Question

If A and B are two events such that $P\left(A\right)=\frac{1}{4};P(A\cup B)=\frac{1}{3}$ and $P\left(B\right)=p$, the value of $p$ if A and B are independent is

• A. $\frac{1}{9}$
• B. $\frac{2}{9}$
• C. $\frac{4}{9}$
• D. $\frac{5}{9}$

Answer 1

Answer: (A)

$\frac{1}{9}$

If A and B are independent, then $P\left(\frac{A}{B}\right)=P\left(A\right)$
$P\left(\frac{A}{B}\right)$$=$$\frac{P(A\cap B)}{P\left(B\right)}$

So, $P(A\cap B)=P\left(A\right)P\left(B\right)=\frac{1}{4}p$

$P(A\cup B)=P\left(A\right)+P\left(B\right)-p(A\cap B)$

$\frac{1}{3}=\frac{1}{4}+p-\frac{1}{4}P$

$P=\frac{1}{9}$