Given that the events $A$ and $B$ are such that $P(A)=\frac{1}{2}, P(A \cup B)=\frac{3}{5}$ and $P(B)=p .$ Find $p$ if they are independent.

Vedclass pdf generator app on play store
Vedclass iOS app on app store

When $A$ and $B$ are independent, $P(A \cap B)=P(A) P(B)=\frac{1}{2} p$

It is known that, $P(A \cup B)=P(A)+P(B)-P(A \cap B)$ $\Rightarrow \frac{3}{5}=\frac{1}{2}+p-\frac{1}{2} p$

$\Rightarrow \frac{3}{5}=\frac{1}{2}+\frac{p}{2}$

$\Rightarrow \frac{p}{2}=\frac{3}{5}-\frac{1}{2}=\frac{1}{10}$

$\Rightarrow p=\frac{2}{10}=\frac{1}{5}$

Similar Questions

Given two independent events $A$ and $B$ such $P(A)$ $=0.3,\, P(B)=0.6 .$ Find $P(A$  or $B)$

If $A$ and $B$ are any two events, then $P(\bar A \cap B) = $

An event has odds in favour $4 : 5$, then the probability that event occurs, is

The probability that a leap year selected at random contains either $53$ Sundays or $53 $ Mondays, is

If $P(A) = 0.25,\,\,P(B) = 0.50$ and $P(A \cap B) = 0.14,$ then $P(A \cap \bar B)$ is equal to