Question

customers arrive at a suburban ticket outlet at a rate of 14 per hour on monday mornings. this can be described by a poisson distribution. selling the tickets and providing general information takes an average of 3 minutes per customer, and varies exponentially. determine the probability that a customer will have to wait more than 15 minutes in the queue. select one: a. 0.006 b. 0.756 c. 0.156 d. 0.656

Answer 1

The probability that a customer has to wait more than 15 minutes is calculated using the exponential distribution formula with a service rate of 20 customers per hour. The result is approximately 0.0067, closest to option A - 0.006.

Step-by-step explanation:

To calculate the probability that a customer has to wait more than 15 minutes, we need to understand the arrival rate and service rate. Given that customers arrive at the rate of 14 per hour, we can find the average time between arrivals:

Calculation of Time Between Arrivals

Since customers arrive at a rate of 14 per hour, on average, one customer arrives every 4.29 minutes (60 / 14).

Service Rate

It is mentioned that selling tickets and providing information takes an average of 3 minutes per customer, so the service rate is 20 customers per hour (60 / 3).

Exponential Distribution

The question also states that the service times vary exponentially. Because we know the service rate is 20 per hour, the probability that a service time exceeds 15 minutes is found using the exponential distribution formula:

`Pr(T > t) = e^(-λt)`, where λ is the service rate and T is the service time.

For a service rate of 20 per hour and T = 15 minutes (or 0.25 hours), this becomes:

`Pr(T > 0.25) = e^(-(20)(0.25)) = e^(-5)` approximately equals to 0.0067, which is not given exactly in the options, but closest to option A - 0.006.

Answer Selection

The correct answer to the probability that a customer has to wait more than 15 minutes is option A, which is approximately 0.006.

Answer 2

To determine the probability that a customer will have to wait more than 15 minutes in the queue, we need to calculate the arrival rate and the service rate.

Given:
- Arrival rate: 14 customers per hour (λ = 14)
- Service rate: 3 minutes per customer (μ = 1/3)

The arrival rate (λ) is the same as the mean of a Poisson distribution. The service rate (μ) is the reciprocal of the mean of an exponential distribution.

To find the probability that a customer will have to wait more than 15 minutes, we need to find the probability that the service time exceeds 15 minutes.

The service time follows an exponential distribution with a mean of 1/μ = 3 minutes.

The probability that the service time exceeds 15 minutes can be calculated using the exponential distribution formula:

P(X > t) = e^(-μt)

where X is the service time and t is the time threshold.

In this case, t = 15 minutes = 15/60 hours = 0.25 hours.

P(X > 0.25) = e^(-1/3 * 0.25) = e^(-0.083) ≈ 0.92

Therefore, the probability that a customer will have to wait more than 15 minutes in the queue is approximately 0.92.

Since none of the given options match the calculated probability, it seems there might be an error in the options provided.