Question

A customer relations manager at observes that a teller can serve, on average, 15 clients in an hour. Calculate the probability that the teller serves: 6 clients in a 30 minute period. Interpret your answer. More than 14 but less than 17 clients in a particular hour. Interpret your answer.

Answer 1

Let $X=$ the number of clients served by a teller: $X\sim Po(\lambda t).$

a) $\lambda=15, t=0.5$

$$\lambda t=15(0.5)=7.5$$

$$P(X=6)=\dfrac{e^{-\lambda t}(\lambda t)^6}{6!}=\dfrac{e^{-7.5}(7.5)^6}{6!}$$

$$\approx0.136718$$

The probability that the teller serves 6 clients in a 30 minute period is $0.136718.$

b) $\lambda=15, t=1$

$$\lambda t=15(1)=15$$

$$P(14<X<17)=P(X=15)+P(X=16)$$

$$=\dfrac{e^{-15}(15)^{15}}{15!}+\dfrac{e^{-15}(15)^{16}}{16!}$$

$$\approx0.1024359+0.0960336\approx0.198470$$

The probability that the teller serves more than 14 but less than 17 clients in a particular hour is $0.198470.$