Common Core Math 2 – Unit 6: Probability
Day 1 - Notes: Sample Spaces, Subsets, and Basic Probability
Sample Space:
List the sample space, S, for each of the following:
a. Tossing a coin:
b. Rolling a six-sided die:
c. Drawing a marble from a bag that contains two red, three
blue, and one white marble:
Intersection of two sets (A B):
Union of two sets (A B):
Example: Given the following sets, find A B and A B
A = {1,3,5,7,9,11,13,15} B = {0,3,6,9,12,15}
A B = ____________________A B =
________________________________________
Venn Diagram:
Example: Use the Venn Diagram to answer the following
questions:
1. What are the elements of set A?
2. What are the elements of set B?
3. Why are 1, 2, and 4 in both sets?
4. What is A B?
5. What is A B?
Example: In a class of 60 students, 21 sign up for chorus, 29
sign up for band, and 5 take both. 15 students in the class are not
enrolled in either band or chorus.
6. Put this information into a Venn Diagram. If the sample
space, S, is the set of all students in the class, let students in
chorus be set A and students in band be set B.
7. What is A B? ______________________
8. What is A B? _____________________
Complement of a set:
Example: Use the Venn Diagram above to find the following:
9. What is AC? _______________ BC? ______________
10. What is (A B)C? ______________________ 11. What is (A B)C?
______________________
Example: A survey of used car salesmen revealed the following
information:
24 wear white patent-leather shoes28 wear plaid trousers
20 wear both of these things2 wear neither of these things
Create a Venn diagram.
White Shoes
Plaid Trousers
12. How many salesmen were surveyed?
13. How many salesmen wore both plaid trousers and white
shoes?
14. How many salesmen didn’t wear plaid trousers?
Two-Way Table:
Categorical data organized in 2 dimensions
Example: Android vs. iPhone
Android
iPhone
Total
Male
Female
Total
15. How many students total from this class prefer the
Android?
16. How many students prefer the iPhone?17. How many students
are females?
18. How many students are males?19. How many students are
females AND
prefer the Android?
20. How many students are males AND prefer the iPhone?
Tree Diagrams:
Example:
A fair coin is flipped twice.
Example: Make a tree diagram to show the different combinations
you could make if you could select between white and chocolate milk
and either chocolate chip or oatmeal cookies.
Fundamental Counting Principal:
Day 1 – Practice
List the sample space for each situation:
1. Rolling a die.
2. Gender possibilities of having three children.
3. The months in which your birthday could fall.
4. A spinner has 4 equally likely regions numbered 1 to 4.
The arrow is spun twice. What is the sample space in this
experiment?
Use the fundamental counting principle to determine the number
of elements in the sample space.
5. There are 2 entry doors and 3 staircases in your
school. How many ways are there to enter the building and go
to the second floor?
6. Ordering a sandwich with the following choices: wheat or rye
bread, ham or turkey, swiss or cheddar cheese.
Andy has asked his girlfriend to make all the decisions for
their date on her birthday. She will pick a restaurant and an
activity for the date. Andy will choose a gift for her. The local
restaurants include Mexican, Chinese, Seafood, and Italian. The
activities she can choose from are Putt-Putt, bowling, and movies.
Andy will buy her either candy or flowers.
7. How many outcomes are there for these three decisions?
8. Draw a tree diagram to illustrate the choices.
A travel agent plans trips for tourists from Chicago to Miami.
He gives them three ways to get from town to town: airplane, bus,
train. Once the tourists arrive, there are two ways to get to the
hotel: hotel van or taxi. The cost of each type of transportation
is given in the table below.
Transportation Type
Cost
Airplane
$350
Bus
$150
Train
$225
Hotel Van
$60
Taxi
$40
9. Draw a tree diagram to illustrate the possible choices for
the tourists. Determine the cost for each outcome.
10. If the tourists were flying to New York, the subway would be
a third way to get to the hotel. How would this change the number
of outcomes? Use the Fundamental Counting Principle to explain your
answer.
11. Tanya went shopping and bought the following items: one red
t-shirt, one blue blouse, one white t-shirt, one floral blouse, one
pair of khaki capri pants, one pair of black pants, one pair of
white capri pants, and one pair of denim shorts. Make a tree
diagram to show the choices she has to wear.
12. At the after school Tiger Club meeting, there were four
drinks you could choose from: orange juice, Coke, Dr. Pepper, and
water. There were three snacks you could choose from: peanuts,
fruit, and cookies. Each student may only have one drink and one
snack.
a) Create a tree diagram showing all possible choices
available.
b) Find the number of possible choices available.
13. There are 3 trails leading to Camp A from your starting
position. There are 5 trails from Camp A to Camp B. How
many different routes are there from the starting position to Camp
B?
14. A jar contains 2 red, 2 green and 1 blue beads. Two beads
are drawn with replacement. Use a tree diagram to illustrate the
outcomes and figure out the number of outcomes where at least one
red bead is drawn.
15. Create your own 3 event tree diagram problem that has 12
outcomes in the sample space. Be creative with your problem
description!
Day 2 – Notes: Basic Probability
Probability of an Event: P(E) =
Your answer can be written as a fraction or a %. (Remember to
write it as a %, you need to multiply the decimal by 100)
· We can use sample spaces, intersections, unions, and
compliments of sets to help us find probabilities of events.
· Note that P(AC) is every outcome except (or not) A, so we can
find P(AC) by finding 1 – P(A)
· Why do you think this works?
Example: An experiment consists of tossing three fair coins.
1. List the sample space for the outcomes of the experiment.
2. Find the following probabilities:
a. P(all heads) _________________b. P(two tails)
_________________
c. P(no heads) _________________d. P(at least one tail)
_____________
e. How could you use complements to find d?
Example: A bag contains six red marbles, four blue marbles, two
yellow marbles and 3 white marbles. One marble is drawn at
random.
3. List the sample space for this experiment.
4. Find the following probabilities:
a. P(red) ____________b. P(blue or white) ____________ c. P(not
yellow) ____________
Note that we could either count all the outcomes that are not
yellow or we could think of this as being 1 – P(yellow). Why is
this?
Example: Given the Venn Diagram below, find the probability of
the following if a student was selected at random:
5. P( blonde hair)
6. P(blonde hair and blue eyes)
7. P(blonde hair or blue eyes)
8. P(not blue eyes)
Mutually Exclusive Events
Suppose you are rolling a six-sided die. What is the probability
that you roll an odd number or you roll a 2?
· Can these both occur at the same time? Why or why not?
Mutually Exclusive Events:
· The probability of two mutually exclusive events occurring at
the same time, P(A and B), is _______.
To find the probability of one of two mutually exclusive events
occurring, use the following formula:
Examples:
1. If you randomly chose one of the integers 1 – 10, what is the
probability of choosing either an odd number or an even number?
Are these mutually exclusive events? Why or why not?
Complete the following statement: P(odd or even) = P(_____) +
P(_____)
Now fill in with numbers: P(odd or even) = _______ + ________ =
___________________________
Does this answer make sense?
2. Two fair dice are rolled. What is the probability of getting
a sum less than 7 or a sum equal to 10?
Are these events mutually exclusive?
Sometimes using a table of outcomes is useful. Complete the
following table using the sums of two dice:
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
3
4
4
5
6
P(getting a sum less than 7 OR sum of 10) =
Mutually Inclusive Events
Suppose you are rolling a six-sided die. What is the probability
that you roll an odd number or a number less than 4?
· Can these both occur at the same time? If so, when?
Mutually Inclusive Events:
Probability of the Union of Two Events: The Addition Rule:
***____________________________________________________________________________________________***
Examples:
1. What is the probability of choosing a card from a deck of
cards that is a club or a ten?
A
B
B
P(choosing a club or a ten) =
2. What is the probability of choosing a number from 1 to 10
that is less than 5 or odd?
(Draw a Venn Diagram of the situation first)
3. A bag contains 26 tiles with a letter on each, one tile for
each letter of the alphabet. What is the probability of reaching
into the bag and randomly choosing a tile with one of the first 10
letters of the alphabet on it or randomly choosing a tile with a
vowel on it?
4. A bag contains 26 tiles with a letter on each, one tile for
each letter of the alphabet. What is the probability of reaching
into the bag and randomly choosing a tile with one of the last 5
letters of the alphabet on it or randomly choosing a tile with a
vowel on it?
5. P(A) = 0.7, P(B) = 0.4, P(A and B) = 0.3
a. Make a Venn Diagram representing the situation.
b. Are A and B mutually exclusive?
d. Find P(A or B)
e. EXTENSION: Find the following probabilities:
P (A)C P(A and B)C P(A or B)C
Day 2 - Homework - Mutually Exclusive and Inclusive Events
1. 2 dice are tossed. What is the probability of obtaining a sum
equal to 6?
2. 2 dice are tossed. What is the probability of obtaining a sum
less than 6?
3. 2 dice are tossed. What is the probability of obtaining a sum
of at least 6?
4. Thomas bought a bag of jelly beans that contained 10 red
jelly beans, 15 blue jelly beans, and 12 green jelly beans. What is
the probability of Thomas reaching into the bag and pulling out a
blue or green jelly bean?
5. A card is chosen at random from a standard deck of cards.
What is the probability that the card chosen is a heart or spade?
Are these events mutually exclusive?
6. 3 coins are tossed simultaneously. What is the probability of
getting 3 heads or 3 tails? Are these events mutually
exclusive?
7. In question 6, what is the probability of getting 3 heads and
3 tails when tossing the 3 coins simultaneously?
8. Are randomly choosing a person who is left-handed and
randomly choosing a person who is right-handed mutually exclusive
events? Explain your answer.
9. Suppose 2 events are mutually exclusive events. If one of the
events is randomly choosing a boy from the freshman class of a high
school, what could the other event be? Explain your answer.
10.
Consider a sample set as . Event A is the multiples of 4, while
event B is the multiples of 5. What is the probability that a
number chosen at random from S will be from both A and B?
11. For question 10, what is the probability that a number
chosen at random will be from either A or B?
12. A card is chosen at random from a standard deck of cards.
What is the probability that the card chosen is a heart or a face
card? Are these events mutually inclusive?
13. What is the probability of choosing a number from 1 to 10
that is greater than 5 or even?
14. A bag contains 26 tiles with a letter on each, one tile for
each letter of the alphabet. What is the probability of reaching
into the bag and randomly choosing a tile with one of the letters
in the word ENGLISH on it or randomly choosing a tile with a vowel
on it?
15. Jack is a student in Bluenose High School. He noticed that a
lot of the students in his math class were also in his chemistry
class. In fact, of the 60 students in his grade, 28 students were
in his math class, 32 students were in his chemistry class, and 15
students were in both his math class and his chemistry class. He
decided to calculate what the probability was of selecting a
student at random who was either in his math class or his chemistry
class, but not both. Draw a Venn diagram and help Jack with his
calculation.
a. Are these events mutually exclusive?
b. Find P(Math or Chemistry)
c. Find P(Math and Chemistry)
d. What is the difference between b and c?
16. Brenda did a survey of the students in her classes about
whether they liked to get a candy bar or a new math pencil as their
reward for positive behavior. She asked all 71 students she taught,
and 32 said they would like a candy bar, 25 said they wanted a new
pencil, and 4 said they wanted both. If Brenda were to select a
student at random from her classes, what is the probability that
the student chosen would want:
Pencil
No Pencil
Total
Candy Bar
No Candy Bar
Total
a. a candy bar or a pencil?
b. neither a candy bar nor a pencil?
Day 3 – Notes: Independent and Dependent Events
Independent Events:
Examples- Landing on heads from two different coins, rolling a 4
on a die, then rolling a 3 on a second roll of the die.
Probability of A and B occurring:
Example: A jar contains three red, five green, two blue and six
yellow marbles. A marble is chosen at random from the jar. After
replacing it, a second marble is chosen. What is the probability of
choosing a green then a red marble?
Dependent Events:
Examples- Picking a blue marble and then picking another blue
marble if I don’t replace the first one.
Example: A jar contains three red, five green, two blue and six
yellow marbles. A marble is chosen at random from the jar. The
marble is not replaced, and a second marble is chosen. What is the
probability of choosing a green marble then a red marble?
Example- Use a tree diagram: A jar contains 1 red, 1 green, 1
blue and 1 yellow marble. A marble is chosen at random from the
jar. The marble is not replaced, and a second marble is chosen.
What is the probability of choosing a green and a red marble (in
any order)?
More Examples:
1. A jar contains three red, five green, two blue and six yellow
marbles. A marble is chosen at random from the jar. A second marble
is chosen. What is the probability of choosing a green then a
yellow marble if the first marble is not replaced?
2. A jar contains three red, five green, two blue and six yellow
marbles. A marble is chosen at random from the jar. A second marble
is chosen. What is the probability of choosing a green marble both
times if the first marble is not replaced?
3. Suppose a card is chosen at random from a deck of cards,
replaced, and then a second card is chosen. Would these events be
independent? How do we know?
What is the probability that both cards are 7s?
4. Suppose a card is chosen at random from a deck, the card is
NOT replaced, and then a second card is chosen from the same deck.
What is the probability that both will be 7s?
This is similar the earlier example, but these events are
dependent? How do we know?
Now find the probability that both cards will be 7s.
5. A box contains 5 red marbles and 5 purple marbles. What is
the probability of drawing 2 purple marbles and 1 red marble in
succession without replacement?
6. In Example 5, what is the probability of first drawing all 5
red marbles in succession and then drawing all 5 purple marbles in
succession without replacement?
Mutually Exclusive
P(A or B) = P(A) + P(B)
-Drawing a king or a queen
-Selecting a male or a female
-Selecting a blue or a red marble
Mutually Inclusive
P(A or B) = P(A) + P(B) - P(overlap)
-Drawing a king or a diamond
-rolling an even sum or a sum greater than 10 on two dice
-Selecting a female from Georgia or a female from Atlanta
Independent
P(A and B) = P(A) ∙ P(B)
WITH REPLACEMNT:
-Drawing a king and a queen
-Selecting a male and a female
-Selecting a blue and a red marble
Dependent
P(A and B) = P(A) ∙ P(B given A)
WITHOUT REPLACEMENT:
-Drawing a king and a queen
-Selecting a male and a female
-Selecting a blue and a red marble
Day 3 – Homework: Independent and Dependent Events
1. Suppose you have a jar of candies: 4 red, 5 purple and 7
green. Find the following probabilities of the following
events:
a. Selecting a red candy. b. Selecting a purple candy.
c. Selecting a green or red candy. d. Selecting any color except
a green candy.
f. If you select one candy, put it back, and select again, what
is the probability of pulling two purple candies?
g. If you select two candies, what is the probability that the
first is red and the second is green?
h. If you select two candies, what is the probability that one
is red and one is green? (hint: be careful! Order does not
matter!)
2. A coin and a die are tossed. Calculate the probability of
getting tails and a 5.
3. In Tania's homeroom class, 9% of the students were born in
March and 40% of the students have a blood type of O+. What is the
probability of a student chosen at random from Tania's homeroom
class being born in March and having a blood type of O+?
4. If a baseball player gets a hit in 31% of his at-bats, what
it the probability that the baseball player will get a hit in 5
at-bats in a row?
5. Two cards are chosen from a deck of cards. The first card is
replaced before choosing the second card. What is the probability
that they both will be clubs?
6. If the probability of receiving at least 1 piece of mail on
any particular day is 22%, what is the probability of not receiving
any mail for 3 days in a row?
7. Assume you have 8 red beads and 12 blue beads in a bag and
you will be drawing 3 beads from the bag one at a time without
replacement. Make a tree diagram that represents this situation if
you pulled three beads and label each branch with the probabilities
of the outcomes.
a. What is the probability of pulling three red beads?
b. What is the probability of pulling one red and two blue beads
in any order?
8. Jenny bought a half-dozen doughnuts, and she plans to
randomly select 1 doughnut each morning and eat it for breakfast
until all the doughnuts are gone. If there are 3 glazed, 1 jelly,
and 2 plain doughnuts, what is the probability that the last
doughnut Jenny eats is a jelly doughnut?
9. Steve will draw 2 cards one after the other from a standard
deck of cards without replacement. What is the probability that he
will draw a heart then a diamond?
10. A class consists of 12 girls and 8 boys. A group of three is
picked for a committee. If the students are picked at random, what
is the probability that they all will be boys?
11. A fair coin is flipped four times.
a. Find the probability that it will land up heads each
time.
b. Find the probability that it land the same way each time.
12. A bag of marbles contains 12 red marbles, 8 blue marbles,
and 5 green marbles. If three marbles are pulled out, find each of
the following probabilities.
a. Pulling out three green marbles without replacement
b. Pulling out three red marbles with replacement
c. Pulling out a red, a green, and a blue, in order, without
replacement
Day 4 – Notes: Conditional Probability
Conditional Probability:
For example:
· What is the probability of selecting a queen given an ace has
been drawn and not replaced.
· What is the probability that a student in the 10th grade is
enrolled in biology given that the student is enrolled in CCM2?
Conditional Probability Formula: The conditional probability of
B given A is expressed as P(B | A)
Examples
1. You are playing a game of cards where the winner is
determined by drawing two cards of the same suit. What is the
probability of drawing clubs on the second draw if the first card
drawn is a club?
2. A bag contains 6 blue marbles and 2 brown marbles. One marble
is randomly drawn and discarded. Then a second marble is drawn.
Find the probability that the second marble is brown given that the
first marble drawn was blue.
3. In Mr. Jonas' homeroom, 70% of the students have brown hair,
25% have brown eyes, and 5% have both brown hair and brown eyes. A
student is excused early to go to a doctor's appointment. If the
student has brown hair, what is the probability that the student
also has brown eyes?
Bus
Walk
Car
Other
Total
9th or 10th
106
30
70
4
11th or 12th
41
58
184
7
Total
4. Suppose we survey all the students at school and ask them how
they get to school and also what grade they are in. The chart below
gives the results. Complete the two-way frequency table:
Suppose we randomly select one student.
a. What is the probability that the student walked to
school?
b. P(9th or 10th grader)
c. P(rode the bus OR 11th or 12th grader)
e. What is P(Walk|9th or 10th grade)?
d. What is the probability that a student is in 11th or 12th
grade given that they rode in a car to school?
5. The manager of an ice cream shop is curious as to which
customers are buying certain flavors of ice cream. He decides to
track whether the customer is an adult or a child and whether they
order vanilla ice cream or chocolate ice cream. He finds that of
his 224 customers in one week that 146 ordered chocolate. He also
finds that 52 of his 93 adult customers ordered vanilla. Build a
two-way frequency table that tracks the type of customer and type
of ice cream.
Vanilla
Chocolate
Total
Adult
Child
Total
a. Find P(vanillaadult)b. Find P(childchocolate)
6. A survey asked students which types of music they listen to?
Out of 200 students, 75 indicated pop music and 45 indicated
country music with 22 of these students indicating they listened to
both. Use a Venn diagram to find the probability that a randomly
selected student listens to pop music given that they listen
country music.
Tree Diagrams
· Earlier in this unit, you learned about tree diagrams.
· Let’s look at an example to see how to use conditional
probability in tree diagrams.
7. You wake up by 6 am for school about 80% of the time. When
you wake up at 6 am, the chances you are on time is about 75% of
the time. When you do not wake up by 6 am, you are late 58% of the
time.
a. Make a tree diagram of this situation. Make event A: woke up
by 6 am, and event B: you were on time to school
b. Find P(BC A). c. Find P(A and B).
d. What is the probability that you are late to school
P(BC)?
e. What is the probability that you woke up by 6 am given that
you were late to school P(A BC)?
Day 4 – Homework: Conditional Probability
Write all probabilities as a percent rounded to one decimal
place, if necessary.
1. Compete the following table using sums from rolling two dice.
Use the table to answer questions 2-5.
1
2
3
4
5
6
1
2
3
4
5
6
2. 2 fair dice are rolled. What is the probability that the sum
is even given that the first die that is rolled is a 2?
3. 2 fair dice are rolled. What is the probability that the sum
is even given that the first die rolled is a 5?
4. 2 fair dice are rolled. What is the probability that the sum
is odd given that the first die rolled is a 5?
5. Steve and Scott are playing a game of cards with a standard
deck of playing cards. Steve deals Scott a black king. What is the
probability that Scott’s second card will be a red card?
6. Donna discusses with her parents the idea that she should get
an allowance. She says that in her class, 55% of her classmates
receive an allowance for doing chores, and 25% get an allowance for
doing chores and are good to their parents. Her mom asks Donna what
the probability is that a classmate will be good to his or her
parents given that he or she receives an allowance for doing
chores. What should Donna's answer be?
7. At a local high school, the probability that a student speaks
English and French is 15%. The probability that a student speaks
French is 45%. What is the probability that a student speaks
English, given that the student speaks French?
8. On a game show, there are 16 questions: 8 easy, 5
medium-hard, and 3 hard. If contestants are given questions
randomly, what is the probability that the first two contestants
will get easy questions?
9. On the game show above, if the first contestant got an easy
question, what is the probability the second contestant will get a
hard question?
Mammals
Birds
Reptiles
Amphibians
Total
Domestic
63
78
14
10
Foreign
251
175
64
8
Total
10. Animals on the endangered species list are given in the
table below by type of animal and whether it is domestic or foreign
to the United States. Complete the table and answer the following
questions.
An endangered animal is selected at random. What is the
probability that it is:
a. a bird found in the United States? b. foreign or a
mammal?
c. domestic? d. a bird given that it is found in the United
States?
e. a bird given that it is foreign?
11. Using the table in #10, fill in the Venn Diagram looking at
the events D: Domestic and R: Reptiles.
R
D
Answer the questions below.
a. P(D and R)b. P(D or R)c. P(RC)
d. P(D⃒ R)e. P(R⃒ D)
b. Write in words what part d. and part e. notation means and
although they appear similar, why are their probabilities
different?
c. Challenge! Explain what P(RC⃒ D) is asking for and then
find:
Day 5 – Notes: Independence
Let’s review! To this point, we have decided whether events are
independent just by thinking about whether there is an overlap
between the two events. Try these:
Which of these sets of events are independent?
1. Tossing two dice and getting a 6 on both of them.
2. Choosing a marble out of a bag, then choosing another marble
without replacing it.
3. Picking a King from a deck of cards and rolling a 4 on a
die.
4. Using numbers 1 – 20, picking an odd number, replacing it,
and picking a number less than 10.
Are they independent??
Let’s consider what customers order at lunch. The data is
displayed in the chart below. Are the events “drinking water”
–event A and “ordering salad” – event B independent?
Orders
salad
Orders sandwich
Drinks water
8
16
Drinks diet soda
6
12
There’s definitely overlap here – there are 8 people who did
both. But does that mean that they’re not independent?
There are two ways to determine mathematically whether items are
independent.
Option 1:
In words: The probability of drinking water times the
probability of ordering a salad has to equal the probability of
doing both things.
Example 1:
Orders salad
Orders sandwich
Total
Drinks water
8
16
Drinks diet soda
6
12
Total
42
Find the probabilities:
P(drinking water) = P(ordering salad) = P(drinking water and
ordering salad) =
Now fit the probabilities into the formula:
P(drinking water) ∙ P(ordering salad) =
P(drinking water and ordering salad)=
Are these two events independent? How do you know?
There’s another way to show independence. It uses the definition
of independence – that the probability of event A will not change
whether event B changes or not. If we translate that into an
equation, we get this:
Option 2:
Example 2: Are the events “drinking water” –event A and
“ordering salad” – event B independent?
In words, we have to show that the probability of ordering salad
is equal to the probability of ordering salad, given that the
person is drinking water. So let’s find the conditional
probability:
Orders salad
Orders sandwich
Total
Drinks water
8
16
24
Drinks diet soda
6
12
18
Total
14
28
42
P(salad)= P(salad|water) =
Using this formula, we see that they’re independent again. That
shouldn’t be much of a surprise – if two events come out to be
independent one way, doesn’t it make sense that they would be
independent the other way, too? In fact, if you checked two other
events from the table, they’ll come out independent, too.
Determine whether ordering diet soda (event A) and ordering a
sandwich (event B) are independent. Do it both ways:
Put it in words:
Find the probabilities:
Compute, Simplify, and Compare:
Write your conclusion in words:
Put it in words:
Find the probabilities:
Compute, Simplify, and Compare:
Write your conclusion in words:
It is important to know both methods to determine independence
since one may be easier than the other.
Example 3:
The Venn diagram shows the number of students enrolled in both
Math 2 and Biology during fall semester:
Are taking Biology (event A) and taking Math 2 (event B)
independent?
Find the probabilities:
Compute, Simplify, and Compare:
Are these two events independent? Write your conclusion in
words:
*you can do the other direction too!*
Find the probabilities:
Compute, Simplify, and Compare:
Are these two events independent? Write your conclusion in
words:
Example 4: The formulas can be used to find probabilities given
that events are independent.
Events A and B are independent.
If P(A) = 0.21, P(B) = 0.34, find P(A and B).
Example 5: Find P(A) if P(B) = 0.8, P(A and B) = 0.40
Day 5: Classwork
1. Tomas selects one of these garments at random. Let A be the
event he selects a blue garment and let B be the event he selects a
shirt. Are these independent events? His closet contains:
1 blue shirt, 1 green shirt, 1 blue hat, 1 green scarf, 1 blue
pair of pants, and 1 green pair of pants.
P(A)= _________ P(B)= __________ __________ P(A | B)= __________
P(B | A)= __________
Does ? Does P(A) = P(A | B)? Does P(B) = P(B | A)?
Vanilla
Chocolate
Total
Adult
52
41
93
Child
26
105
131
Total
78
146
224
2. Determine whether age and choice of ice cream are independent
events. (you can pick either age or either flavor as event A and
event B)
Event A: __________________ Event B: _________________
3. A survey of consumers in Mudville showed 10% were
dissatisfied with their plumbing jobs. Half of these complaints
dealt with Mr. Badwrench, who does 40% of the plumbing jobs in
Mudville. What is the probability that a randomly selected consumer
has an unsatisfactory plumbing job, given that the plumber was Mr.
Badwrench. Are bad plumbing jobs and Mr. Badwrench independent?
Smoker
Nonsmoker
Total
Lung cancer
0.12
0.03
0.15
No lung cancer
0.04
0.81
0.85
Total
0.16
0.84
1
4. Are having lung cancer and smoking independent? (the numbers
in these tables are probabilities – they work the same way!)
5. Are selecting a 2 and selecting a club from a standard deck
independent?
6. Carrie is a kicker on her rugby team. She estimates that her
chances of scoring on a penalty kick during a game are 75% when
there is no wind, but only 60% on a windy day. The weather forecast
gives a 55% probability of windy weather today. Are the events
“Carrie scores on a penalty kick” and “it’s windy” independent?
Makes
the kick
Misses
the kick
Total
Windy
Not windy
Total
1
Day 5 – Homework: Independence
1. Determine which of the following are examples of independent
or dependent events.
a. Rolling a 5 on one die and rolling a 5 on a second die.
b. Choosing a cookie from the cookie jar and choosing a jack
from a deck of cards.
c. Selecting a book from the library and selecting a book that
is a mystery novel.
d. Going to the beach and bringing an umbrella.
e. Choosing an 8 from a deck of cards, replacing it, and
choosing a face card.
f. Choosing a jack from a deck of cards and choosing another
jack, without replacement.
Use one of the two independence rules to determine if the
following events are independent:
2. An elementary class consists of 12 boys and 10 girls. Six of
the boys and 5 of the girls are wearing jeans. Are the events
“wearing jeans” and gender independent?
3. There is a 34% chance that a person selected at random is a
regular smoker of cigarettes and an 18% chance that a person
selected at random has emphysema. The percentage of the population
who are both regular smokers and have emphysema is 14%. Are being a
smoker and having emphysema independent?
4. A large lending institution gives both adjustable and fixed
rate mortgages on residential property. It breaks residential
property into three categories: single- family, condo, and
multi-family. The accompanying table, sometimes called a joint
probability table displays the probabilities for the problem
instead of frequencies (numbers):
Single-Family
Condo
Multi-Family
Adjustable
.40
.21
.09
Fixed
.10
.09
.11
a. What is the probability of having a fixed rate mortgage?
b. What is the probability of having an adjustable mortgage and
owning a condo?
c. What is the probability of having a fixed mortgage or owning
a single-family home?
d. What is the probability of having a fixed mortgage if you own
a multi-family dwelling?
e. What is the probability of having an adjustable mortgage if
you own a multi-family dwelling?
f. Are owning a condo and having an adjustable mortgage
independent?
5. A survey of students at Smartville High regarding the
cafeteria food showed the following:
I like the food
I dislike the food
No opinion
Freshman
60
105
54
Sophomore
36
121
32
Junior
26
134
23
Senior
16
142
21
Is class independent of liking the food in the cafeteria?
6. The table below shows the proportions of a population that
have given hair color and eye color combinations.
Eye Color
Hair Color
Brown
Blonde
Red
Total
Blue
0.17
0.21
0.02
0.40
Brown
0.21
0.13
0.01
0.35
Green
0.07
0.03
0.15
0.25
Total
0.45
0.37
0.18
1.00
a. Use the table to determine whether having red hair and green
eyes are independent.
b. Use the table to determine whether having blonde hair and
brown eyes are independent.
7. The diagram below shows the number of days it rained and
snowed in January in Boone, NC. Based on the diagram, are the
events of having snow and having rain independent?
snow
3
5
8
15
rain
rain
8. A survey of 57 tenth graders was done to determine which
subject was their favorite. The results are shown in the table
below:
Math
English
Social Studies
Science
Total
female
8
6
10
6
30
male
10
4
9
4
27
total
18
10
19
9
57
Does it appear, based on the data, that the preference for math
as a favorite subject and gender are independent? Show your
analysis and explain your findings.
9. The probability that a person is left handed is 12%, the
probability that the person has brown eyes is 42%, and the
probability of being left handed with brown eyes is 2%. Are these
two events independent? Support your answer.
Day 6 – Notes – Experimental Probability vs. Theoretical
Probability
Experimental probability: when you do the experiment
Theoretical probability: what should happen in an ideal
situation.
How can you tell which is experimental and which is theoretical
probability?
Theoretical
Toss a coin and getting a head or a tail is 1 out of 2
sides.
Experimental
You toss a coin 10 times and record a head 3 times, a tail 7
times.
Example: Identify the type of probability for each
situation.
a. A bag contains 3 red marbles and 3 blue marbles.
b. You draw a marble out of a bag, record the color, and replace
the marble. After 6 draws, you record 2 red marbles.
Simulation:
Some of our simulations can be done quite easily using actual
probability tools like dice or spinners. Some situations, though,
will require us to use a random number generator or
a table of random digits.
Steps for Conducting a Simulation
1. State the problem or describe the experiment
2. State the assumptions
3. Determine how to represent outcomes
4. Simulate many repetitions
5. State your conclusions
Experimental vs. Theoretical Probability Investigation
Situation: A number cube is rolled 10 times.
Think about it! Do you think you are about to roll a “fair”
number cube? How do you know? What results would a fair number cube
have?
Sample Space: The set of all possible outcomes. For example, the
sample space of tossing a coin is {Heads, Tails} because these are
the only two possible outcomes.
Experimental Probability: When conducting an experiment, the
probability of that event happening in your experiment. P(Event) =
Number of times the event occurs
Total number of trials
Simulation Probability: Similar to experimental probability, but
instead of the person doing the experiment, a calculator/computer
program simulates the process. To see a simulation of flipping a
coin, see this website:
http://bcs.whfreeman.com/ips4e/cat_010/applets/Probability.html
Theoretical Probability: In theory, the probability of an event
occurring. The probability of an event is the ratio or the number
of favorable outcomes to the total possible outcomes. P(Event) =
Number or favorable outcomes Total possible outcomes
Simulation: Have each person in your group do the following.
Press the MATH button, scroll right to PRB, choose option
5:randInt( . To put restrictions 1 through 6, type in randInt(1,6).
And hit Enter 10 times – log your results in the 2nd column below.
Once your group is finished, tally the entire group’s tally in the
3rd column.
Experimental: Have each person in your group toss a dice 10
times. Log your results in the 2nd column. Once your group is
finished, tally the entire group’s tally in the 3rd column.
List the sample space of this experiment:
______________________________________________
What is the probability that..
-use the third column on the previous page to answer the 1st and
2nd blank columns)
-write your probability both as a fraction and as a percent.
Experimental – look at the group column in the 1st table to
answer.
Simulation – look at the group column in the 2nd table to
answer.
Theoretical – use your brain! Remember:
# favorable outcomes
# total outcomes
Rolled a 3?
Rolled a 5?
Rolled a 3 or a 5?
Rolled an even number?
Did NOT roll a 4?
Writing in Math
Compare the three columns of each situation. Were they close or
not? Would you expect them to be close? If they aren’t close,
why?
Let’s try more with Theoretical Probability! Here is the sample
space for rolling two six-sided number cubes:
How many total possible outcomes from rolling two six-sided
number cubes:
Find the following theoretical probabilities:
a. Probability of rolling a 1 and a 4 = b. P(sum of 8) =
c. P(doubles) = d. P(product of 12) =
Day 6 – Homework
Amanda used a standard deck of 52 cards and selected a card at
random. She recorded the suit of the card she picked, and then
replaced the card. The results are in the table below.
Diamonds
Hearts
Spades
Clubs
1. Based on her results, what is the experimental probability of
selecting a heart?
2. What is the theoretical probability of selecting a heart?
3. Based on her results, what is the experimental probability of
selecting a diamond or a spade?
4. What is the theoretical probability of selecting a diamond or
a spade?
5. Compare these results, and describe your findings.
6. Dale conducted a survey of the students in his classes to
observe the distribution of eye color. The table shows the results
of his survey.
Eye color
Blue
Brown
Green
Hazel
Number
12
58
2
8
a. Find the experimental probability distribution for each eye
color.
P (blue) = ______ P (brown) = _____ P (green) = _____ P (hazel)=
______
b. Based on the survey, what is the experimental probability
that a student in Dale’s class has blue or green eyes?
c. Based on the survey, what is the experimental probability
that a student in Dale’s class does not have green or hazel
eyes?
d. If the distribution of eye color in Dale’s grade is similar
to the distribution in his classes, about how many of the 360
students in his grade would be expected to have brown eyes?
7. Your sock drawer is a mess! You just shove all of your socks
in the drawer without worrying about finding matches. Your aunt
asks how many pairs of each color you have. You know that you have
32 pairs of socks, or 64 individual socks in four different colors:
white, blue, black, and tan. You do not want to count all of your
socks, so you randomly pick 20 individual socks and predict the
number from your results.
Color of sock
White
Blue
Black
Tan
Number of socks
12
1
3
4
a. Find the experimental probability of each:
P (white) = P (blue) = P (black) = P (tan) =
b. Based on your experiment, how many socks of each color are in
your drawer? Show your work!
White = Blue = Black = Tan =
c. Based on your results, how many pairs of each sock are in
your drawer?
White = Blue = Black = Tan =
d. Your drawer actually contains 16 pairs of white socks, 2
pairs of blue socks, 6 pairs of black socks, and 8 pairs of tan
socks. How accurate was your prediction?
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