Question

Probability of solving a particular question by person $A$ is $\frac{1}{3}$ and probability of solving that question by person $B$ is $\frac{2}{5}$. What is the probability of solving that question by atleast one of them?

• A. $\frac{3}{5}$
• B. $\frac{7}{9}$
• C. $\frac{2}{5}$
• D. $\frac{2}{3}$

Answer 1

Answer: (A)

$\frac{3}{5}$

Given, $P\left(A\right)=\frac{1}{3}$ and $P\left(B\right)=\frac{2}{5}$

$\therefore P\left(\overline{A}\right)=1-P\left(A\right)=1-\frac{1}{3}=\frac{2}{3}$
and $P\left(\overline{B}\right)=1-P\left(B\right)=1-\frac{2}{5}=\frac{3}{5}$

Now, $P\left(atleastoneofthem\right)$

$=P(A\cap \overline{B})+P(\overline{A}\cap B)+P(A\cap B)$
$=P\left(A\right)\cdot P\left(\overline{B}\right)+P\left(\overline{A}\right)\cdot P\left(B\right)+P\left(A\right)\cdot P\left(B\right)$
[$\because A$ and $B$ are independent events]

$=\frac{1}{3}\cdot \frac{3}{5}+\frac{2}{3}\cdot \frac{2}{5}+\frac{1}{3}\cdot \frac{2}{5}$

$=\frac{3}{15}+\frac{4}{15}+\frac{2}{15}=\frac{9}{15}=\frac{3}{5}$.