Let $\mathrm{A}, \mathrm{B}$ and C be three events, which are pairwise independent and…

Let $\mathrm{A}, \mathrm{B}$ and C be three events, which are pairwise independent and $\overline{\mathrm{E}}$ denote the complement of an event E . If $\mathrm{P}(\mathrm{A} \cap \mathrm{B} \cap \mathrm{C})=0$ and $\mathrm{P}(\mathrm{C})\gt0$, then $\mathrm{P}((\overline{\mathrm{A}} \cap \overline{\mathrm{B}}) / C)$ is equal to
  1. $\quad \mathrm{P}(\mathrm{A})+\mathrm{P}(\overline{\mathrm{B}})$
  2. $\quad \mathrm{P}(\overline{\mathrm{A}})-\mathrm{P}(\overline{\mathrm{B}})$
  3. $\quad \mathrm{P}(\overline{\mathrm{A}})-\mathrm{P}(\mathrm{B})$
  4. $\quad \mathrm{P}(\overline{\mathrm{A}})+\mathrm{P}(\overline{\mathrm{B}})$

Solution

Given that $\mathrm{A}, \mathrm{B}$ and C are pairwise independent. $\begin{aligned} \therefore \quad & P(A \cap B \cap C)=0 \\ & \Rightarrow P(A) \cdot P(B) \cdot P(C)=0 \\ & \Rightarrow P(A) \cdot P(B)=0...(i) \\ & P((\bar{A} \cap \bar{B}) / C)=\frac{P(\bar{A} \cap \bar{B} \cap C)}{P(C)} \\ & =\frac{P(\bar{A}) \cdot P(\bar{B}) \cdot P(\bar{C})}{P(\bar{C})} \\ & =[1-P(A)] \cdot[1-P(B)] \\ & =1-P(A)-P(B)+P(A) P(B) \\ & =P(\bar{A})-P(B)...[From(i)] \end{aligned}$

Asked in: MHT CET 2024 (11 May Shift 2)

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