Assume that the duration in minutes of a telephone conversation follows the exponential distribution $f(x) = \frac{1}{5}e^{-\frac{x}{5}}, x \geq 0$. Find the probability that the conversation will exceed five minutes.

Assume that the duration in minutes of a telephone conversation follows the exponential distribution $f(x) = \frac{1}{5}e^{-\frac{x}{5}}, x \geq 0$. The probability that the conversation will exceed five minutes is 

(A) $\dfrac{1}{e}$ 

(B) $1 - \dfrac{1}{e}$ 

(C) $\dfrac{1}{e^2}$ 

(D) $1 - \dfrac{1}{e^2}$ 

(GATE 2007) 

Answer: (A) $\dfrac{1}{e}$ 

Explanation: 

The required probability is 

$P(5 < X <\infty)$ 

$= \int\limits_{5}^{\infty}f(x)dx$ 

$= \int\limits_{5}^{\infty}\frac{1}{5}e^{-\frac{x}{5}}dx$ 

$= \dfrac{1}{e}$