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What Do You Expect Investigation 2
A C E
2.1
Applications | Connections | Extensions
Applications
1. A bucket contains one green block, one red block, and two yellow blocks. You choose one block from the
bucket.
a. Find the theoretical probability that you will choose each color.
P(green) =
P(yellow) =
P(red) =
b. Find the sum of the probabilities in part (a).
c. What is the probability that you will not choose a red block? Explain how you found your answer.
d. What is the sum of the probability of choosing a red block and the probability of not choosing a red
block?
2. A bubble-gum machine contains 25 gumballs. There are 12 green, 6 purple, 2 orange, and 5 yellow gumballs.
a. Find each theoretical probability.
P(green) =
P(purple) =
P(orange) =
P(yellow) =
b. Find the sum.
P(green) + P(Purple) + P(orange) + P(yellow) =
1
c. Write each of the probabilities in part (a) as a percent.
P(green) =
P(purple) =
P(orange) =
P(yellow) =
d. What is the sum of all the probabilities as a percent?
e. What do you think the sum of the probabilities
for all the possible outcomes must be for any situation?
Explain.
3. Bailey uses the results from an experiment to calculate the probability of each color of block being chosen from
a bucket. He says P(red) = 35%, P(blue) = 45%, P(yellow) = 20%. Jarod uses theoretical probability because
he knows how many of each color block is in the bucket. He says P(red) = 45%, P(blue) = 35%, and P(yellow)
= 20%. On Bailey’s turn, he predicts blue. On Jarod’s turn he predicts red. Neither boy makes the right
prediction.
a. Did the boys make reasonable predictions based on their own probabilities? Explain.
b. Did they do something wrong with their
calculations? Explain.
14. Find numbers that make the sentence true.
=
=
2
2.2
4. A bag contains two white blocks, one red block, and three purple blocks. You choose one block from the
bag.
a. Find each probability.
P(white) =
P(red) =
P(purple) =
b. What is the probability of not choosing a white block? Explain how you found your answer.
2;
3
b.
the probability of choosing a white block is 1 , so the probability of not
3
choosing a white block is 1 1  2 .
3
3
c. Suppose the number of blocks of each color is doubled. What happens to the probability of
choosing each color?
c. The probabilities are the same. There will be 4 white blocks, 2 red blocks, and 6 purple blocks,
so P(white) = 4 = 1 ; P(red)  2 , or
12
12
3
1
;
6
P(purple) 
6
,
12
or
1
2
d. Suppose you add two more blocks of each color to the original bag. What happens to the
probability of choosing each color?
d. P(white)  4 , or 1 ; P(red)  3 , or 1 ; P(purple)  5
3
4
12
12
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e. How many blocks of which colors should you add to the original bag to make the probability of
choosing a red block equal to ½?
Answers may vary. Possible answer :Add 4 red blocks. Then there are 5 red blocks, 2 white blocks, and 3
purple blocks, for a total of 10 blocks. And P(red)  5 , or 1 .
10
2
5. A bag contains exactly three blue blocks. You choose a block at random. Find each probability.
a. P(blue)
3
3
1
b. P(not blue) = 0
c. P(yellow)= 0
3
6. A bag contains several marbles. Some are red, some are white, and some are blue. You count the marbles
and find the theoretical probability of choosing a red marble is . You also find the theoretical probability
of choosing a white marble is
.
a. What is the least number of marbles that can be in the bag?
Since the probability of choosing a white marble is
3
10
, there must be at least 10 marbles in the bag.
b. Can the bag contain 60 marbles? If so, how many of each color does it contain?
Yes. Possible answer: Let’s assume the bag contains 60 marbles. Since the
probability of choosing red is 1 , then 1 of the proposed 60 marbles, or
5
5
12 marbles, would be red. Since 3 of the 60 marbles must be white, 18 would
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be white. This gives us a total of 30 marbles, so the other 30 marbles would
be blue. Since we get whole numbers of marbles when we do this division,
the bag could contain 60 marbles.
c. If the bag contains 4 red marbles and 6 white marbles, how many blue marbles does it contain?
Possible answer: Since 3 of the marbles in the bag are white, and 6 marbles are white, we need to
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answer this question: 3 of what number equals 6? The answer is 20. This works
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because P(red) 1 , and 1 of 20 = 4 red marbles. Since 4 marbles are red
5
5
and 6 are white, there are 20 – 10 = 10 blue marbles in the bag
d. How can you find the probability of choosing a blue marble?
Possible answer: You could add the other two probabilities (of red and white) and subtract the result from 1:
1 3
2
3
5
5
5
+
=
+
=
, and 1 
 , or 1 . So the probability of choosing a blue marble is 1 .
2
2
5 10 10 10 10
10 10
7. Decide whether each statement is true or false. Justify your answers.
a. The probability of an outcome can be 0.
True. The outcome must be impossible (such as rolling a 7 on a number cube).
b. The probability of an outcome can be 1.
True. The outcome must be absolutely certain (such as rolling a number less than 7 on a number cube).
c. The probability of an outcome can be greater than 1.
False. All probabilities are between 0 (impossible) and 1 (absolutely certain), inclusive.
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For Exercises 18-25, estimate the probability that the given event occurs. Any probability must be between 0
and 1 (or 0% and 100%). If an event is impossible, the probability it will occur is 0, or 0%. If an event is
certain to happen, the probability it will occur is 1, or 100%.
NOTE: Numbers 18,19, and 21 will changes based on the student. 20,22,23,24,25 should be very
similar for everybody.
The event is
Impossible.
The event is
certain to happen
0
¼
½
¾
1
0%
25%
50%
75%
100%
18. You are absent from school at least one day
during the school year.
Probably above 75% depending on your situation
23. The sun sets tonight.
100%
19. You have pizza for lunch one day this week.
Depends on you
20. It snows on July 4 this year in Mexico.
24. You take a turn in a game by tossing four coins.
The result is all heads.
Close to 0
Below 5%
21. You get all the problems on your next math test
correct.
Depends on the student
25. You toss a coin and get 100 tails in a row.
22. The next baby born in your local hospital is a
Close to zero
girl.
Around 50%
5
2.3
8. Patricia and Jean design a coin-tossing game. Patricia suggests tossing three coins. Jean says they
can toss one coin three times. Are the outcomes different for the two situations? Explain.
The outcomes are the same for the two situations. We can see this by identifying the three coins in the first case with the
three tosses in the second. Coin 1 can be heads or tails, just as the first toss can be, and so on. Tossing a coin three
times or tossing three coins at once does have the same number of equally likely outcomes. The outcomes include
HHH, TTT, THT, HTH, TTH, HHT, THH, and HTT. Note: Some students may answer no for this question, which is fine
as long as their reasoning is correct. They may say that the outcomes for tossing three coins at once are three heads,
three tails, two heads and one tail, or two tails and one head. These descriptions do describe the outcomes,
though notice that these outcomes are not equally likely. This must be the case when determining theoretical
probability. If the order of heads and tails matters in the game, then tossing one coin 3 times makes
it easier to keep track of this. That is, if HTT is considered different from THT in their game, then tossing one coin 3
timeswill make spotting the difference easier.
9. Pietro and Eva are playing a game in which they toss a coin three times. Eva gets a point if no two consecutive
toss results match (as in H – T – H). Pietro gets a point if exactly two consecutive toss results match (as in H – H –
T). If all three toss results match, no one scores a point. The first player to get 10 points wins. Is this a fair game?
Explain. If it is not a fair game, change the rules to make it fair.
This is not a fair game. There are two winning outcomes for Eva (THT and HTH), but four winning outcomes for Pietro
(TTH,HHT, THH, and HTT). Two outcomes (HHH and TTT) have no winners. The game can be made fair by
changing the point scheme. Eva can be awarded twice as many points as Pietro for each winning combination.
10. Silvia and Juanita are designing a game. A player in the game tosses two number cubes. Winning
depends on whether the sum of the two numbers is odd or even. Silvia and Juanita make a tree diagram of
possible outcomes.
Number Cube 1
Number Cube 2
Outcome
odd
odd
even
odd
even
even
a. List all the outcomes for the sums.
odd, odd; odd, even; even, odd; even,even
b. Design rules for two-player game that is fair.
Possible answer: If the sum of the two number cubes is even, Player 1 scores a point. If the sum of
the two number cubes is odd, Player 2 scores a point.
c. Design rules for a two-player game that is not fair.
Possible answer: If the product of the two number cubes is even, Player 1scores a point. If the
product of the two number cubes is odd, Player 2 scores a point.
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d. How is this situation similar to tossing two coins and seeing if the coins match or don’t match?
As with tossing a coin, considering even and odd on a number cube gives two equally likely outcomes. The game
suggested in part (b) is like the game in Problem 1.3, One More Try. In this game, a match is two evens or two
odds, which has a sum that is even. No match is one of each, which has an odd sum. The games that students
might imagine for part (b) might not be as easily connected to the two-coins game.
26. Karen and Mia play games with coins and number cubes. No matter which game they play, Karen loses
more often than Mia. Karen is not sure if she just has bad luck or if the games are unfair. The games are
described in this table. Review the game rules and complete the table.
Game
Game 1
Roll a number cube.

Karen scores a point if the roll is even.

Mia scores a point if the roll is odd.
Game 2
Roll a number cube

Karen scores a point if the roll is a multiple of
4.

Mia scores a point if the roll is a multiple of 3.
Game 3
Toss two coins.

Karen scores a point if the coins match.

Mia scores a point if the coins do not match.
Game 4
Roll two number cubes.

Karen scores a point if the number cubes
match.

Mia scores a point if the number cubes to not
match.
Game 5
Roll two number cubes.

Karen scores a point if the product of the two
7
Can
Karen
Game
Karen
Likely to
Fair or
Win?
Win?
Unfair?
numbers is 7.

Mia scores a point if the sum of the two
numbers is 7.
2.4
11. Melissa is designing a birthday card for her sister. She has a blue, a yellow, a pink, and a green sheet of
paper. She also has a black, a red, and a purple marker. Suppose Melissa chooses one sheet of paper and
one marker at random.
a. Make a tree diagram to find all the possible color combinations.
b. What is the probability that Melissa chooses pink paper and a red marker?
P(pink, red) = 1
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c. What is the probability that Melissa chooses blue paper? What is the probability she does not
choose blue paper?
P(blue paper) = 1 ; P(not blue paper) = 3 , or 1 – P(blue) = 1 – 1 = 3
4
4
4 4
d. What is the probability that she chooses a purple marker?
P(purple marker) = 4 , or 1
12
3
8
12. Lunch at school consists of a sandwich, a vegetable, and a fruit. Each lunch combination is equally likely
to be given to a student. The students do not know what lunch they will get. Sol’s favorite lunch is a
chicken sandwich, carrots, and a banana.
School Lunch Menu
Sandwiches Vegetables
Fruit
Chicken
Carrots
Apple
Hamburger
Spinach
Banana
Turkey
a. Make a tree diagram to determine how many different lunches are possible. List all the possible
outcomes.
b. What is the probability that Sol gets his favorite lunch? Explain your reasoning.
The probability of Sol getting his favorite lunch is 1 . Since the cook is not paying any attention to how
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she puts the lunches together, and there are equal numbers of all kinds of sandwich, vegetable, and
fruit, all of the 12 combinations are equally likely
c. What is the probability that Sol gets at least one of his favorite lunch items? Explain.
The probability of Sol getting at least one of his favorite things is 10 , or 5 . Only 2 of the 12 combinations of items,
12
6
hamburger-spinach-apple and turkey- spinach-apple don’t contain at least one of his favorite things.
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13. Suppose you spin the pointer of the spinner at the right once and roll the number cube. (The numbers on
the cube are 1, 2, 3, 4, 5, and 6.)
a. Make a tree diagram of the possible outcomes of a
spin of the pointer and roll of the number cube
1
2
b. What is the probability that you get a 2 on both the
spinner and the number cube? Explain your reasoning.
Since (2, 2) is one of 12 equally likely possibilities, the probability is 1 .
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c. What is the probability that you get a factor of 2 on both the spinner and the number cube?
Since the factors of 2 are 1 and 2, the only possibilities are (1, 1), (1, 2), (2, 1), and (2, 2). Thus there are 4 ways out
of 12 equally likely outcomes, so the probability is 4 , or 1 .
3
12
d. What is the probability that you get a multiple of 2 on both the number cube and the spinner?
The multiples of 2 in the data are 2, 4, and 6, so the only possibilities are (2, 2), (2, 4), and (2, 6). Thus there are 3
ways out of 12 equally likely outcomes, so the probability is 3 , or 1 .
4
12
10
32. Koto’s class makes the line plot shown below. Each mark represents the first letter of the name of a
student in her class
First Letters of Names
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Letter
Suppose you choose a student at random from Koto’s Class.
a. What is the probability that the student’s name
begins with a J?
b. What is the probability that the student’s name
begins with a letter after F and before T in the alphabet?
c. What is the probability that you choose Koto?
d. Suppose two new students Melvin and Theo, join
the class. You now choose a student at random from
the class. What is the probability that the student’s
name begins with J?
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Homework What Do You Expect Investigation 3
A C E
3.1
Applications | Connections | Extensions
Applications
2. The cooks at Kyla’s school make the spinners below to help them choose the lunch menu. They let the
students take turns spinning. For parts (a) – (c), decide which spinner you would choose. Explain your
reasoning.
a. Your favorite lunch is pizza.
b. Your favorite lunch is lasagna.
c. Your favorite lunch is hot dogs.
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3. When you use each of the spinners below, the two possible outcomes are landing on 1 and landing on 2.
Are the outcomes equally likely? If not, which outcome has a greater theoretical probability? Explain.
a.
1
2
4. Molly designs a game for a class project. She makes the three spinners shown. She tests to see which one
she likes best for her game. She spins each pointer 20 times and writes down her results, but she forgets to
record which spinner gives which set of data. Match each spinner with one of the data sets. Explain your
answer.
First data set: 1, 2, 3, 2, 1, 1, 2, 1, 2, 2, 2, 3, 2, 1, 2, 2, 2, 3, 2, 2,
Second data set: 2, 3, 1, 1, 3, 3, 3, 1, 1, 2, 3, 2, 2, 2, 1, 1, 1, 3, 3, 3
Third data set: 1, 2, 3, 3, 1, 2, 2, 2, 3, 2, 1, 2, 2, 2, 3, 2, 2, 3, 2, 1
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For 15 – 20, Complete the following table. Write each probability as a fraction, decimal, or percent.
Fraction
15.
¼
16.
1
Decimal
Percent
25%
/8
17.
33 1/3 %
18.
10%
19.
.166666666….
20.
0.05
21. The cooks at Kyla’s school let students make spinners to determine the lunch menu.
a. Make a spinner for which the chance of selecting lasagna is 25%, the chance of selecting a
hamburger is 16 2/3%, and the chance of selecting a tuna sandwich is 33 1/3%. The last choice is
hot dogs.
b. What is the chance of selecting hot dogs?
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25%