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If $A$ and $B$ are events such that $P(A \cup B)=0.5, P(\bar{B})=0.8$
and $P(A / B)=0.4$, then what is $P(A \cap B)$ equal to?
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and $P(A / B)=0.4$, then what is $P(A \cap B)$ equal to?
Solution:
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Verified Answer
The correct answer is:
$0.08$
Let $P(A \cup B)=0.5, P(\bar{B})=0.8, P\left(\frac{A}{B}\right)=0.4$
$\quad P(\bar{B})=1-P(B)$
$\Rightarrow 0.8=1-P(B)$
$\Rightarrow P(B)=1-0.8=0.2$
Now, $P\left(\frac{A}{B}\right)=\frac{P(A \cap B)}{P(B)}$
$\Rightarrow P(B) \times P\left(\frac{A}{B}\right)=P(A \cap B)$
$\Rightarrow 0.4 \times 0.2=P(A \cap B)$
$\Rightarrow 0.08=P(A \cap B)$
$\quad P(\bar{B})=1-P(B)$
$\Rightarrow 0.8=1-P(B)$
$\Rightarrow P(B)=1-0.8=0.2$
Now, $P\left(\frac{A}{B}\right)=\frac{P(A \cap B)}{P(B)}$
$\Rightarrow P(B) \times P\left(\frac{A}{B}\right)=P(A \cap B)$
$\Rightarrow 0.4 \times 0.2=P(A \cap B)$
$\Rightarrow 0.08=P(A \cap B)$
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