# What is Conditional Probability?

The ** conditional probability** of an event

*B*is the probability that the event will happen given the knowledge that an event

*A*has already happened. This probability is written

*P(B|A)*, notation for the

*probability of B given A*. In the case where events

*A*and

*B*are

*independent*, the conditional probability of event

*B*given event

*A*is simply the probability of event

*B*, that is

*P(B)*.

If events *A* and *B* are not independent, then the probability of the *intersection of A and B* (the probability that both events occur) is defined by

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

From this definition, the conditional probability *P(B|A)* is easily obtained by dividing by *P(A)*:

The equation above is of course only valid if *P(A) *is not equal to 0. If the formula looks daunting, the following video will help you understand it. Thanks to Khan Academy. This one is great.

The Bayes’ Theorem gives a way of calculating P(A|B) given the knowledge that P(B|A) has already happened. Bayes’s formula is defined as follows:

Sample problem in my next post.

Good and interesting, except… You break the rule we pain so much to try to inculcate our pupils: You have popped an element in – the letter C, in the terminal formula by Bayes – which you have omited to introduce! Grr! 😉