Question

a local restaurant has 28 tables that can each seat a party of up to four people. the restaurant does not take reservations. parties arrive at a rate of 16 per hour and on average a party uses a table for 90 minutes (for simplicity assume that all parties have up to four people). if no tables are available, customers wait in the bar. what is the probability that an arriving party waits for less than 15 minutes before getting seated? (you can assume there is enough space in bar for customers to wait, and customers are willing to wait as long as it takes)

Answer 1

To find the probability that an arriving party waits for less than 15 minutes before getting seated, we can calculate the cumulative distribution function (CDF) of the exponential distribution at 15 minutes.

Step-by-step explanation:

To find the probability that an arriving party waits for less than 15 minutes before getting seated, we need to calculate the waiting time until the next available table.

On average, one customer arrives every 2 minutes, so the waiting time can be modeled using the exponential distribution. The average waiting time is 90 minutes per party, so the parameter for the exponential distribution is 1/90.

To find the probability of waiting less than 15 minutes, we calculate the cumulative distribution function (CDF) of the exponential distribution at 15 minutes.

Using the formula for the CDF, P(X <= x) = 1 - e^(-λx), where λ is the rate parameter and x is the waiting time, we can calculate the probability.

Plugging in the values, we get P(X <= 15) = 1 - e^(-15/90)

≈ 0.155.