Question

what is the probability that both events b and c will occur? now, find the probability of event c. p(c)=\frac{?}{5}

Explanation:

Step1: Calculate the probability of \(A\) and \(C\)

Use the formula for the probability of two - step events \(P(A\cap C)=P(A)\times P(C|A)\). Here \(P(A)=\frac{1}{4}\) and \(P(C|A)=\frac{3}{5}\), so \(P(A\cap C)=\frac{1}{4}\times\frac{3}{5}=\frac{3}{20}\)

Step2: Calculate the probability of \(B\) and \(C\)

Use the formula \(P(B\cap C)=P(B)\times P(C|B)\). Here \(P(B)=\frac{3}{4}\) and \(P(C|B)=\frac{3}{5}\), so \(P(B\cap C)=\frac{3}{4}\times\frac{3}{5}=\frac{9}{20}\)

Step3: Calculate \(P(C)\)

Since \(C=(A\cap C)\cup(B\cap C)\) and \(A\) and \(B\) are mutually - exclusive (a single trial can't result in both \(A\) and \(B\)), by the addition rule for mutually - exclusive events \(P(C)=P(A\cap C)+P(B\cap C)\)

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Answer:

\(\frac{3}{5}\)