rolling a die?
b) rolling two equal numbers when rolling two regular dice?
Hint: Start by making an organized list or a chart of the outcomes to find the probability.
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability
I-49
Sample solution:
a) H1 H2
H3
H4
H5
H6
T1
T2
T3
T4
T5
T6
There are 12 outcomes, of which 3 satisfy the requirements, so the probability is 3/12 or 1/4. So,
you can generate the numbers 1 to 4 and use 1 to represent getting heads and an even
number.
b) 1,1 1,2 1,3 1,4 1,5 1,6
2,1 2,2 2,3 2,4 2,5 2,6, …
Continuing in this way, there are six outcomes that correspond to rolling two equal numbers. So
the probability is 6/36 or 1/6. You can generate the numbers 1 to 6 and use 1 to represent
getting two equal numbers.
2. a) How would you use a random number generator to simulate …
i) getting exactly 2 heads when tossing 4 coins?
ii) getting at least 2 heads when tossing 4 coins?
iii) getting at most 2 tails when tossing 4 coins?
b) Compare your answers to part a.ii) and iii). What do you notice?
Sample solution: a) Make an organized list of the 16 outcomes:
1: HHHH
2: HHHT
3: HHTH
4: HHTT
5: HTHH
6: HTHT
7: HTTH
8: HTTT
9: THHH
10: THHT
11: THTH
12: THTT
13: TTHH
14: TTHT
15: TTTH
16: TTTT
i) The numbers that represent exactly 2 heads are 4, 6, 7, 10, 11, and 13.
ii) The numbers that represent at least 2 heads are 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, and 13.
iii) The numbers that represent at most 2 tails are 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, and 13.
b) The answers are the same because, out of 4 tosses, at least 2 heads means the same thing
as at most 2 tails.
3. How could you use a standard deck of cards to simulate getting exactly 2 heads when
tossing 3 coins?
Sample answer: Number the outcomes from 1 to 8:
1: HHH
2: HHT
3: HTH
4: HTT
5: THH
6: THT
7: TTH
8: TTT
Use the hearts suit from 1 to 8. Shuffle the cards and pick the top card. If it is 2, 3, or 5, then
that counts as getting exactly 2 heads.
4. Use two or three different online random number generators to simulate the probability of
picking an even number when picking a random number from 1 to 5. Which generator gets the
closest to the theoretical probability after 100 trials? After 1,000 trials? After 10,000 trials?
NOTE: The actual theoretical probability is 40%.
I-50
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability SP7-9
Simulating Repeated Experiments
Pages 213–215
Standards: 7.SP.C.8c
Goals:
Students will design and use a simulation to determine probabilities of compound events.
Prior Knowledge Required:
Can model probabilities of simple events using random number generators
Knows that more trials produce more reliable estimates of the theoretical probability
Vocabulary: empirical probability, event, experiment, likely, model, outcome, probability,
random number generator, theoretical probability
Materials:
BLM 200 Random Pairs from 1 to 6 (p. I-64)
BLM Random Triples from 1 to 5 (p. I-65)
BLM Simulating Probability (p. I-66, optional)
access to an online random number generator, or a paper clip and BLM Spinner from
1 to 3 (p. I-67), BLM Spinner from 1 to 4 (p. I-68), and BLM Spinner from 1 to 5 (p. I-69)
for each student
Using random number generators to model probabilities of compound events. Remind
students that they can use a random number generator to model real-life events. Tell students
that you want to model rolling two dice. ASK: How many outcomes are there? (36) SAY: So you
could tell the number generator to randomly pick a number from 1 to 36, but then you would
have to keep track of which number represents which outcome. It is easier to tell the computer
to pick two numbers from 1 to 6, or to pick one number twice. That does the same thing as
rolling two dice, or rolling one die twice. So the outcome “the random number generator picks 1
and 4” represents rolling 1 and 4.
Exercises: Use a chart to find the probability of the event when you roll two dice.
a) the sum is 7
b) the two numbers are consecutive
c) the sum is odd
Answers: a) 6/36 = 1/6, b) 10/36 = 5/18, c) 18/36 = 1/2
Provide students with BLM 200 Random Pairs from 1 to 6. Have students complete the BLM
for at least one of the three events in Question 1, and then check with a partner that they got the
same answer. The BLM provides 200 pairs of numbers randomly chosen from 1 to 6. It asks
students to find the empirical probabilities of the three events from the preceding exercises, and
to compare the empirical probabilities to the theoretical probabilities. Students who don’t have
time to do all three can copy from a partner’s sheet, but all students should do at least one
event. (1. A. 15.5% and 6/36, B. 23.5% and 10/36, C. 50.5% and 18/36; 2. yes, yes, yes)
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability
I-51
(MP.5) It can be tedious to find theoretical probabilities of compound events using a tree
diagram, chart, or organized list. When students finish BLM 200 Random Pairs from 1 to 6,
point out that the empirical probabilities are not exactly the same as the theoretical probabilities,
but they are close. Tell students that this is particularly useful when you don’t know the
theoretical probability and you want to estimate it by doing many repeated experiments.
SAY: The more repeated experiments you do, the more confident you can be that your results
are close to the actual theoretical probability. Write on the board:
A class has 12 girls and 18 boys. The class picks one student at random each week for
three weeks to win a prize. What is the probability that all three students are boys?
ASK: What fraction of the class is girls? (12/30) What is that in lowest terms? (2/5) Write on
the board:
1
2
Girl
3
4
5
Boy
SAY: The class picks a student at random three times, so that means we are picking three
numbers from 1 to 5. There are a lot of ways of picking 3 numbers from 1 to 5. You could make
a tree diagram, but it would have 5 × 5 × 5 = 125 outcomes. ASK: How about a chart? Would
that work? (no, we would need to put 3 directions on the chart, but our charts have only two
directions) SAY: Making an organized list of 125 outcomes would be a lot of work, so let’s use
computers instead to estimate the probability rather than find it exactly. We can use a computer
to randomly generate lists of 3 numbers from 1 to 5 and determine which ones satisfy all 3
numbers representing boys, as shown on the board. To find 10 such groups of 3 numbers, tell
the computer to generate 30 numbers and put them into groups of three. Write on the board:
1,5,4
1,5,4
5,2,4
1,3,3
2,3,4
3,1,1
2,5,2
1,5,2
3,2,2
3,5,5
Ask a volunteer to circle all the examples in which all three students chosen are boys. (circle
only 3,5,5) SAY: So our experimental probability is 1/10 or 10% that all three students are boys.
Now provide students with BLM Random Triples from 1 to 5. Tell students that you used a
random number generator to choose 150 numbers from 1 to 5, and you put them in groups of 3,
so the computer found 50 groups of 3 numbers. Have students write a checkmark beside the
triples that show that all three students chosen are boys (i.e., each number in the triple is a 3, 4,
or 5). The experimental probability from the BLM is 20%. Tell students that the true theoretical
probability is 21.6%. ASK: Did we get closer to the true probability by using a larger sample?
(yes) SAY: Another sample of 10 might well have gotten 20% probability, but it could also have
gotten 30% or 40% or 0%. By choosing 50 numbers, we know we won’t be off by as much as
we were with 10 numbers.
(MP.3, MP.6) ASK: What if, instead of picking the children at random, we picked based on scores
on a music test? Then could we still use a random number generator to estimate the probability of
all children being boys? (no, because boys might be better or worse than girls at music, and that
I-52
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability will change the probability. If more boys are good at music, the probability will be higher than
random; if more girls are good at music, the probability will be lower than random)
(MP.4) Exercises: If there is a 40% chance of a flood each year, how would you use a random
number generator to estimate the probability of the given event?
a) In the next four years, there will be a flood in at least two years.
b) In the next five years, there will be a flood in at least two consecutive years.
c) There will not be a flood in the next six years.
Sample answers: Use 1 and 2 to represent the occurrence of a flood, and 3 to 5 to represent
no flood. Then: a) pick four numbers from 1 to 5—are at least two of them a 1 or 2?; b) pick five
numbers from 1 to 5—are there two consecutive numbers that are either 1 or 2?; c) pick six
numbers from 1 to 5—are they all 3, 4, or 5?
Have students find an empirical probability for at least one of the exercises above. (Faster
students can do more.) Students can use a random number generator if they have one. Get
students started on part a) by writing on the board:
Choose ______ numbers from 1 to 5.
Format them in ______ columns.
SAY: If you want to do 10 simulations, you have to pick 40 numbers, because each simulation
uses 4 numbers. ASK: How many columns should you format them in? (4)
If students do not have access to a random number generator, they can use BLM Spinner from
1 to 5. Make sure all problems are represented when all students finish. Students can use BLM
Simulating Probability to structure their answers and pool their results with other students, so
that they can estimate the probabilities.
Problems with different contexts but the same structure. Write on the board:
A soccer player scores a goal in 40% of the games. What is the probability
that the player will score a goal in at least two of the next four games?
ASK: Which of the exercises above would have the same answer as this one? (part a))
SAY: You would still use the numbers from 1 to 5, and you would still pick four numbers and see
if at least two of them are a 1 or a 2.
(MP.2, MP.8) Exercises: Which part in the exercises above has the same answer as the given
problem?
a) 40% of students in a class are boys. Students are picked at random once a week for five
weeks. Estimate the probability that a boy will be chosen in at least two consecutive weeks.
b) 40% of blood donors have Type O blood. What is the probability that none of the first six
donors asked have Type O blood?
Bonus: A player scores a goal in 60% of the games. What is the probability that in the next five
games, the player will go two consecutive games without scoring?
Answers: a) part b), b) part c), Bonus: part b)
Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability
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NOTE: In Question 6 on AP Book 7.1 p. 215, students are asked to use a random number
generator. If students do not have access to online random number generators, they can use
BLM Spinner from 1 to 3 for part a), BLM Spinner from 1 to 4 for part b), and BLM Spinner
from 1 to 5 for part c). Provide students with all three spinners and have them decide which
spinner is appropriate for each part.
Extensions
(MP.1, MP.2) 1. Create two problems with the same structure and the same answer, but
different contexts.
Sample answer:
1. A baseball player gets a hit 30% of the time at bat, and has a batting average of .300. What is
the probability that the player doesn’t get a hit in any of the next 3 times at bat?
2. There is a 30% chance of a hurricane each year. What is the probability that there will not be
a hurricane in the next 3 years?
(MP.1, MP.4) 2. A cereal company puts one of four animal cards in each cereal box—a cat, a
dog, a bird, and a fish—each with equal likelihood. If a customer collects all four animals, the
customer wins a free box of cereal.
a) Customer A buys 4 boxes. Customer B buys 5 boxes, and Customer C buys 6 boxes. Design
and use a simulation using a spinner or a random number generator to estimate the probability
that each customer gets a free box of cereal.
b) Does the probability of winning increase as a customer buys more boxes?
Answer: Use a spinner with four equal parts. For customer A, spin it 4 times, recording which
part it lands on each time. Are all four parts represented? Do this 100 times, recording how
many times all four parts are represented. For customer B, spin it 5 times and check whether all
four parts are represented. Do this 100 times. For customer C, spin it 6 times instead of 5 times.
The theoretical probabilities are 3/32 for four boxes, 15/64 for five boxes, and 195/512 for six
boxes. So students should find that the probability increases as the customer buys more boxes.
(MP.1, MP.4) 3. How could you generate two random numbers to simulate choosing a red face
card (J, Q, or K of hearts or diamonds) from a regular deck of 52 cards?
Answer: Choose one number from 1 to 13, and another number from 1 to 2. Use 11, 12, and 13
to represent the face cards, and use 1 to represent red.
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Teacher’s Guide for AP Book 7.1 — Unit 8 Statistics and Probability