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Elementary Probability Theory

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Copyright © Cengage Learning. All rights reserved.

Section 5.2 Some Probability

Rules—Compound Events

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Focus Points

• Compute probabilities of general compound events.

• Compute probabilities involving independent events or mutually exclusive events.

• Use survey results to compute conditional probabilities.

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Conditional Probability and Multiplication Rules

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Conditional Probability and Multiplication Rules

Independent events

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Conditional Probability and Multiplication Rules

Conditional probability P(A|B)

If the events are dependent, then we must take intoaccount the changes in the probability of one event caused by the occurrence of the other event.

The notation P(A, given B) denotes the probability that event A will occur given that event B has occurred.

This is called a conditional probability.

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Conditional Probability and Multiplication Rules

We read P(A, given B) as “probability of A given B.” If A and B are dependent events, then P(A) P(A, given B) because the occurrence of event B has changed the probability that event A will occur.

A standard notation for P(A, given B) is P(A | B).

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Conditional Probability and Multiplication Rules

Multiplication rules of probability

We will use either formula (5) or formula (6) according to the information available.

Formulas (4), (5), and (6) constitute the multiplication rules of probability.

They help us compute the probability of events happening together when the sample space is too large for convenient reference or when it is not completely known.

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Conditional Probability and Multiplication Rules

Note: For conditional probability, observe that the multiplication rule

P(A and B) = P(B) P(A | B)

can be solved for P(A | B), leading to

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Example 4 – Multiplication Rule, Independent Events

Suppose you are going to throw two fair dice. What is the probability of getting a 5 on each die?

Solution Using the Multiplication Rule:

The two events are independent, so we should use formula (4).

P(5 on 1st die and 5 on 2nd die)

= P(5 on 1st) P(5 on 2nd)

To finish the problem, we need to compute the probability of getting a 5 when we throw one die.

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Example 4 – Solution

There are six faces on a die, and on a fair die each is equally likely to come up when you throw the die. Only one face has five dots, so by formula (2) for equally likely outcomes,

Now we can complete the calculation.

P(5 on 1st die and 5 on 2nd die) = P(5 on 1st) P(5 on 2nd)

cont’d

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Example 4 – Solution

Solution Using Sample Space:

The first task is to write down the sample space. Each die has six equally likely outcomes, and each outcome of the second die can be paired with each of the first. The sample space is shown in Figure 5-2.

cont’d

Sample Space for Two Dice Figure 5-2

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Example 4 – Solution

The total number of outcomes is 36, and only one is favorable to a 5 on the first die and a 5 on the second. The 36 outcomes are equally likely, so by formula (2) for equally likely outcomes,

P(5 on 1st and 5 on 2nd) =

The two methods yield the same result. The multiplication rule was easier to use because we did not need to look at all 36 outcomes in the sample space for tossing two dice.

cont’d

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Conditional Probability and Multiplication Rules

Procedure:

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