Q64 / 180 · KCET 2022 · Mathematics

If $A$ and $B$ are two independent events such that $P(\bar{A})=0.75, P(A \cup B)=0.65$ and $P(B)=x$, then find the value of $x$.

  • A
    $\frac{5}{14}$
  • Correct answer
    B
    $\frac{8}{15}$
  • C
    $\frac{9}{14}$
  • D
    $\frac{7}{15}$

Solution

Given, $P(\bar{A})=0.75, P(A \cup B)=0.65$ and $P(B)=x$ $P(A)=1-P(\bar{A})=1-0.75=0.25$ Also, $A$ and $B$ are independent. $$ \begin{aligned} & \Rightarrow P(A \cap B)=P(A) \cdot P(B)=0.25 x \\ & \text { Using } P(A \cup B)=P(A)+P(B)-P(A \cap B), \\ & \Rightarrow \quad 0.65=0.25+x-0.25 x \Rightarrow 0.40=0.75 x \\ & \Rightarrow \quad x=\frac{40}{75}=\frac{8}{15} \end{aligned} $$

Hence, the correct answer is option: 'B'.

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