If A and B are two independent events such that $\mathrm{P}\left(\mathrm{A}^{\prime}\right)=0.75^{\circ},…

If A and B are two independent events such that $\mathrm{P}\left(\mathrm{A}^{\prime}\right)=0.75^{\circ}, \mathrm{P}(\mathrm{A} \cup \mathrm{B})=0.65$ and $\mathrm{P}(\mathrm{B})=\mathrm{p}$, then value of $p$ is
  1. $\frac{9}{14}$
  2. $\frac{7}{15}$
  3. $\frac{5}{14}$
  4. $\frac{8}{15}$

Solution

$\begin{array}{ll} & \mathrm{P}\left(\mathrm{~A}^{\prime}\right)=0.75 \Rightarrow \mathrm{P}(\mathrm{~A})=0.25 \\ & \mathrm{P}(\mathrm{~A} \cup \mathrm{~B})=\mathrm{P}(\mathrm{~A})+\mathrm{P}(\mathrm{~B})-\mathrm{P}(\mathrm{~A} \cap \mathrm{~B}) \\ \therefore \quad & 0.65=0.25+\mathrm{p}-\mathrm{P}(\mathrm{~A}) \cdot \mathrm{P}(\mathrm{~B}) \end{array}$ $\ldots[\because \mathrm{A} \& \mathrm{~B}$ are independent $]$ $\begin{array}{ll} \therefore & 0.4=p-0.25 \mathrm{p} \\ \therefore & 0.4=0.75 \mathrm{p} \\ \therefore & \mathrm{p}=\frac{0.4}{0.75}=\frac{8}{15} \end{array}$

Asked in: MHT CET 2024 (11 May Shift 1)

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