Choose the correct answer from the given four options. You are gi… - Vidyadip

Question

Choose the correct answer from the given four options.
You are given that A and B are two events such that $\text{P}(\text{B})=\frac{3}{5},\text{P}\Big(\frac{\text{A}}{\text{B}}\Big)=\frac{1}{2}$ and $\text{P}(\text{A}\cup\text{B})=\frac{4}{5},$ then P(A) equals:

$\frac{3}{10}$
$\frac{1}{5}$
$\frac{1}{2}$
$\frac{3}{5}$

✓

Answer

$\frac{1}{2}$

Solution:
We have, $\text{P}(\text{B})=\frac{3}{5},\text{P}\Big(\frac{\text{A}}{\text{B}}\Big)=\frac{1}{2}$ and $\text{P}(\text{A}\cup\text{B})=\frac{4}{5}$
$\therefore\text{P}(\text{A}\cap\text{B})=\Big(\frac{\text{A}}{\text{B}}\Big)\cdot\text{P}(\text{B})$
$=\frac{1}{2}\cdot\frac{3}{5}=\frac{3}{10}$
Now $\text{P}(\text{A}\cup\text{B})=\text{P}({\text{A}})+\text{P}({\text{B}})\cdot\text{P}(\text{A}\cap\text{B})$
$\Rightarrow\frac{4}{5}=\text{P}(\text{A})+\frac{3}{5}-\frac{3}{10}$
$\therefore\text{P}(\text{A})=\frac{4}{5}-\frac{3}{5}+\frac{3}{10}=\frac{1}{2}$

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