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True or False: If A and B are two events such that P(A)>0 and P(A)+P(B)>1,then\[P(B|A)\geq 1-\frac{p(B')}{P(A)}\]

1 Answer

Toolbox:
  • If $A$ and $B$ are two events associated with an experiment the condiitional probability that $B$ occures given that $A$ has already occured given by
  • $P(B/A)=\Large\frac{p(A\cap B)}{P(A)}$
  • $P(A\cap B)=P(A)+P(B)-P(A\cup B)$
$\Large P(B/A)=\frac{P(A)+P(B)-P(A\cup B)}{P(A)}$
Also $P(A)+P(B)>1$
$\Rightarrow P(A)>1-P(B)$
$\Rightarrow P(A)>1-P(\bar{B})$
$\Rightarrow\Large\frac{P(\bar{B})}{P(A)}<1$
$P(B/A)=\Large\frac{P(A)}{P(A)}+\frac{P(B)-P(A\cup B)}{P(A)}$
$=\Large1+\frac{1-P(B')-P(A\cup B)}{P(A)}$
=$\Large 1-\frac{P(B')}{P(A)}+\frac{1-P(A\cup B)}{P(A)}$
$\Rightarrow\Large P(B/A)\leq 1-\frac{P(B')}{P(A)}$
The given statement is False.
answered Jun 13, 2013 by poojasapani_1
 

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