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JEE Advance2007
Paper-2
Question
Let $\mathrm{E}^{\mathrm{c}}$ denote the complement of an event E . Let $\mathrm{E}, \mathrm{F}, \mathrm{G}$ be pairwise independent events with $\mathrm{P}(\mathrm{G})>0$ and $\mathrm{P}(\mathrm{E} \cap \mathrm{F} \cap \mathrm{G}) =0$. Then $\mathrm{P}\left(\mathrm{E}^{\mathrm{C}} \cap \mathrm{F}^{\mathrm{C}} \mid \mathrm{G}\right)$ equals
Select the correct option:
A
$\mathrm{P}\left(\mathrm{E}^{\mathrm{C}}\right)+\mathrm{P}\left(\mathrm{F}^{\mathrm{C}}\right)$
B
$P\left(E^C\right)-P\left(F^C\right)$
C
$P\left(E^C\right)-P(F)$
D
$\mathrm{P}(\mathrm{E})-\mathrm{P}\left(\mathrm{F}^{\mathrm{C}}\right)$
✓ Correct! Well done.
✗ Incorrect. Try again or view the solution.
Solution
$P\left(\frac{E^c \cap F^c}{G}\right)=\frac{P\left(E^c \cap F^c \cap G\right)}{P(G)}=\frac{P(G)-P(E \cap G)-P(G \cap F)}{P(G)}$$\begin{aligned} & =\frac{P(G)(1-P(E)-P(F))}{P(G)} \quad[\because P(G) \neq 0] \\ & =1-P(E)-P(F) \\ & =P\left(E^c\right)-P(F)\end{aligned}$
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Mathematics
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