Question

In an experiment, the probability that event B occurs is , and the probability that event A occurs given that event B occurs is 3 7) What is the probability that events A and B both occur? Simplify any fractions.

Answer 1

We have to use the conditional probability formula:

`P(A|B)=(P(A\cap B))/(P(B))`

Where P(A|B) is the probability that A occurs given that B occurs, P(B) is the probability that B occurs, and P(A∩B) is the probability that both events A and B occur.

In this case, since we are asked for the probability that events A and B both occur, we need to solve the equation for P(A∩B):

`P(A\cap B)=P(A|B)\cdot P(B)`

And the information we have about the problem is:

`\begin{gathered} P(A|B)=(3)/(7) \\ P(B)=(2)/(9) \end{gathered}`

We substitute this into the formula for P(A∩B):

`P\mleft(A\cap B\mright)=(3)/(7)\cdot(2)/(9)`

Solving the multiplication of fractions:

`\begin{gathered} P\mleft(A\cap B\mright)=(3\cdot2)/(7\cdot9) \\ P\mleft(A\cap B\mright)=(6)/(63) \end{gathered}`

And finally, we simplify the fraction by dividing both numbers in the fraction by 3:

`P\mleft(A\cap B\mright)=(2)/(21)`

Answer: 2/21