a) \(P(A \cap B)=P(A)P(B/A)=\dfrac{1}{3}\dfrac{1}{4}=\dfrac{1}{12}\)
b) \(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
Por lo tanto: \(P(B)=P(A\cup B)-P(A)+P(A\cap B)=\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{12}=\dfrac{1}{4}\)
c) \(P(\bar B/A)=\dfrac{P(\bar B \cap A)}{P(A)} =\dfrac{P(A)-P(A \cap B)}{P(A)}=\dfrac{\dfrac{1}{3}-\dfrac{1}{12}}{\dfrac{1}{3}}=\dfrac{3}{4}\)
De otra forma: \(P(\bar B/A)=1-P(B/A) =1-{\dfrac{1}{4}}=\dfrac{3}{4}\)
d) Por la regla del producto: \(P(\bar A \cap \bar B)=P(\bar A / \bar B)·P(\bar B)\)
Despejando:
\(P(\bar A / \bar B)=\dfrac{P(\bar A \cap \bar B)}{P(\bar B)}=\dfrac{P(\overline{A \cup B})}{P(\bar B)}=\dfrac{1-P(A \cup B)}{1-P( B)}=\dfrac{1-\frac{1}{2}}{1-\frac{1}{4}}=\dfrac{2}{3}\)
