Given
P(A)=0.35
P(A')=0.65
P(B|A)=0.5
P(B|A')=0.7
Then, the probability of event A, given that event B has occurred is given by:
[tex]P(A|B)=\frac{P(A)\cdot P(B|A)}{P(A)\cdot P(B|A)+P(A^{\prime})\cdot P(B|A^{\prime})}[/tex]
Substitute the given probabilities:
[tex]P(A|B)=\frac{(0.35)\cdot(0.5)}{(0.35)\cdot(0.5)+(0.65)\cdot(0.7)}[/tex]
Multiplying:
[tex]\begin{gathered} P(A|B)=\frac{0.175}{0.175+0.455} \\ P(A|B)=\frac{0.175}{0.63}=0.278 \end{gathered}[/tex]
Answer: 0.278
