Learning new information can change which outcomes are still possible. Conditional probability describes the probability of an event after another event is known to have occurred.

The probability of $A$ given $B$ is written

$$P(A\mid B).$$

When $P(B)>0$,

$$P(A\mid B)=\frac{P(A\cap B)}{P(B)}.$$

The event $B$ becomes the new reference set, and only the part of $A$ lying inside $B$ remains relevant.

Example

Suppose a card is drawn from a standard $52$-card deck. Let $A$ be the event “the card is an ace” and $B$ the event “the card is a spade”.

Once we know the card is a spade, there are only $13$ possible cards left, one of which is the ace of spades. Therefore

$$P(A\mid B)=\frac1{13}.$$

Multiplication rule

Rearranging the definition gives

$$P(A\cap B)=P(A\mid B)P(B).$$

Equally,

$$P(A\cap B)=P(B\mid A)P(A).$$

This rule is useful when a process unfolds in stages.

Probability trees and tables

A probability tree records successive conditional probabilities along branches. Multiplying probabilities along one path gives the probability of that complete sequence of events.

Contingency tables represent the same information through counts or proportions. In both cases, conditioning means restricting attention to the row, column or branch corresponding to the known event.