The probability that event B occurs is 1/4
[tex]P(B)=\frac{1}{4}[/tex]The probability that event A occurs given that event B occurs is 1/3
[tex]P(A|B)=\frac{1}{3}[/tex]What is the probability that events A and B both occur?
[tex]P(A\: and\: B)=\text{?}[/tex]Recall that the conditional probability is given by
[tex]P(A|B)=\frac{P(A\: and\: B)}{P(B)}_{}[/tex]Re-writing the above formula for P(A and B)
[tex]P(A\: and\: B)=P(A|B)\cdot P(B)[/tex]So, the probability that events A and B both occur is
[tex]\begin{gathered} P(A\: and\: B)=P(A|B)\cdot P(B) \\ P(A\: and\: B)=\frac{1}{3}\cdot\frac{1}{4} \\ P(A\: and\: B)=\frac{1}{12} \end{gathered}[/tex]Therefore, the probability that events A and B both occur is 1/12
