Question

The amount of time (in minutes) that a party hat to wait to be seated in a restaurant has an exponential distribution with a mean 15. Find the probability that it will take between 10 and 20 minutes to be seated for a table. If a party has already waited 10 minutes for a table, what is the probability it will be at least another 5 minutes before they are seated? If the restaurant decides to give a free drink to the top 10% of customers who have to wait the longest, at least how long will a party have to wait in order to get a free drink?

Answer 1

The distribution given here is:

as the mean is the reciprocal of its parameter.

a) The probability that it will take between 10 and 20 minutes to be seated for a table is computed here as :

Therefore 0.2498 is the required probability here.

If a party has already waited 10 minutes for a table, the probability it will be at least another 5 minutes before they are seated is computed as the probability that the waiting time is more than 5 minutes ( as exponential distribution follows a memoryless property )

Therefore 0.7165 is the required probability here.

Let the waiting time for top 10% of customers who have to wait the longest be K. Then, we have here:

Taking natural log both sides, we get here:

Therefore 34.54 minutes is the required waiting time here.