Question

Let \(A\) and \(B\) two events such that \(P(A \cap B)=\frac{1}{6}, P(A \cup B)=\frac{31}{45}\) and \(P(\bar{B})=\frac{7}{10}\) then
(A) \(A\) and \(B\) are independent
(B) \(A\) and \(B\) are mutually exclusive
(C) \(\mathrm{P}\left(\frac{\mathrm{A}}{\mathrm{B}}\right)<\frac{1}{6}\)
(D) \(P\left(\frac{B}{A}\right)<\frac{1}{6}\)

Answer 1

Ans : (A)
Hint : \(P(\bar{B})=\frac{7}{10} \Rightarrow P(B)=\frac{3}{10}\) \(P(A \cup B)=P(A)+P(B)-P(A \cap B) \Rightarrow P(A)=\frac{5}{9}\)
\(\therefore \mathrm{P}(\mathrm{A}) \times \mathrm{P}(\mathrm{B})=\frac{5}{9} \times \frac{3}{10}=\frac{1}{6}=\mathrm{P}(\mathrm{A} \cap \mathrm{B})\) \(\Rightarrow \mathrm{A}, \mathrm{B}\) are independent

Answer 2

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Answer 3

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