Question

For all parts of this problem, P(Ā)=0.75 , P(B)=0.3 , and P(B∣A)=0.15

Answer 1

12. I assume $\bar A$ denotes the complement of A, so

$P(A) = 1 - P(\bar A) = 0.25$

Then by definition of conditional probability,

$P(B\mid A) = \dfrac{P(A\cap B)}{P(A)} = \dfrac{P(A\cap B)}{0.25} = 0.15 \\\\ \implies \boxed{P(A\cap B) = 0.0375}$

13. By inclusion/exclusion,

$P(A\cup B) = P(A) + P(B) - P(A\cap B) = 0.25 + 0.3 - 0.0375 \\\\ \implies \boxed{P(A\cup B) = 0.5125}$