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Conditional probability is the measure of the likelihood of an event B occurring, given that event A has occurred previously. The probability of B given A is written as P(B|A). It can be calculated by dividing the probability of the intersection of A and B by the probability of A.
P(B|A)=P(AandB)/P(A),where P(A)≠0
It is worth noting that usually P(B|A) and P(A|B) are not equal, meaning that conditional probability is not reversible. For example, let A be the event of a prime number and B be the event of an odd number. The probability that a prime number is odd is almost 1, but the reverse, an odd number being prime, is much smaller. P(B|A)≠ P(A|B)
Assuming that event A has occurred, the sample space is reduced to A.
This means that the probability that event B can happen is reduced to the outcomes in the intersection of A and B, that is, to those outcomes in A⋂ B.
The possible outcomes are given by P(A) and the favorable outcomes by P(A⋂ B). Therefore, the conditional probability formula can be obtained using the probability formula.
P(B|A)=P(AandB)/P(A)