If P(B)=3 5 ,P(A|B)=1 2 and P(A )=4 5 , then P( A )+P( A )= - Vidyadip

MCQ

If $\text{P(B)}=\frac{3}{5},\text{P}(\text{A}|\text{B})=\frac{1}{2}$ and $\text{P}(\text{A}\cup\text{B})=\frac{4}{5},$ then $\text{P}(\overline{\text{A}\cap\text{B}})+\text{P}(\overline{\text{A}}\cap\text{B})=$

A
$\frac{1}{5}$
B
$\frac{4}{5}$
C
$\frac{1}{2}$
✓
$1$

✓

Answer

Correct option: D.

$1$

$\text{P(B)}=\frac{3}{5},\text{P}\Big(\frac{\text{A}}{\text{B}}\Big)=\frac{1}{2},\text{P}\Big({\text{A}}\cup{\text{B}}\Big)=\frac{4}{5}$

Consider,

$\text{P}\Big(\frac{\text{A}}{\text{B}}\Big)=\frac{1}{2}$

$\Rightarrow\ \frac{\text{P}(\text{A}\cap\text{B})}{\text{P(B)}}=\frac{1}{2}$

$\Rightarrow\ \frac{\text{P}(\text{A}\cap\text{B})}{\frac{3}{5}}=\frac{1}{2}$

$\Rightarrow\ \text{P}(\text{A}\cap\text{B})=\frac{3}{10}$

$\text{P}(\overline{\text{A}\cup\text{B}})+\text{P}(\overline{\text{A}}\cap\text{B})$

$=1-\text{P}(\text{A}\cap\text{B})+\text{P(B)}-\text{P}(\text{A}\cap\text{B})$

$=1-\frac{3}{10}+\frac{3}{5}-\frac{3}{10}$

$=1$

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