Respuesta :
We have to use the conditional probability formula:
[tex]P(A|B)=\frac{P(A\cap B)}{P(B)}[/tex]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):
[tex]P(A\cap B)=P(A|B)\cdot P(B)[/tex]And the information we have about the problem is:
[tex]\begin{gathered} P(A|B)=\frac{3}{7} \\ P(B)=\frac{2}{9} \end{gathered}[/tex]We substitute this into the formula for P(A∩B):
[tex]P\mleft(A\cap B\mright)=\frac{3}{7}\cdot\frac{2}{9}[/tex]Solving the multiplication of fractions:
[tex]\begin{gathered} P\mleft(A\cap B\mright)=\frac{3\cdot2}{7\cdot9} \\ P\mleft(A\cap B\mright)=\frac{6}{63} \end{gathered}[/tex]And finally, we simplify the fraction by dividing both numbers in the fraction by 3:
[tex]P\mleft(A\cap B\mright)=\frac{2}{21}[/tex]Answer: 2/21
