Evaluate P ( A B ) if 2P(A) = P (B) = 5 13 and P ( A|B ) = 2 5 - Vidyadip

Question

Evaluate ${P\left( {A \cup B} \right)}$ if 2P(A) = P (B) = $\frac{5}{{13}}$ and $P\left( {A|B} \right) = \frac{2}{5}$

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Answer

Given: 2 P(A) = P (B) = $\frac{5}{{13}}$, $P\left( {A|B} \right) = \frac{2}{5}$
$\therefore $ P (A) = $\frac{5}{{26}}$
Now, $P\left( {A \cap B} \right) = P\left( {B|A} \right)$ . P (B) = $\frac{2}{5} \times \frac{5}{{13}} = \frac{2}{{13}}$
and $P\left( {A \cup B} \right)$ = P(A) + P (B) – $P\left( {A \cap B} \right)$ = $\frac{5}{{26}} + \frac{5}{{13}} - \frac{2}{{13}} = \frac{{11}}{{26}}$

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