Statistics 3.2: Conditional Probability and the Multiplication Rule

Objective 1: I can find conditional probabilities.

In this section, you will learn to find the probability that 2 events occur in sequence. But before you can do this, you must know how to find ______. A conditional probability is the probability of an event occurring given that ______

______. The conditional probability of event B given that event A has already occurred is denoted ______and is read “______”.

Example 1:

A) Two cards are selected in sequence from a standard deck. Find the probability that the second card is a queen, given that the first card is a king. (Assume the king is not replaced.)

Solution: Because the first card drawn was a king, there are still ____ queens in the deck. But the king is not replaced which means there are only _____ cards remaining to choose from. This event

has the probability of We would write that as P

B) The table below shows the results of a study in which researchers examined a child’s IQ and the presence of a specific gene in the child. Find the probability that a child has a high IQ given that the child has the gene.

gene / gene not / Total
present / present
High IQ / 33 / 19 / 52
Normal IQ / 39 / 11 / 50
Total / 72 / 30 / 102

Solution: There are _____ children who have the gene present (1st column). Out of those 72, ____

have a high IQ. So the probability of this event is

We would write that as

TIY 1:

1) Find the probability that a child does not have the gene.

2) Find the probability that a child does not have the gene, given that the child has a normal IQ.

3) Find the probability that a child has the gene present, given that they have a normal IQ.

Objective 2: I can distinguish between independent and dependent events.

In some experiments, one event does not affect the probability of another event. For example, when you roll a die and flip a coin, neither event affects the outcome of the other. These events are called ______.

Two events are independent if ______

______.

Using probability notation: Two events, A and B, are independent if

*Events that are not independent are ______.

*To determine if A and B are independent, first calculate ____. Then calculate ______. If the values are ______, then the events are ______. If

Example 2: Classify the events as independent or dependent events.

A) Selecting a king from a standard deck, not replacing it, and then selecting a queen from the deck.

Solution: P(B|A) = and P(B) = . Keeping the king out of the deck changes the probability for

drawing a queen. These events are ______.

B) Tossing a coin and getting a head, and then rolling a 6-sided die and obtaining a 6.

Solution: P(B|A) = and P(B) = . Tossing the coin does not affect the outcome when you roll

the die. These events are ______.

C) Driving over 85 miles per hour (A), and then getting in a car accident (B).

TIY 2: Decide whether the events are independent or dependent.

1) Smoking a pack of cigarettes per day and developing emphysema, a chronic lung disease.

2) Exercising frequently and having a 4.0 grade point average.

3) Rolling an even number on a die and then rolling a 4.


Objective 3: I can use the multiplication rule to find probabilities.

To find the probability of two events occurring in sequence, use the multiplication rule. The probability that two events A and B will occur in sequence is

If events A and B are independent, then the rule can be simplified to

This rule can be extended for more than 2 events as well.

Example 3: Use the Multiplication Rule to find each probability.

1) Two cards are selected without replacement from a standard deck of cards. Find the probability of selecting a King then selecting a Queen.

2) A coin is tossed and a die is rolled. Find the probability of getting a head, then rolling a 6.

TIY 3:

1) The probability that a salmon swims successfully through a dam is 0.85. Find the probability that two salmon swim successfully through the dam.

2) Two cards are selected from a standard deck without replacement. Find the probability that both cards are hearts.

3) Two cards are selected from a standard deck with replacement. Find the probability that both cards are hearts.

4) Find the probability of rolling 2 even numbers in a row on a six-sided die.

Read Examples 4 and 5, pg 152 and 153.

TIY 4: The probability that the knee surgery is successful is now 90%.

1. Find the probability that three knee surgeries are successful.

2. Find the probability that at least one of the knee surgeries is successful.

TIY 5: In a jury pool, 65% of the people are female. Of these 65%, one out of four works in a health field.

1. Find the probability that a randomly selected person from the jury pool is female and works in a health field.

2. Find the probability that a randomly selected person from the jury pool is female and does not work in a health field.

Extra Examples:

This is #13 from page 154.

In the general population, one woman in eight will develop breast cancer. Research has shown that 1 woman in 600 carries a mutation of the BRCA gene. Eight out of 10 women with this mutation develop breast cancer.

A) Find the probability that a randomly selected woman will develop breast cancer given that she has a mutation of the BRCA gene.

B) Find the probability that a randomly selected woman will carry the mutation of the BRCA gene and will develop breast cancer.

C) Are the events of carrying this mutation and developing breast cancer independent or dependent? Explain.

This is #15 from page 155.

The table shows the results of a survey in which 146 families were asked if they own a computer and if they will be taking a summer vacation this year.

A) Find the probability that a randomly selected family is not taking a summer vacation this year.

B) Find the probability that a randomly selected family owns a computer.

C) Find the probability that a randomly selected family is taking a summer vacation this year given that they own a computer.

D) Find the probability a randomly selected family is taking a summer vacation this year and owns a computer.

E) Are the events of owning a computer and taking a summer vacation this year independent or dependent? Explain.