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  • You are given that A and B are two events such that P(B)= 3|5, P(A | B) = 1|2 and P(A ∪ B) = 4|5, then P(A) equals A. 3|10 B. 1|5 C. 1/2 D. 3|5

You are given that A and B are two events such that P(B)= 3|5, P(A | B) = 1|2 and P(A ∪ B) = 4|5, then P(A) equals
A. 3|10
B. 1|5
C. 1/2
D. 3|5

Answers (1)

Given

\begin{aligned} &P(B)=\frac{3}{5}, P(A \mid B)=\frac{1}{2} \text { and } P(A \cup B)=\frac{4}{5}\\ &\text { As we know, }\\ &\mathrm{P}(\mathrm{A} \mid \mathrm{B}) \times \mathrm{P}(\mathrm{B})=\mathrm{P}(\mathrm{A} \cap \mathrm{B}) \text { [Property of conditional Probability] }\\ &\Rightarrow \frac{1}{2} \times \frac{3}{5}=P(A \cap B)\\ &\Rightarrow P(A \cap B)=\frac{3}{10}\\ &\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})[\text { Additive Law of Probability }]\\ &\therefore \frac{4}{5}=P(A)+\frac{3}{5}-\frac{3}{10}\\ &\Rightarrow P(A)=\frac{4}{5}-\frac{3}{5}+\frac{3}{10}\\ &\Rightarrow P(A)=\frac{1}{5}+\frac{3}{10}\\ &\Rightarrow P(A)=\frac{2+3}{10}\\ &\Rightarrow P(A)=\frac{5}{10}=\frac{1}{2} \end{aligned}

Hence, the correct option is C

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infoexpert22

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