Question

consider a situation in which ( p(a)=\frac{1}{8}, p(c)=\frac{1}{4} ), and ( p(a \text{ and } b)=\frac{1}{12} ). what is ( p(b \text{ and } c) )?

Explanation:

Step1: Use the property of independent events

If \(A\) and \(B\) are independent events, then \(P(A\cap B)=P(A)\times P(B)\). Given \(P(A)=\frac{1}{8}\) and \(P(A\cap B)=\frac{1}{12}\), we can find \(P(B)\) by the formula \(P(B)=\frac{P(A\cap B)}{P(A)}\).

$$P(B)=\frac{\frac{1}{12}}{\frac{1}{8}}=\frac{1}{12}\times\frac{8}{1}=\frac{2}{3}$$

Step2: Assume \(B\) and \(C\) are independent (if no information about dependence - default for such problems without contrary info)

If \(B\) and \(C\) are independent events, then \(P(B\cap C) = P(B)\times P(C)\). Given \(P(C)=\frac{1}{4}\) and \(P(B)=\frac{2}{3}\)

$$P(B\cap C)=\frac{2}{3}\times\frac{1}{4}=\frac{2}{12}=\frac{1}{6}$$

Answer:

\(\frac{1}{6}\)