Question

A switch board can handle only 4 telephone calls per minute. If the incoming calls per

minute follow poission distribution with parameter 3, find the probability that the switch

board is over taxed in any one minute?

Answer 1

Let $X=$ the number of telephone calls per minute: $X\sim Po(\lambda).$

$$P(X=x)=\dfrac{e^{-\lambda}\cdot \lambda^x}{x!}$$

Given $\lambda=3.$

$$P(X>4)=1-P(X=0)-P(X=1)$$

$$-P(X=2)-P(X=3)-P(X=4)$$

$$=1-\dfrac{e^{-3}\cdot 3^0}{0!}-\dfrac{e^{-3}\cdot 3^1}{1!}$$

$$-\dfrac{e^{-3}\cdot 3^2}{2!}-\dfrac{e^{-3}\cdot 3^3}{3!}-\dfrac{e^{-3}\cdot 3^4}{4!}$$

$$=1-\dfrac{e^{-3}}{8}(8+24+36+36+27)\approx0.1847$$

The probability that the switch board is over taxed in any one minute is 0.1847.