Conditional Probability

Theorem: If A and B are two dependent events then the probability of occurrence of A given that B has already occurred and is denoted by P(A/B) is given by

Similarly, the probability of occurrence of B given that A has already occurred is given by

Proof: Let S be the sample space.
Then, we have

Interchange A and B in equation (i), we get

Example: Find the probability of drawing a heart on each of two consecutive draws from well shuffled-packs of cards if the card is not replaced after the draw.

Solution: Let event A is a heart on the first draw, and event B is a heart on the second draw.

When we get a heart on the first draw, the second draw has 51 outcomes and 12 are favorable.