Question

1 In an experiment, the probability that event B occurs is and the probability that event A occurs given 1 that event B occurs is - 3 What is the probability that events A and B both occur? Simplify any fractions.

Answer 1

The probability that event B occurs is 1/4

`P(B)=(1)/(4)`

The probability that event A occurs given that event B occurs is 1/3

`P(A|B)=(1)/(3)`

What is the probability that events A and B both occur?

`P(A\: and\: B)=\text{?}`

Recall that the conditional probability is given by

`P(A|B)=(P(A\: and\: B))/(P(B))_{}`

Re-writing the above formula for P(A and B)

`P(A\: and\: B)=P(A|B)\cdot P(B)`

So, the probability that events A and B both occur is

`\begin{gathered} P(A\: and\: B)=P(A|B)\cdot P(B) \\ P(A\: and\: B)=(1)/(3)\cdot(1)/(4) \\ P(A\: and\: B)=(1)/(12) \end{gathered}`

Therefore, the probability that events A and B both occur is 1/12