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  • Let A and B be two events such that P(A) = 3/8, P(B) = 5/8 and P(A ∪ B) = 3/4. Then P(A | B).P(A′ | B) is equal to A.2/5 B. 3/8 C. 3/20 D. 6/25

Let A and B be two events such that P(A) = 3/8, P(B) = 5/8 and P(A ∪ B) = 3/4. Then P(A | B).P(A′ | B) is equal to
A.2/5
B. 3/8
C. 3/20
D. 6/25

 

Answers (1)

\begin{aligned} &\text { Given- }\\ &\mathrm{P}(\mathrm{A})=3 / 8, \mathrm{P}(\mathrm{B})=5 / 8 \text { and } \mathrm{P}(\mathrm{A} \cup \mathrm{B})=3 / 4\\ &\text { As we know, } \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})\\ &P(A \cap B)={\frac{3}{8}}+\frac{5}{8}-\frac{3}{4}\\ &\mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{1}{4}\\ &\text { Therefore, } \mathrm{P}(\mathrm{A} \mid \mathrm{B})=\frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{\mathrm{P}(\mathrm{B})}, \text { then }\\ &\mathrm{P}(\mathrm{A} \mid \mathrm{B})=2 / 5\\ &\text { For, } \mathrm{P}\left(\mathrm{A}^{\prime} \mid \mathrm{B}\right) \end{aligned}

\\ P\left(A^{\prime} \mid B\right)=\frac{P\left(A^{\prime} \cap B\right)}{P(B)} \\ P\left(A^{\prime} \mid B\right)=\frac{P(B)-P(A \cap B)}{P(B)} \\ P\left(A^{\prime} \mid B\right)=\frac{\frac{5}{8}-\frac{1}{4}}{\frac{5}{8}} \\ P\left(A^{\prime} \mid B\right)=3 / 5 \\ \text { Therefore, } P(A \mid B) \cdot P\left(A^{\prime} \mid B\right)=\left(\frac{3}{5} \times \frac{2}{5}\right) \\ \text { Hence, } P(A \mid B) \cdot P\left(A^{\prime} \mid B\right)=6 / 25

Option D is correct.

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