INTEGRATED MATH
QTR 3
PROBABILITY
INVESTIGATION 3
INDEPENDENT EVENTS
NAME _____________________________
DATE Tuesday March 24th 2015
HOUR _______
The Multiplication Rule for Independent Events
Graphical representations of data or quantitative relationships can reveal important
underlying patterns. Making a “picture” of a mathematical situation can often help you
understand that situation better.
An area model is simply a graphical way of representing all the possible outcomes of a
situation. If we were to create an area model to represent the chances that a child would
have freckles if both parents have one freckle gene and one no-freckles gene, it would
look like this:
Parent 1 – freckles gene
Parent 1 – no-freckles gene
Parent 2 – freckles gene
Child has freckles
Child has freckles
Parent 2 – no-freckles gene
Child has freckles
Child does not have freckles
From this graphical ‘view’ of the situation, we can easily see the chances that a child will
not have freckles if both parents have one freckles gene and one no-freckles gene  ¼.
How can we calculate that mathematically?
Situation 1 – Shower Singing
About half of all U.S. adults are female.
According to a survey published in
USA Today, three out of five (or 60%)
adults sing in the shower.
1a. Suppose an adult from the United
States is selected at random. From the
information above, do you think that
the probability that the person is a
female and sings in the shower is equal
3
3
3
to , greater than , or less than ?
5
5
5
b.
Now examine the situation using the area model shown below.
Explain why there are two rows labeled “No”
for “Sings in Shower” and three labeled “Yes.”
Male
Yes
What assumption does this model make about
singing habits of males and females?
Yes
Yes
No
No
c.
On this area model, shade in the region that
represents the event: female and sings in the
shower.
d. What is the probability that an adult selected at
random is a female and sings in the shower?
Your answer should be a fraction.
e. What is the probability that an adult selected at
random is a male and does not sing in the shower?
Your answer should be a percent.
Female
Situation 2 - Dice
Consider this problem: What is the probability that it takes exactly two rolls of a pair of
dice before getting doubles for the first time?
2a. Calculate the probability of rolling doubles (same number on each die) on a pair of
6-sided dice. You may want to reference #3 from WS 1 to help with this problem.
2b. Explain why it makes sense to label the rows of the area model as shown below.
Doubles
Not
Doubles
Not
Doubles
Not
Doubles
Not
Doubles
Not
Doubles
b. On this area model, label the six columns to represent the possible outcomes on the
second roll of a pair of dice.
c. On this area model, lightly shade the squares that represent the event not getting
doubles on the first roll and getting doubles on the second roll.
d. What is the probability of not getting doubles on the first roll and then getting
doubles on the second roll? Your answer should be in unsimplified fraction form.
e. Use your area model to find the probability that you will not get doubles either time.
Your answer should be in unsimplified fraction form.
Situation 3 - Practice
Make an area model to help you determine the probabilities of each situation below.
Be sure to include correct labels and shading.
3a. Use area models to answer the question: About 80% of Americans pour shampoo
into their hand rather than directly onto their hair. What is the probability that both your
teacher and the President of the U.S. pour shampoo into their hand before putting it on
their hair? You need to consider how many rows and columns are necessary here.
Write your answer as a percent.
3b. Use area models to answer the question: About 25% of Americans put catsup
directly on their fries, rather than on the plate. What is the probability that both your
school principal and your favorite celebrity put catsup directly on their fries? Write your
answer as a decimal.
Situation 4 – Multiplication Rule
4a. For the previous situations, the pairs of events in each of those problems are
independent events: knowing whether one of the events occurs does not change the
probability that the other event occurs.
b. Suppose A and B are independent events. The probability symbolically is:
P(A and B)
The notation P(A and B) is read: “the probability of event A and event B occurring."
When A and B are independent events, then P(A and B) = P(A) ∙ P(B)
State this Multiplication Rule in words. (A.K.A. Write the rule as a written sentence)
Often a probability problem is easier to understand if it is written in words that are more
specific than the words the original problem uses. For example, consider the problem:
What is the probability of taking exactly two tries to roll doubles?
You could express and calculate this probability in the following manner:
P(don’t roll doubles first try and do roll doubles second try)
= P(don’t roll doubles first try )  P(do roll doubles second try )
= P(no dbls )  P(do get dbls)
5 1
=   
6 6
5
=
36
5. For each event below, show how you could mathematically compute the probability
without making an area model (we are making an assumption of independence here).
Show your work (as usual). Be careful counting outcomes!!!!!
a. Find P(male and doesn’t sing in the shower). Your final answer should be an
unsimplified fraction and then a decimal.
b. Find P(female and sings in the shower). Your final answer should be a simplified
fraction.
c. Find P(not doubles on the first roll and not doubles on the second roll). Your final
answer should be an unsimplified fraction and then a percent.
6a. Suppose Shiomo is playing a game in which he needs to roll a pair of dice and get
doubles and then immediately roll the dice again and get a sum of six. He wants to
know the probability that this will happen. (this is written as P(doubles,sum of six))
Which of the following best describes the probability Shiomo wants to find?
Option 1: P(gets doubles first roll or gets sum of six second roll)
Option 2: P(gets doubles first roll and gets sum of six second roll)
Option 3: P(gets doubles and sum of six)
b. How many total outcomes are there when rolling two dice? Hint: Reference WS 1.
c. Explain why the Multiplication Rule is appropriate to use here.
d. What is the probability? Your answer should be a simplified fraction. Note: When
counting rolls that have a sum of six, 5-1 is considered different than 1-5)
e. Find P(doubles, sum of six, sum of eleven). Answer as a simplified fraction.
7a. Suppose A, B, and C are three independent events. Write a rule for calculating
P(A and B and C). You may want to use your method in 6e.
b. According to the National Center for Education Statistics, the percentage of students
who are homeschooled in the U.S. is 2%. If you pick 10 students at random in the
U.S., find P(none homeschooled). Show the multiplication and give your answer as
a decimal rounded to the nearest thousandth.
c. You are randomly picking two cards from a standard deck (52 total cards) one at a
time, replacing it each time and thoroughly reshuffling each time.
Find P(heart, ten). Your answer should be simplified fraction.
d. In the United States, about 105 boys are born for every 100 girls. What is the
probability (to the nearest hundredth of a percent) that a family with two children
will have an older girl and a younger boy? Is this probability different than the
probability that the family will have an older boy and a younger girl? Explain your
reasoning.
e. In the game Monopoly, Jenny is sent to jail. She wants to know the probability that
she will fail to roll doubles in three tries. Calculate such a probability to the nearest
percent.
Male
Yes
Yes
Yes
No
No
Doubles
Not
Doubles
Not
Doubles
Not
Doubles
Not
Doubles
Not
Doubles
Female