CONDITIONAL PROBABILITY Consider two events, A and B. Suppose we know that B has occurred. This knowledge may change the probability that A will occur. We denote by P(A|B) the conditional probability of event A given that B has occurred.

To obtain a formula for P(A|B), let us refer to the following figure:

Note that the knowledge that B has occurred effectively reduces the sample space from S to B. Therefore, interpreting probability as the area, P(A|B) is the proportion of the area of B occupied by A:

Example- Tossing Two Dice: Conditional Probability An experiment consists of tossing two fair dice which has a sample space of 6x6=36 outcomes. Consider two events: A={Sum of dice is 4 or 8} and B={Sum of dice is even}

The sum of 4 or 8 can be achieved in eight ways with two dice, so A consists of the following elements: A={(1,3), (2,2), (3,1), (2,6), (3,5), (4,4), (5,3), (6,2)} The sum of the dice is even when both have either even or odd outcomes, so B contains the following pairs: B={(1,1), (1,3), (1,5), (3,1), (3,3), (3,5), (5,1), (5,3), (5,5), (2,2), (2,4), (2,6), (4,2), (4,4), (4,6), (6,2), (6,4), (6,6)}

Thus A consists of 8 outcomes, while B consists of 18 outcomes: (2,2), (2,4), (2,6), (4,2), (4,4), (4,6), (6,2), (6,4), (6,6)} Furthermore, A is a subset of B.

Assuming that all outcomes are equally likely, the conditional probability of A given B is: