Lesson 4-3 Independent Events

Math 2HonorsName______

Learning Goals:

I can define and/or identify the following: complement, independent event, dependent event.
I can illustrate the concept of independence using everyday examples of independent events.
I can predict if two events are independent, explain my reasoning, and check my statement by

calculating P(A and B) and P(A)xP(B).

I can explain and provide an example to illustrate that for two independent events, the probability of the events occurring together are the product of the probability of each event.

Some physical characteristics, such as freckles, eyelash length, and the ability to roll one’s tongue up from the sides, are determined in a relatively simple manner by genes inherited from one’s parents. Each person has two genes that determine whether or not he or she will have freckles, one inherited from the father and one from the mother. If a child inherits a “freckles” gene from either parent or from both parents, the child has freckles. In order not to have freckles, the child must inherit a “no-freckles” gene from both parents. This explains why the gene for freckles is calleddominant, and the gene for no freckles is called recessive. A parent with two freckles genes must pass on a freckles gene to the child; a parent with two no-freckles genes must pass on a no-freckles gene. If a parent has one the each, the probability is that he or she will pass on the freckles gene, and the probability is that he or she will pass on the no-freckles gene.

Think About This Situation

Consider the chance of inheriting freckles.

In what sense does the gene for freckles “dominate” the gene for no freckles?

What is the probability that a child will have freckles if both parents do not have freckles?

What is the probability that a child will not have freckles if each parent has one freckles gene and one no-freckles gene?

Can you determine the probability that a child of freckled parents also will have freckles?

About half of all U.S. adults are female. According to a survey

published in USA Today, three out of five adults sing in the shower.

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 to , greater that , or less than ? Explain.

Now examine the situation using the area model shown below.

Explain why there are two rows labels “No” for

“Sings in Shower” and three labeled “Yes.”

What assumption does this model make about

singing habits of males and females?

Shade in the region that represents the event: female and sings in the shower.

What is the probability that an adult selected at random is a female and sings in the shower?

What is the probability that an adult selected at random is a male and does not sing in the shower?

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?
Explain why it makes sense to label the rows of the area model as shown below. Label the six columns to represent the possible outcomes on the second roll of a pair of dice.

Shade the squares that represent the event not getting doubles on the first roll andgetting doubles of the second roll.
What is the probability of not getting doubles on the first roll and then getting doubles on the second roll? ______
Use your area model to find the probability that you will get doubles both times. ______
Use your area model to find the probability that you will not get doubles either time.______

Make an area model to help you determine the probabilities that a child will or will not have freckles when each parent has one freckles gene and one no-freckles gene.

What is the probability that the child will not have freckles? ______

Compare your answer to Part a with your class’ answer to Part c of the Think About This Situation.

4.Use the below squares to make an area model to answer these questions. Show your work!

a.About 25% of Americans put ketchup directly on their french fries rather than on the plate (and then dip the fries). What is the best estimate for the probability that both Mr. Mackar and Mr. Shirey put ketchup directly on their plate? (be sure to label the columns!)

b.About 84% of Americans pour shampoo into their hand rather than directly onto their hair. What is the best estimate of the probability that both Mr. Mackar and Mr. Shirey pour shampoo into their hand before putting it onto their hair? Hint: you should not make a fraction out of .84 – just use the decimal and make the length of each side of the square 1 unit.

Area model for part (a):Area model for part (b)

Mackar

Fry

Plate

Shirey

Plate

5.Look back at the situations described in numbers (1) through (4). The pairs of events in each of thoseproblems are (or were assumed to be) independent events. Two events are independent if knowingwhether or not one of the events occurs does not change the probability that the other will occur.

Is it reasonable to assume that the events described in number (2) are independent? Explain in terms of the above definition?

Give an explanation as to why the event in number (4a) could possibly NOT be independent.

How could you compute the probabilities in number (4) WITHOUT constructing an area model. Think about this question in terms of how you compute area. . .

Definition 1a:Events A and B are independent events if and only if

The phrase if and only if means that the definition must work in both directions, which means that:

Definition 1b: if and only if events A and B are independent events.

Definition 1b tells us that, if two events are independent, then to find the probability of BOTH of them happening at the same time is

6.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.

Which of the following best describes the probability Shiomo wants to find?

Option 1: P(gets doubles on the first roll or gets a sum of six on the second roll)

Option 2: P(gets doubles on the first roll and gets a sum of six on the second roll)

Option 3: P(gets doubles and a sum of six)

Explain why the Multiplication Rule can be used to find the probability that this sequence of two events will happen. What is the probability?

7.A modification of the game in Problem 6 involves rolling a pair of dice three times. In this modified game, Shiomo needs to roll doubles, then a sum of six, and then a sum of eleven.

Find the probability that this sequence of three events will happen.

Suppose A, B, and C are three independent events. Write a rule for calculating P(A and B and C) using the probabilities of each individual event.

Write the Multiplication Rule for calculating the probability that each of four independent events occurs.

8. For each of the following questions, explain whether it is reasonable to assume that the events are independent. Then, if it applies, use the Multiplication Rule to answer the question.

What is the probability that a sequence of seven flips of a fair coin turns out to be exactly HTHTTHH?

What is the probability that a sequence of seven flips of a fair coin turns out to be exactly TTTTTTH?

According to the National Center for Education Statistics, 27.9% of public school students live in a small town or rural area. If you select 5 students at random, what is the probability that they all live in a small town or rural area?

According to the National Center for Education Statistics, the percentage of students who are homeschooled in the United States is 2.2%. If you pick 10 students at random in the United States, what is the probability that none of the 10 are homeschooled?

Refer to Part d. You pick a family with two school-age children at random in the United States. What is the probability that both children are homeschooled?