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.)

What is the probability that a service time is less than or equal to one minute?

What is the probability that a service time is between 45 seconds and one minute?

Suppose the manager of the restaurant is considering instituting a policy such that if the time it takes to serve you exceeds six 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?

Answer 1

From the given data,

Let X be the random variable indicating the service time of a fast food restaurant.

We are given that   with (the average service time)

The probability distribution function of X is given by,

Q.What is the probability that a service time is less than or equal to one minute?

For this question, we have to calculate

   (0 to 1)

A.Hence, the probability that a service time is less than or equal to one minute is 0.3788.

ALITER:

We know that the distribution function of X is

Q.What is the probability that a service time is between 45 seconds and one minute?

Since service time is in 45 seconds, we have to convert it to minutes, so that we make it 45/60.

Here, we use

  

A.Hence, the probability that a service time is between 45 seconds and one minute is 0.0785.

Note: We can also do the problem using integrals.

Q.Suppose the manager of the restaurant is considering instituting a policy such that if the time it takes to serve you exceeds six minutes, your food is free. What is the probability that you will get your food for free?

The probability that you will get your food for free is the probability that the serving time is greater than six minutes, i.e.,

A.Hence, the probability that you will get your food for free is 0.0574.

Q. If the manager is okay with 1% chance of customers receiving free food, does the policy seem like a good idea?

The actual probability for the customers receiving free food is 0.0574.

The probability that the manager is okay with is 0.01.

Since 0.01< 0.0574,

We conclude that the policy seems like a good idea.