TABLE OF CONTENTS Sometimes we are concerned with probabilities about some portion of the sample space rather than the entire sample space. Here are two examples. The probability that a person has an annual income over $100,000 would be different than the probability that a college graduate has an annual income over $100,000. As of June 27, 2000, John Vander Wal had a season batting average of .292, but his batting average against left-handed pitchers was .188. In both of these examples, we are reducing the sample space. For the first example, the reduced sample space consists of college graduates and in the second example the reduced sample space consists of left-handed pitching opposition. The following example illustrates the concept of conditional probability.
Example 2.1. One chip is selected at random from a box containing five chips numbered 1, 2, 3, 4, 5. So S = {1 , 2 , 3 , 4 , 5} is an equally probable sample space. What is the probability of selecting a 1? If we let B = {1}, then 
. Now suppose we are told that the outcome is an odd number. Let A = {1 , 3 , 5}. We are given that the outcome is in A. Now what is the probability of getting a 1? The answer is 1/3. We write this as 
. The vertical line separating events B and A means given and P(
