Conditional Probability

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If A and B are two events, the probability that A will occur given B has already occurred is written as P(B|A), which reads probability B given A. It is called the conditional probability of B given A.

If A and B are independent, the occurrence of B will not affect the probability of A, and vice versa, in which case:

P(B|A) = P(B) and P(A|B) = P(A)

The probability of both A and B occurring, denoted P(A B) is given by:

but if A and B are independent;

This can be extended to several events, e.g.:

and if A, B, and C are independent:

Example

A box contains 5 yellow balls and 2 green balls. What is the probability that three balls randomly taken from the box (without replacement) will all be yellow?

A = first ball is yellow

B = second ball is yellow

C = third ball is yellow

i.e. 5 yellow balls in a box of 7

i.e. 4 yellow balls left in a box of 6

i.e. 3 yellow balls left in a box of 5

Thus:

If the balls were replaced after each draw, then each draws results would be independent of the others, and we would have P(A) = P(B) = P(C ) = 5/7, and