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Two Basic Rules of Probability

Module by: Susan Dean, Barbara Illowsky, Ph.D.. E-mail the authors

Summary: This module introduces the multiplication and addition rules used when calculating probabilities.

The Multiplication Rule

If AA and BB are two events defined on a sample space, then: P(A AND B) = P(B)P(A|B)P(A AND B)=P(B)P(A|B).

This rule may also be written as : P(A|B)=P(A|B)= P(A AND B) P(B) P(A AND B) P(B)

(The probability of AA given BB equals the probability of AA and BB divided by the probability of BB.)

If A and B are independent, then P(A|B) = P(A)P(A|B) = P(A). Then P(A AND B) = P(A|B) P(B)P(A AND B) = P(A|B) P(B) becomes P(A AND B) = P(A) P(B)P(A AND B)=P(A) P(B).

The Addition Rule

If A A and BB are defined on a sample space, then: P(A OR B) = P(A) + P(B) - P(A AND B)P(A OR B)=P(A)+P(B)-P(A AND B).

If AA and BB are mutually exclusive, then P(A AND B) = 0P(A AND B)= 0. Then P(A OR B) = P(A) + P(B) - P(A AND B)P(A OR B)=P(A)+P(B)-P(A AND B) becomes P(A OR B) = P(A) + P(B)P(A OR B)= P(A)+P(B).

Example 1

Klaus is trying to choose where to go on vacation. His two choices are: AA = New Zealand and BB = Alaska

  • Klaus can only afford one vacation. The probability that he chooses AA is P(A) = 0.6P(A) = 0.6 and the probability that he chooses BB is P(B) = 0.35P(B)= 0.35.
  • P(A and B) = 0P(A and B) = 0 because Klaus can only afford to take one vacation
  • Therefore, the probability that he chooses either New Zealand or Alaska is P(A OR B) = P(A) + P(B) = 0.6 + 0.35 = 0.95P(A OR B) =P(A) + P(B)=0.6+0.35 =0.95. Note that the probability that he does not choose to go anywhere on vacation must be 0.050.05.

Example 2

Carlos plays college soccer. He makes a goal 65% of the time he shoots. Carlos is going to attempt two goals in a row in the next game.

AA = the event Carlos is successful on his first attempt. P(A) = 0.65P(A)=0.65. BB = the event Carlos is successful on his second attempt. P(B) = 0.65P(B)=0.65. Carlos tends to shoot in streaks. The probability that he makes the second goal GIVEN that he made the first goal is 0.90.

Problem 1

What is the probability that he makes both goals?

Problem 2

What is the probability that Carlos makes either the first goal or the second goal?

Problem 3

Are AA and BB independent?

Problem 4

Are A A and BB mutually exclusive?

Example 3

A community swim team has 150 members. Seventy-five of the members are advanced swimmers. Forty-seven of the members are intermediate swimmers. The remainder are novice swimmers. Forty of the advanced swimmers practice 4 times a week. Thirty of the intermediate swimmers practice 4 times a week. Ten of the novice swimmers practice 4 times a week. Suppose one member of the swim team is randomly chosen. Answer the questions (Verify the answers):

Problem 1

What is the probability that the member is a novice swimmer?

Problem 2

What is the probability that the member practices 4 times a week?

Problem 3

What is the probability that the member is an advanced swimmer and practices 4 times a week?

Problem 4

What is the probability that a member is an advanced swimmer and an intermediate swimmer? Are being an advanced swimmer and an intermediate swimmer mutually exclusive? Why or why not?

Problem 5

Are being a novice swimmer and practicing 4 times a week independent events? Why or why not?

Example 4

Studies show that, if she lives to be 90, about 1 woman in 7 (approximately 14.3%) will develop breast cancer. Suppose that of those women who develop breast cancer, a test is negative 2% of the time. Also suppose that in the general population of women, the test for breast cancer is negative about 85% of the time. Let B B = woman develops breast cancer and let NN = tests negative. Suppose one woman is selected at random.

Problem 1

What is the probability that the woman develops breast cancer? What is the probability that woman tests negative?

Problem 2

Given that the woman has breast cancer, what is the probability that she tests negative?

Problem 3

What is the probability that the woman has breast cancer AND tests negative?

Problem 4

What is the probability that the woman has breast cancer or tests negative?

Problem 5

Are having breast cancer and testing negative independent events?

Problem 6

Are having breast cancer and testing negative mutually exclusive?

Glossary

Independent Events:
The occurrence of one event has no effect on the probability of the occurrence of any other event. Events A and B are independent if one of the following is true: (1). P ( A | B ) = P ( A ) ;P( A | B)=P(A); (2) P ( B | A ) = P ( B ) ;P( B | A)=P(B); (3) P ( A and B ) = P ( A ) P ( B )P(AandB)=P(A)P(B).
Mutually Exclusive:
An observation cannot fall into more than one class (category). Being in more than one category prevents being in a mutually exclusive category.
Sample Space:
The set of all possible outcomes of an experiment.

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