Question

Nite Time Inn has a toll-free telephone number so that customers can call at any time to make a reservation. A typical call takes about 4 minutes to complete, and the time required follows an exponential distribution. Find the probability (by hand and using MS Excel) that a call takes: 3 minutes or less.

exceed 5 minutes

Answer 1

a.

Under exponential distribution, parameter $\lambda$ tells the probability that some process will last a given amount of time or less. In this case, a typical call takes about 4 minutes to complete, so

$1\div\lambda=4$

$\lambda=1\div4 = 0.25$

The probability that a call takes 3 minutes or less is:

$P(times\eqslantless T) = 1-e^{-\lambda*T}$

$P(times \eqslantless3) = 1 - e^{-0.25*3}=1-e^{-0.75}$

In MS Excel, EXP function is used to determine the value of e^-0.75. As per MS Excel, e^-0.75 is 0.472.

$P(times \eqslantless3) = 1 - 0.472 =0.528$

This means there is 52.8% probability that a call takes 3 minutes or less.

b.

The probability that a call exceed(takes longer than) 5 minutes is:

$P(time\eqslantless5)+P(time\eqslantgtr5)=1$

$P(time>5) = 1-P(time\eqslantless5)$

$P(time>5) = 1-(1-e^{-0.25*5} )$

$P(time>5) = 1-1+e^{-1.25}$

In MS Excel, EXP function is used to determine the value of e^-1.25. As per MS Excel, e^-1.25 is 0.287.

$P(time>5) =1-1+0.287 = 0.287$

So, it can say that there is 52.8% probability of a call exceeding 5 minutes.