Conditional Probability
Calculate probabilities when extra information is known.
Formula
\[P(A\mid B)=\frac{P(A\cap B)}{P(B)}\]
Read this as “the probability of A given B”. The event B has already happened, so the sample space is restricted to B.
Worked example
In a group, \(P(A)=0.6\), \(P(B)=0.5\), and \(P(A\cap B)=0.3\). Find \(P(A\mid B)\).
\[P(A\mid B)=\frac{P(A\cap B)}{P(B)}=\frac{0.3}{0.5}=0.6.\]
Independence connection
If A and B are independent, knowing B happened does not change the probability of A:
\[P(A\mid B)=P(A)\]