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STD 12 - 13. probability — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 13. probability1 Mark
MCQ
Let $A, B$ and $C$  be three events, which are pair-wise independence and $\bar E$  denotes the complement of an event $E$ . If $P(A \cap B \cap C) = 0$  and  $P(C) > 0,$ then $P[(\bar A \cap \bar B)|\,C]$ is equal to
  • A
    $P(A)\, + \,P(\bar B)$
  • B
    $P(\bar A)\, - P(\bar B)$
  • ✓
    $P(\bar A)\, - P(B)$
  • D
    $P(\bar A)\, + P(\bar B)$
✓

Answer

Correct option: C.
$P(\bar A)\, - P(B)$
c
${\rm{ Here, P}}(\bar A \cap \bar B|{\rm{C}}) = \frac{{P(\bar A \cap \bar B \cap C)}}{{P\left( C \right)}}.$

$ = \frac{{P(C) - P(A \cap C - P(B \cap C) + P(A \cap B \cap C))}}{{P(C)}}$

$=1-\frac{P(A) \cdot P(C)+P(B) \cdot P(C)}{P(C)} $

$(\because P(A \cap B \cap C)=0)$

$=1-\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{B})$

$=\mathrm{P}(\bar{A})-\mathrm{P}(\mathrm{B})$

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