1. 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.

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

3. 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:

4. 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}

5. 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)}

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.

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