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Algebra 2 Probability Unit
Unit or Topic
Standards or
Goals
Understand independence and conditional probability and use them to interpret data • Use the rules of probability to compute probabilities of compound events in
a uniform probability model Using Probability to Make Decisions • Calculate expected values and use them to solve problems • Use probability to evaluate
outcomes of decisions
Algebra 2 Probability Unit
Unit or Topic
Title:
Learning Objectives
Current Teaching Design
Classroom Activities
Online Activities
TO-DO
What will students be able to do?
List every activity that you
currently complete in your
traditional classroom situation to
teach this unit.
Based on what you have learned so
far, what instruction, activities and
assessment will you continue to
complete in the classroom? Place
an X in this column next to that
item.
Based on what you have learned so far,
what instruction, activities and
assessment will you no move to the
online environment? Place an X in this
column next to that item.
What items must you complete in
order to finish the creation of this
unit. If any of the items to the right
must be modified for online
delivery list it here. For example,
create a short podcast, find a
YouTube video, write a discussion
question, re-write directions for an
activity so it can take place online.
Determine sample spaces
for common items used in
probability (coins, dice,
playing cards) and
calculate simple
probabilities based on
these items.
Calculate unions and
intersections of events
based off of a table of dice
rolls; use Venn diagrams to
find probabilities.
Guided notes (see
below); using
manipulatives (dice,
coins, playing cards) to
determine possible
number of outcomes
https://www.youtube.com/
watch?v=MZgbECeJrOM&list
=PLm5rQvUDala4e9BsPGCkGPrJhpiLlDCg&index=1
Guided notes (see
below)
https://www.youtube.com/
watch?v=i24PI2aN73c&list=
PLm5rQvUDala4e9BsPGCkGPrJhpiLlDCg&index=2
Use multiplication to count
the number of
arrangements of items
(license plates, # of
possible multiple choice
answers on a test, etc.)
Guided notes (see
below)
X Student activity : Venn
Diagrams Activity (see
below)
https://www.youtube.com/
watch?v=5CZO7fuPD4w&list
=PLm5rQvUDala4e9BsPGCkGPrJhpiLlDCg&index=3
Create activity using
Venn Diagrams applet at
http://www.pitt.edu/~up
jecon/MCG/STAT/Java/V
ennGame.xhtml.
Use permutations to count
the number of ways of
arranging a subset of items
from a set of items (9
person baseball lineup
from a team of 15 players,
etc).
Use combinations to count
the number of ways to
select items from a set of
items (# of different
Powerball combinations,
etc).
Use tables to determine if
2 events are independent
(1 event has no impact on
the other) or mutually
exclusive (both events
cannot happen at the same
time)
Guided notes (see
below)
https://www.youtube.com/
watch?v=Gcp02ox5D8A&list
=PLm5rQvUDala4e9BsPGCkGPrJhpiLlDCg&index=4
Guided notes (see
below)
https://www.youtube.com/
watch?v=razL7T_JKTs&list=P
Lm5rQvUDala4e9BsPGCkGPrJhpiLlDCg&index=5
Guided notes (see
below)
https://www.youtube.com/
watch?v=Uk6Caz8fLuU&list=
PLm5rQvUDala4e9BsPGCkGPrJhpiLlDCg&index=6
Calculate an expected
value from a probability
distribution chart
(expected winnings when
playing a scratchoff lottery
game, expected number of
girls in a family of 5 kids,
etc)
Design a simulation based
on a real life probability
(simulate a game where
LeBron James shoots 10
free throws, etc)
Calculate a binomial
probability, where either
an event happens or it
doesn’t (probability of
getting exactly 4 questions
out of 10 right on a
Guided notes (see
below)
https://www.youtube.com/
watch?v=3SnvFf0nhcs&list=
PLm5rQvUDala4e9BsPGCkGPrJhpiLlDCg&index=7
Guided notes (see
below)
https://www.youtube.com/
watch?v=1RhFwD44Jr8&list=
PLm5rQvUDala4e9BsPGCkGPrJhpiLlDCg&index=8
Guided notes (see
below)
https://www.youtube.com/
watch?v=WNFVf0TMnLA&lis
t=PLm5rQvUDala6ek5hCJBgs
CRCLsNwXAxv6&index=1
true/false test, etc)
Calculate binomial
probabilities where more
than 1 outcome is possible
(probability of getting at
least 4 questions out of 10
right on a true/false test,
etc)
Graph a binomial
distribution, showing all
possible values (show a
histogram or bar graph
displaying the probabilities
of getting 0, 1, 2, …, 10
questions right on a 10
question true/false test,
etc.)
Use a standard normal bell
curve to estimate
probabilities (finding
probability of getting a test
grade within 1 standard
deviation of the mean, etc)
Use data described by a
normal bell curve to find
probabilities (finding
proportion of adults who
are taller than 6 feet, etc)
Classroom Assessment
Guided notes (see
below)
https://www.youtube.com/
watch?v=f1GldvGu0k&list=PLm5rQvUDala6e
k5hCJBgsCRCLsNwXAxv6&in
dex=2
Guided notes (see
below)
https://www.youtube.com/
watch?v=BKm_kHVN7Ww&li
st=PLm5rQvUDala6ek5hCJBg
sCRCLsNwXAxv6&index=3
x Student activity: Binomial
Probability Distributions (see
below)
Guided notes (see
below)
Guided notes (see
below)
https://www.youtube.com/
watch?v=4xtt3ZoRkcg&list=P
Lm5rQvUDala6ek5hCJBgsCR
CLsNwXAxv6&index=4
https://www.youtube.com/
Create an activity based
watch?v=Fuc1PLp3oJg&list= on the normal probability
PLm5rQvUDala6ek5hCJBgsC graphs applet at
RCLsNwXAxv6&index=5
http://davidmlane.com/h
X Supplemental activity to
yperstat/z_table.html
teach students how to
calculate normal
probabilities
Students have daily assignments (worksheets, book assignments) for each of the 13 learning objectives that will
be self-graded and reported to the instructor.
Formative assessments for learning objectives 1-4, 5-8, and 9-11 are given. These can be self-graded.
Summative assessments for learning objectives 1-8 and 9-13 will be given. These are graded by the instructor.
Online Assessment
Formative assessments for learning objectives 1-4, 5-8, and 9-11 could be adapted and turned into online
assessments.
Venn Diagrams Activity and Binomial Probability Distributions worksheets were originally created in portrait view. Due to the
landscape view of this document, there were some formatting problems.
Name ____________________________________________________ Date
_____________
Venn Diagrams Activity
1.
Watch this brief video on how to set up and inspect a Venn diagram at
https: //www.youtube.com/watch?v=k55rNoXMn2I
For #2-9, let P(A) = 0.40, P(B) = 0.65, and P(A and B) = 0.25.
2.
Fill in the 4 regions of the Venn diagram at the right
with appropriate probabilities.
3.
What should the sum of these 4 probabilities equal?
4.
Find P(B and not A). _________
Shade the region that corresponds
with this probability below.
5.
Find P(A and not B). _________
Shade the region that corresponds
with this probability below.
6.
Find P(A or B). __________
Shade the region that corresponds
with this probability below.
8.
7.
Find P(not A and not B). ________
Shade the region that corresponds
with this probability below.
Check to see if your shading for #4-7 is correct by using using the Venn diagram applet
at http://www.pitt.edu/~upjecon/MCG/STAT/Java/VennGame.xhtml. If there are any
errors, correct them and recalculate the probabilities. You can use this applet when you
get to problems #10-13 as well.
For #9-13, let’s say that in a survey of high school students, 40% of the students were
currently taking algebra, 30% of the students were taking biology, and 12% of the students
were taking both algebra and biology. Let A = students taking algebra and B = students taking
biology.
9.
Fill in the 4 regions of the Venn diagram at the right
with appropriate probabilities.
10.
Find P(A or B). _________
Shade the region that corresponds
11.
Find P(not A).
_________
Shade the region that corresponds
12.
with this probability below.
with this probability below.
Find P(not B). __________
13.
Shade the region that corresponds
with this probability below.
Find P(not A and B). ________
Shade the region that corresponds
with this probability below.
Name ______________________________________________ Date
___________________
Binomial Probability Distribution Graphs
For #1-4, consider a multiple choice test with 10 questions. Each question has four possible
answers: A, B, C, and D. A lazy student decided not to study for this test, and now has to
guess on every single question. Let’s see how well this student may end up performing.
1.
Using the formula for calculating binomial probabilities, P(x = k) = nCk pk q n-k , to
find the probability that the student gets exactly half of the questions right on the test.
Show the values substituted into the formula.
2.
Using the binompdf command on your calculator, find the probability that the student
gets exactly half of the questions right on the test. Show the command that appeared on
the calculator screen.
3.
Go to http://homepage.stat.uiowa.edu/~mbognar/applets/bin.html to access the binomial
distribution applet.
You should see a screen that looks like this:
Type in the values for n and p in
the appropriate boxes, and type in the value you used for k in #1 in the box that reads
“x=”. Then click in the empty pink box. What is the number that appears in the box?
4.
A histogram that displays the probabilities of getting any of 0 through 10 questions right
appears on the website. If you click on a bar, the probability for that event appears on
the screen.
The tallest bar corresponds with what x value?
What was the probability of that event?
Sketch this histogram here →
5.
There are 4 athletes that are shooting free throws at a basketball hoop. The players
range in ability levels. The first athlete makes a basket in 25% of all free throws taken,
the second athlete makes 45%, the third athlete makes 65%, and the fourth athlete
makes 85% of all free throws taken. The athletes each attempt 12 free throws. Using
the binomial distribution applet, sketch the histograms for each of these athletes that
shows the probabilities of making a basket in any of 0 through 12 free throws attempted.
First Athlete
Second Athlete
Third Athlete
6.
Fourth Athlete
Describe how the shapes of the graphs change from the first to the fourth athlete, as
their probabilities of success increase.
7.
Consider two workers who are in charge of making cookies for a bakery. The first
worker is skilled, so 70% of her cookie trays are baked perfectly. The second worker is
new to the job, so only 30% of her cookie trays are baked perfectly. They each make 8
trays of cookies.
Using the binomial distribution applet, sketch the histograms for each of these bakers
that shows the probabilities of getting any of 0 through 8 trays cooked perfectly.
First Worker
Second Worker
8.
In #7, the graph of the first worker is a reflection of the graph of the second worker.
What is the relationship between graphs like these that are reflections of each other?
9.
Visit the website http://www.mathbootcamps.com/common-shapes-of-distributions/.
List the names of the 5 different shapes of histograms listed on this site.
10.
It is estimated that about 10% of all pennies that are still in circulation were minted
before 1960. Graph the probability distributions for samples of different amounts of
pennies using the binomial distribution applet.
Histogram for a sample of 10 pennies
Histogram for a sample of 50 pennies
Histogram for a sample of 100 pennies
Histogram for a sample of 500 pennies
11.
What happens to the shapes of the histograms as the sample size increases?
Guided Notes are below. They are not formatted correctly on landscape view. For correct formatting, change to portrait view.
Name __________________________________________________ Date ________________________
Probability Notes 1 (FST 7-1)--Basic Principles of Probability
Finding sample spaces
Outcome--the result of an experiment
Sample space--the set (or list) of all possible outcomes of an experiment
The situation
Counting the red Skittles in a
packet containing 10 Skittles
Tossing 1 coin
Tossing 2 coins
Tossing 3 coins
Rolling 2 dice
The sample space
# of outcomes
in the sample
space
Finding outcomes
The situation
The event
Tossing 3 coins
Getting exactly 2 heads
Rolling 2 dice
Getting a sum of 6
Rolling 2 dice
Getting a sum of 13
A list of the outcomes for this event
Calculating probabilities
Probability--the study of chance
The probability that an event E occurs =
# of outcomes in the event
# of outcomes in the sample space
To abbreviate “the probability that an event E occurs,” we write P(E).
Probabilities can equal 0, 1, or any
 number in between.
What is the probability of tossing 3 coins and
getting exactly 2 heads?
What is the probability of rolling 2 dice and getting
a sum of 6?
What is the probability of rolling 2 dice and getting
a sum of 13?
If 60% of students in class are female, find the
probability that a person chosen at random from
this class is a male.
Calculating relative frequencies
Relative frequency--the fraction of times an event occurs in real life data collection
If you flip 1 coin, the P(tail) = _________
However, if you flip a coin twice, it is possible to get no tails.
If this occurs, the relative frequency of tails flipped is:
____________
Usually, relative frequency is approximately equal to the corresponding probability.
Name __________________________________________________ Date ________________________
Probability Notes 2 (FST 7-2)--Addition Counting Principles
Calculate probabilities or number of outcomes in a dice rolling event
Let A = the event of rolling a sum of 7 with the two dice
P(A) = probability of rolling a sum of 7 =
___________
P(not A) = probability of not rolling a sum of 7 = ___________
N(A) = number of ways of rolling a sum of 7 =
Calculating intersections
: “intersect”, AND

Let A = the first die shows a 4.
Let B = the 2nd die is a multiple of 2.
Find P(A B).
Answer: ________________________

___________
Let A = the 1st die is a multiple of 3.
Let B = the sum of the dice is 7.
Find N(A B).
Answer: ________________________

Let A = the 1st die is larger than 4
Let B = the sum of the dice is 3.
Find PA B).

Answer: ______________________
If P(A B) = 0, we say A and B are mutually exclusive events.
This means that A and B have no events in common.
Calculating unions

: “union”, OR
Formulas for calculating unionts:

P(A
N(A

Let A = the first die shows a 4.
Let B = the 2nd die is a multiple of 2.
Find P(A B) using a formula.
Answer: ________________________

 B) = P(A) + P(B) - P(A  B)
 B) = N(A) + N(B) - N(A  B)


Let A = the 1st die is a multiple of 3.
Let B = the sum of the dice is 7.
Find N(A B) using a formula.
Answer: ________________________

Let A = the 1st die is larger than 4
Let B = the sum of the dice is 3.
Find PA B) using a formula.

Answer: ______________________
Use a Venn Diagram to answer probability questions
Numbers within circle M represent the percent of
people who are fans of the Mudhens. Numbers
within circle K represent the percent of people who
are fans of the Knights.
Use the Venn diagram at the right to find the
following probabilities.
P(M) =
P(M

 K) =
P(M

 K) =
P(not M) =
P(neither M nor K) =
Name __________________________________________________ Date ________________________
Probability Notes 3 (FST 7-3)--Multiplication Counting Principles
Calculate number of ways of choosing one element from each of several sets
1.)
On Fridays, Mr. Herek likes to show his school spirit by wearing green. He has 2 different green shirts
(light and dark), 3 different green ties (school logo, dark green, stripes), and 2 styles of pants (khaki and
black) that go well with these shirts and ties. How many different outfits
consisting of a green shirt,
green tie, and pants are possible?
Tree diagram:
Multiplication Counting Principle: The number of ways to choose one element from set A and one
element from set B is N(A) N(B) .
Using Multiplication Counting Principle:

2.)
At the movie theatre, there are 4 different sizes of popcorn and 12 different varieties of candy that
can be purchased. How many different ways can a person order a popcorn and a box of candy?
Calculate number of arrangements with replacement
3.)
Some license plates in Michigan consist of 3 letters
by 4 numbers. How many different license plate
combinations of this form are possible?
4.)
A test consists of 4 true/false questions and 6 multiple choice questions, where each multiple
choice question has 5 choices.
a.)
5.)
followed
How many different answer combinations are possible?
b.)
What is the probability of answering all of the questions correctly?
If a fair 6-sided die is rolled, how many possible outcomes are there if you roll the die 8 times?
Each of the examples we just solved involved the replacement of values. This means that we can reuse
certain values in our combinations.
For the license plate example, we are allowed to repeat letters and numbers. A license plate does not have
to have 3 different letters and 4 different numbers. For the test example, you can answer more than 1
question true (or false), and you can repeat the same letter for the multiple choice answers (like B). For the
dice rolling example, the same number can be rolled over and over again.
When there are n choices for each of k elements, there are nk arrangements with replacement.
Calculate number of arrangements without replacement
5.)
Abby, Betsy, Christina, Danielle, and Esther are running the 100 meter dash. The timers will
determine which place the athletes finish. How many different orders can the athletes finish?
6.)
On a matching test, students are given the first 10 presidents of the United States in the left hand
column and a list of the first 10 vice presidents of the United States in the right hand column.
Students have to match up the president with the correct vice president. How many different
arrangements could students form?
Examples 5 and 6 involved making selections without replacement. This means that we cannot reuse values
in our arrangements.
For the racing example, if Abby finishes in first place, she cannot finish in 2nd place also. For the matching
quiz example, if once a student matches George Washington to a vice president, the student cannot match
up George Washington to another vice president. We cannot reuse George Washington.
There are n! arrangements of n elements without replacement.
Name __________________________________________________ Date ________________________
Probability Notes 4 (FST 7-4)—Permutations
Calculate permutations where all elements are arranged
Permutations:
Arrangements of a sequence
Permutations involve choosing values/letters/numbers without replacement.
Take the letters in the word SUM.


MSU is a permutation (or arrangement) of the word SUM.
MUM is not a permutation of the word SUM, because once the M is chosen the first time, it cannot be
used again.
1.)
Al, Bo, Ed, Flo, and Jo are competing for 1st through 5th places in a race. How many different
finishes of these 5 athletes are possible?
2.)
How many arrangements of 4 pictures on a wall are possible?
3a.)
How many permutations of the letters in the word SUM are possible?
b.)
List these permutations—here are 2:
SUM, MSU, …
If all elements are being arranged, there are n! permutations of n different objects.
Calculate permutations where some elements are arranged
4.)
Al, Bo, Ed, Flo, and Jo are now competing for a medal in a race. The winner will get a gold
medal, 2nd place will get a silver medal, and 3rd place will get a bronze medal. How many
different ways can the 3 medalists be determined?
Option 1: n(n-1)(n-2)… __________ x ___________ x ____________ = __________
Number
of
athletes
that could
win gold.
Number
Number of
of
athletes
athletes
that could
that could
win
win silver,
bronze,
assuming
assuming
the gold
the gold
medalist
and silver
can’t.
medalists
On the next side, we’ll look at a formula that you can usecan’t.
to get the same answer.
Option 2:
, where n=total # of elements and r=# of elements to arrange.
Since there are a total of 5 athletes and we are arranging 3 of them, we say
5P3
5.)
=
Consider the word MATH.
5a.)
Make a list of all of the 2 letter permutations that can be formed from the word
MATH—here are 2:
MA, MT, ….
5b.)
Using Option 1 (multiplication of descending numbers), how many permutations can
be formed consisting of 2 letters from the word MATH?
5c.)
6.)
7.)
Using Option 2 (permutations formula with factorials), how many permutations can
be formed consisting of 2 letters from the word MATH?
There are 12 players on a basketball team. The positions on a basketball team are point
guard, shooting guard, small forward, power forward, and center. H
6a.)
Using Option 1, how many different lineups consisting of a different player at each
position are possible?
6b.)
Answer the same question using Option 2.
A teacher wants to take a class picture. The class has 15 girls and 10 boys. The students
will be lined up from left to right in a single row.
7a.)
How many ways can the students be arranged from left to right?
7b.)
How many ways can the students be arranged if all of the girls are on the left and all
of the boys are on the right?
7c.)
How many ways can the students be arranged if all of the girls are on one side of the
room and all of the boys are on the other side of the room?
Name __________________________________________________ Date ________________________
Probability Notes 5 (FST 8-6)—Combinations
Distinguish between permutations and combinations
Permutations are arrangements of items from a set. ORDER MATTERS.
If Al, Bea, Carl, and Deb are running for president and vice president, there are 4P2 or 4x3=12
arrangements of a different president and vice president (president listed first).
If AB is a possible arrangement, list the other 11 different 2-letter permutations using ABCD.
Combinations are the number of ways of choosing items from a set. ORDER DOESN’T MATTER.
If two people from Al, Bea, Carl, and Deb are chosen to attend a conference, there are 5 ways to
choose these people. Here, AB means the same thing as BA; in both cases, Al and Bea are chosen, so
we only need to include one of these “combinations.”
List the 6 different 2-letter combinations using the letters ABCD.
Calculate combinations
nCr
represents the number of ways that r items can be chosen from n items.
C = combinations = “choose”
n
Cr 
n!
(n  r )!r !
Use the formula above to verify that there are 6 different 2-letter combinations of ABCD.
1.)
In the Michigan Lottery game Classic Lotto 47, players win the jackpot if they match the 6 numbers
drawn by machine from a pool of 47 different numbers labeled 1 through 47. The machine does not
necessarily draw the numbers in numerical order; however, as long as the player selects the same
numbers, the player wins the jackpot. How many different 6 number combinations are possible?
2.)
How many ways could you choose 4 of your 10 friends to attend a concert with you?
3.)
How many ways could you choose 4 of your 10 friends to attend a concert with you, but you definitely
want Bart and Lisa to be two of these friends?
4.)
Using the letters in the word MATH (and assuming order doesn’t matter),
a.)
how many ways can you choose 1 letter?
List them:
b.)
how many ways can you choose 2 letters?
List them:
c.)
how many ways can you choose 3 letters?
List them:
d.)
how many ways can you choose 4 letters?
List them:
5.)
In a game of poker, a player is dealt five cards from a deck of 52 playing cards. How many different
poker hands are possible?
6.)
In a game of poker, what is the probability of being dealt a flush of all hearts? (Note: there are 13
hearts in a deck of cards).
7.)
In a game of poker, what is the probability of being dealt a flush of any suit—hearts, diamonds,
spades, or clubs?
Name __________________________________________________ Date ________________________
Probability Notes 6 (FST 7-5)--Independent Events
Independent events compared with mutually exclusive events
Independent events
P(A) P(B)
--meet the criteria P(A
--the result of event A does
the result of event B
Mutually exclusive events --meet the criteria P(A  B) =
--events A and B cannot occur
same time
1
1
2
3
2
3
4
 B) =
not affect

0
at the
 mutually exclusive. Events can be neither of the two.
Events can be independent. Events can be
Events cannot be both independent and mutually exclusive.
For #1-3, are the events independent, mutually exclusive, or neither?
1.
A = spinner shows 3
B = sum of die and spinner is 5
4
5
6
1
2
3
4
1
1
2
2
3
3
4
4
1
2
3
4
5
6
2.
A = spinner shows 2
B = sum of die and spinner is 4
3.
A = spinner shows 4
B = sum of die and spinner is 3
1
1
2
2
3
3
4
4
5
5
6
6
4.
The probability of having to stop at a red light at Zimmer Road is 0.15. The probability of
having to stop at a red light downtown is 0.40. The probability of having to stop at a red
light at both intersections is 0.06. Are getting stopped at the Zimmer Road and downtown
intersections by a red light independent events?
5.
Students in grades 9-11 that were exercising after school were surveyed to determine if
they were running or lifting weights. The frequency distribution is shown in the table.
a.
What is the probability that a
randomly chosen student from this
group was running?
6.
run
lift
weights
9th
24
16
10th 11th
36
27
24
18
b.
What is the probability that a randomly chosen student was a 10th grader?
c.
What is the probability that a randomly chosen student was both running and a 10th
grader?
d.
Are grade and type of exercise independent events?
In a deck of cards, if you randomly pick a card, replace it, and pick another, what is the
probability that both cards are red?
Name __________________________________________________ Date ________________________
Probability Notes 7 (FST 7-6)--Probability Distributions
Probability distributions
Consider a family with 3 children. Let B = boy and G = girl. List all of the outcomes in the sample
space.
Let x = the number of girls in the family. In the table below, write down the probability that the family has x
= 0, 1, 2, or 3 girls, based on how often these values occurred in the sample space you listed.
x
P(x)
0
1
2
3
This chart is called a probability distribution. It lists each possible value for x and each value’s
corresponding probability. Two things you should notice about the values in a probability distribution:
1.)
Each probability is a value between 0 and 1.
2.)
The sum of the probabilities equals 1.
Graph a histogram and a scatterplot of this probability distribution.
Expected value (or mean of the probability distribution)
Expected value =  
n
x  p(x )
i
i
i1
For the family of 3 children, the mean number of girls (or expected number of girls) equals:

Name __________________________________________________ Date ________________________
Probability Notes 8 (FST 7-7)--Designing Simulations
Simulations with a random digit table
A simulation is a model of a real life situation where we use probabilities to predict future outcomes.
Example 1: In basketball, a player makes 60% of their free throws. Let’s pretend that this player is
attempting 10 baskets, and we want to simulate (or predict) how many shots the player makes.
a)
Turn in your book to page 874. We will use numbers in the random digit table starting at
Row 1. Write down the first ten 1-digit numbers on Row 1.
___
b)
___
___
___
___
___
___
___
___
___
There are 10 different possible 1-digit numbers (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10).
If we chose 60% of these numbers, that would be 6 out of the 10 numbers.
Let’s use these 6 numbers: 0, 1, 2, 3, 4, 5. These numbers represent “makes.”
The other 4 number--6, 7, 8, 9 --- will represent “misses.”
How many of the 10 numbers you wrote down in part (a) are between 0 and 5? _______
You have just simulated 10 free throws! The number of makes in this simulation is the
number you wrote down in part (b).
c)
Now, write down the first ten 1-digit numbers on Row 2.
___
d)
___
___
___
___
___
___
___
___
___
How many of the 10 numbers you wrote down in part (c) are between 0 and 5? ________
Notice that your answers for parts (b) and
(d) are different.
You don’t always get the same results for each trial of a simulation.
Example 2: In a family, assume that the birth of a boy or a girl is equally likely (50% probability for each).
Let’s use simulations to estimate the probability of having exactly 1 girl in a family of 5 children.
a)
To simulate a family of 5 kids, let’s choose the first 5 numbers from Row 3 of the random
digit table. We will let 50% of these numbers (0, 1, 2, 3, 4) represent a boy, and the other
50% of these numbers (5, 6, 7, 8, 9) represent a girl. Write down these five numbers.
_____ _____ _____ _____ _____
b)
How many of the five numbers from (a) represent girls?
c)
Write down the next 5 numbers in Row 3: _____
_____
_____ _____ _____ _____
How many of the five numbers from (c) represent girls? _____
d)
How many of the next 5 numbers represent girls? _____ the next 5? ____
the next 5? _____ the next 5? _____ the next 5? _____ the next 5? ____
the next 5? _____
the next 5? _____
e)
If you take the results of the 10 trials from parts b, c, and d, what percent of these 10 trials
resulted in the family having exactly 1 girl?
_______
Name __________________________________________________ Date ________________________
Probability Notes 9 (FST 8-9)--Binomial Probabilities
Definition of a binomial experiment
A binomial experiment has the following features:
1.)
There are repeated situations, called trials.
2.)
Each trial has 2 possible outcomes. We often say these outcomes are success and failure.
3.)
The trials are independent; what happens on one trial has no effect on what happens on the
next trial.
4.)
Each trial has the same probability of success.
5.)
The experiment has a fixed number of trials; before we start the experiment, we know
exactly how many times we are going to run the experiment.
Why are the following experiments not binomial?
a.)
You draw 3 cards from a deck of cards with replacement. You record the suit of the card
(heart, diamond, spade, or club).
b.)
You typically hit the bullseye on a dartboard 30% of the time. You throw darts until you hit
the bullseye for the first time.
c.)
30 names are placed in a hat. The names include 10 of your friends. 4 names are drawn.
Success is having a friend’s name be drawn.
Probability in a binomial setting
There is a 25% chance of getting a multiple choice question on a test correct, if each question has
4 possible choices. On a four question multiple choice test, let’s find the probability of getting
exactly 2 questions right. Assume independence between answers.

List the ways could you get exactly 2 questions right.

Find the probability of each of these scenarios happening
Example: R W W R (#1 Right, #2 and #3 Wrong, #4 Right)
Example: For R W W R, 0.25 x 0.75 x 0.75 x 0.25 = .03515625

Add these probabilities together.
Shortcut for finding probability in a binomial setting
In a binomial experiment with n trials, a probability of success p in each trial, and a probability of
failure q = 1-p, the probability of getting exactly k successes is
P(k successes ) =
n
Ck  p k  q n  k
 If there is a 25% chance of getting a multiple choice question on a test correct,
and each
question has 4 possible choices, find the probability of getting exactly 2
questions correct on a
four question multiple choice test by using a formula.

In the scenario above, use a formula to find the probability of getting exactly
0 questions correct
1 question correct
3 questions correct
4 questions correct
Probability distributions
 Using the values above, show the probability distribution for the number of
to the 4 question multiple choice test.
x
p(x)

Graph this probability distribution.
correct answers
Name __________________________________________________ Date ________________________
Probability Notes 10 (FST 8-9)--Binomial Probabilities II
Binomial probability distributions review
The probability of winning a round of Rock-Paper-Scissors is 1/3. You play 4 games against an opponent.

Find the probability of winning exactly 1 game, using the
nCk
pk q
n-k
formula.
 Find the probability of winning exactly 1 game, using the binompdf(n, p, k) command on your
You should get the same answer that was found in the first problem.

Construct a probability distribution for this experiment, where x is the number of games won.

Graph this probability distribution.
calculator.

What is the probability of winning
at least 3 games?
at least 1 game?
no more than 2 games?
Name __________________________________________________ Date ________________________
Probability Notes 11 (FST 10-1)--Binomial Probability Distributions
Graphs of binomial probability distributions
NBA basketball player Jeremy Lin makes 80% of the free throws he attempts. He attempts 6 free throws.

Show the probability distribution for the number of free throws made in 6 attempts.
Use the shortcut formula binompdf(n,p,k) to quickly obtain these probabilities.
X
0
P(X)

1
Graph this distribution.
2
3
4
5
6
Kenneth Faried makes 60% of the free
throws he attempts. Let’s say he attempts
6 free throws.
Show the probability
distribution for the number of free throws
made in 6 attempts. Use the shortcut
binompdf(n,p) to obtain the probabilities.
Then graph the distribution.
X
0
P(X)

1
2
3
4
5
6
Andre Drummond makes 40% of the free
throws he attempts. Let’s say he attempts
6 free throws. Show the probability
distribution for the number of free throws
made in 6 attempts. Use the shortcut
binompdf (n,p) to obtain the probabilities.
Then graph the distribution.
X
0
P(X)
1
2
As p decreased from 0.80 to 0.60 to 0.40, what was the effect on the graphs?
3
4
5
6
Name ___________________________________________________________ Date ___________________________
Probability Notes 12 (FST 10-5)—The Standard Normal Distribution
Evaluate probabilities and shade curves using the Normal Distribution Table.
1.
P( z < 0)
2.
P( z < 1.16)
3.
P(z < 2.4)
4.
P(z < - 1.16)
Answer: ____________
Answer: ____________
Answer: ____________
Answer: ____________
5.
6.
7.
8.
P(z > 0)
Answer: ____________
P( z > 1.16)
Answer: ____________
P(z > 2.4)
Answer: ____________
P(z > -1.16)
Answer: ____________
9.
P(0 < z < 1.16)
Answer: ____________
13.
14.
10.
P(0 < z < 2.4)
Answer: ____________
11.
P(-1.16 < z < 1.16)
Answer: ____________
12.
P(-2.4 < z <2.4)
Answer: ____________
What percent of data is within…
1 standard deviation of the
mean, or P(-1 < z <1 )?
2 standard deviations of the
mean, or P(-2 < z < 2)?
3 standard deviations of the
mean, or P(-3 < z < 3)?
Answer: _____________
Answer: _______________
Answer: ______________
Find the value of c that satisfies each equation.
P(z < c) = .8790
P(z < c) = .0351
P (z >c) = .7291
Answer: c = _______________
Answer: c = _______________
P(z < ___________) = .8790
Answer: c = _______________
Name __________________________________________________ Date ________________________
Probability Notes 13 (FST 10-6)--Other Normal Distributions
Standardizing variables
The mean height of an adult male is 70 inches with a standard deviation of 3 inches. Heights of adult males
are approximately normally distributed. What proportion of adult males are taller than 6 foot tall?
*
Label the graph at the right.
The mean is located at the center of the bell curve.
Each tick mark represents a standard deviation.
*
Draw a vertical line where a 6 foot tall male is located.
Shade in the direction that would represent males
taller than 6 foot.
*
To find this probability, we need to know exactly how many tick marks away from the mean that
the 6 foot tall mark is. This value is also known as z, the standardized score. We use this formula
to find z:
z
x m
s , where m = mean, s = standard deviation, x = variable, and z = standardized score
Use this formula to find the z score for a 6 foot tall man.

*
Let’s label this location on a standard normal bell curve.
*
Now use Appendix D to find the probability of this event
happening.
*
If there are 100 men at the supermarket, approximately how many of them would be taller than 6
foot tall?
The average length of popular songs on YouTube is 192 seconds, with a standard deviation of 25 seconds.
Assume that the distribution of these songs is approximately normal. What proportion of songs are shorter
than 3 minutes long?
Here are the 5 steps I want to see on your homework and test for work:
P(x < 180)
Write what the problem is asking for in symbols:
Find z by substituting into the
z
x m
s
z
formula:
P(z < -0.48)
Rewrite what the problem is asking using z value:
Find the probability using Appendix D:

180 192
 0.48
25

Sketch a normal curve showing appropriate shading:
=.3156
What proportion of songs are less than 2:30?
What proportion of songs are between 3 minutes and 4 minutes long?