Question

In a benchmark study, a fast food restaurant had an average service time of 2.1 minutes. Assume that the service time for the fast food restaurant has an exponential distribution. (Round your answers to four decimal places.) (a) What is the probability that a service time is less than or equal to one minute? (b) What is the probability that a service time is between 30 seconds and one minute? (c) Suppose the manager of the restaurant is considering instituting a policy such that if the time it takes to serve you exceeds four minutes, your food is free. What is the probability that you will get your food for free? If the manager is okay with 1% chance of customers receiving free food, does the policy seem like a good idea? The policy seems like a good idea as we would expect less than 1% of customers to receive free food. The policy does not seem like a good idea as we would expect more than 1% of customers to receive free food.

Answer 1

Here the service time for the fast food restaurant has an exponential distribution.

so if t is the service time then

f(t) =   e ; t > 0

where = 2.1 mins

f(t) = (1/2.1) e^-t/2.1 ; t > 0

F(t) = 1 - e^-t/2.1 ; t > 0

(a) Pr(t < 1 mins) = F(1) = 1 - e^-1/2.1 = 0.3789

(b) Pr(30 seconds < t < 1 mins) = Pr(0.5 mins < t < 1 mins) = F(1) - F(0.5) = [1 - e^-1/2.1] - [1 - e^-0.5/2.1]

= 0.3789 - 0.2119 = 0.1670

(c) Here first we will find the probability that service time is greater than 4 minutes.

Pr(t > 4 minutes) = 1 - Pr(t < 4 minutes) = 1 - F(4) = 1 - [1 - e^-4/2.1] = 0.1489

so here probability that 14.89% times i will get my food for free.

The policy does not seem like a good idea as we would expect more than 1% of customers to receive free food.