Question

$A$ and $B$ are two independent events. The probability that both $A$ and $B$ occur is $1/6$ and the probability that at least one of them occurs is $\frac{2}{3}$. The probability of the occurrence of $A=$............ if $P\left(A\right)=2P\left(B\right)$.

• A. $\frac{2}{9}$
• B. $\frac{4}{9}$
• C. $\frac{5}{9}$
• D. $\frac{5}{18}$

Answer 1

The correct option is C $\frac{5}{9}$

Here given $P(A\cap B)=\frac{1}{6},P(A\cup B)=\frac{2}{3},P\left(A\right)=2P\left(B\right)$
Thus using $P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$
$\Rightarrow \frac{2}{3}=P\left(A\right)+\frac{1}{2}P\left(A\right)-\frac{1}{6}\Rightarrow P\left(A\right)=\frac{5}{9}$