A) \[P\,\left( \frac{A}{B} \right)\ge \frac{P(A)+P(B)-1}{P(B)},\,\,P(B)\ne 0\] is always true
B) \[P\,(A\cap \bar{B})=P(A)-P(A\cap B)\] does not hold
C) \[P\,(A\cup B)=1-P(\bar{A})\,P(\bar{B}),\] if A and B are disjoint
D) None of these
Correct Answer: A
Solution :
We know that \[P(A/B)=\frac{P(A\cap B)}{P(B)}\] Also we know that \[P(A\cup B)\le 1\] \[\Rightarrow P(A)+P(B)-P(A\cap B)\le 1\] \[\Rightarrow P(A\cap B)\ge P(A)+P(B)-1\] \[\Rightarrow \frac{P(A\cap B)}{P(B)}\ge \frac{P(A)+P(B)-1}{P(B)}\] \[\Rightarrow P(A/B)\ge \frac{P(A)+P(B)-1}{P(B)}\]
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