If A and B are two events such that P(A|B)=p,P(A)=p,P(B)=1 3 and … - Vidyadip

MCQ

If A and B are two events such that $\text{P}(\text{A}|\text{B})=\text{p},\text{P(A)}=\text{p},\text{P(B)}=\frac{1}{3}$ and $\text{P}(\text{A}\cup\text{B})=\frac{5}{9},$ then p =

A
$\frac{2}{3}$
B
$\frac{3}{5}$
✓
$\frac{1}{3}$
D
$\frac{3}{4}$

✓

Answer

Correct option: C.

$\frac{1}{3}$

$\text{P}\Big(\frac{\text{A}}{\text{B}}\Big)=\text{p},\text{P(A)}=\text{p},\text{P(B)}=\frac{1}{3},\text{P}(\text{A}\cup\text{B})=\frac{5}{9}$

Consider,

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

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

$\Rightarrow \frac{\text{P(A)}+\text{P(B)}-\text{P}(\text{A}\cup\text{B})}{\text{P(B)}}=\text{P}$

$\Rightarrow\frac{\text{p}+\frac{1}{3}-\frac{5}{9}}{\frac{1}{3}}=\text{p}$

$\Rightarrow\text{p}+\frac{1}{3}-\frac{5}{9}=\frac{\text{p}}{3}$

$\Rightarrow\frac{-2}{9}=\frac{\text{p}}{3}-\text{p}$

$=\frac{-2}{3}\text{p}=\frac{-2}{9}$

$\Rightarrow\text{p}=\frac{1}{3}$

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