Question

A business man is expecting two telephone calls. Mr Walia may call any time between $2$ p.m and $4$ p.m. while Mr Sharma is equally likely to call any time between $2.30$ p.m. and $3.15$ p.m. The probability that Mr Walia calls before Mr Sharma is:

• A. $\frac{1}{18}$
• B. $\frac{1}{9}$
• C. $\frac{1}{6}$
• D. None of these

Answer 1

The correct option is C $\frac{1}{6}$

The probability of getting a telephonic call $\frac{1}{2}$

Let $E_1$ be the event of getting a call from Mr. Walia between $2$ pm and $4$ pm.

Let $E_2$ be the event of getting a call from Mr. Sharma between $2:30$ pm and $3:15$ pm.

$\Rightarrow P\left(E_1\right)=\frac{1}{3}$ $P\left(E_2\right)=\frac{2}{3}$

$\Rightarrow P\left(\frac{E}{E_1}\right)=P\left(\frac{E}{E_2}\right)=\frac{1}{2}$

$=\frac{1}{4}$

$\Rightarrow P\left(\frac{E}{E_1}\right)=\frac{P\left(E_1\right)P\left(\frac{A}{E_1}\right)}{P\left(E_1\right)P\left(\frac{A}{E_1}\right)+P\left(E_2\right)P\left(\frac{A}{E_2}\right)}$

$=\frac{\frac{1}{4}\times \frac{1}{2}}{\frac{1}{3}\times \frac{1}{2}+\frac{2}{3}\times \frac{1}{2}}$

$=\frac{1}{6}$

Hence, the answer is $\frac{1}{6}.$