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  • Fill in the blanks in the following question: If A and B are two events such that P(A|B)=p

Fill in the blanks in the following question:

If A and B are two events such that

\mathrm{P}(\mathrm{A} \mid \mathrm{B})=\mathrm{p}, \mathrm{P}(\mathrm{A})=\mathrm{p}, \mathrm{P}(\mathrm{B})=\frac{1}{3}  and P(A \cup B)=\frac{5}{9} , then p =

Answers (1)

\begin{aligned} &p=1 / 3\\ &\text { We know }\\ &P(A \mid B)=\frac{P(A \cap B)}{P(B)}\\ &=\frac{\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cup \mathrm{B})}{\mathrm{P}(\mathrm{B})}\ \end{aligned}

\\ p=\frac{p+\frac{1}{3}-\frac{5}{9}}{\frac{1}{3}}\\ p=\frac{9 p+3-5}{3}\\ 3 p=9 p-2\\ \Rightarrow 6 p=2\\ \therefore p=1 / 3

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