Conditional probability

Summary

Definition

• If \(A\) and \(B\) are two events then the symbol \(P(A|B)\) denotes the probability of \(A\) given that we know that \(B\) is true and it is called the conditional probability of \(A\) given \(B\) .

Example

• \(S={1,2,3,4,5,6}\) and \(A={1,2},B={1,3}\) are two events. Suppose that each outcome is equally likely .
• If \(B\) is true, then the outcome of the experiment is 1 with 50% probabilty and 3 with 50% probability. Therefore, \(P(A|B)=1/2\) .

Example

• Suppose that \(B⊂A\) . Then \(P(A|B)=1\) . If \(B⊂A\) then \(A\) must be true if \(B\) is true.

Conditional probability, formula

The conditional probability is calculated from the following formula ( \(P\left( B \right)≠0\) )

\[P(A|B)= \frac{P(A∩B)}{P(B)}\]