Conditional Probability and Statistical Independence

Lecture 4

The new notation is a conditional probability. That is, it gives the probability of one event under the condition that we know another event. You can read the bar | as “given the information that.”

Formula for P()

To find the conditional probability that event A occurs given that event B occurs, divide the probability that both A and B occur by the probability that B occurs, that is,

We assume that .

Example Let’s define two events:

= the woman chosen is young, ages 18 to 29

= the woman chosen is married

The probability of choosing a young woman is

.

The probability that we choose a woman who is both young and married is

.

The conditional probability that a woman is married when we know she is under age 30 is

.

Independent Events

Two events A and B that both have positive probability are independent if

.

When events A and B are independent, it is also true that

.

Events that are not independent are said to be dependent.

Probability of Intersection of Two Independent Events

If events A and B are independent, the probability of the intersection of A and B equals the product of the probabilities of A and B; that is

.

The converse is also true: If

,

then events A and B are independent.

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