Question

In: Statistics and Probability

Students arrive at a local bar at a mean rate of 30 students per hour. Assume...

Students arrive at a local bar at a mean rate of 30 students per hour. Assume that the bouncer waits X (minutes) to card the next student. That is, X is the time between two students arriving at the bar. Then we know that X has approximately an exponential distribution.

(a) What is the probability that nobody shows up within the 2 minutes after the previous customer?

(b) What is the probability that the next student arrives in the third minute, knowing that nobody has shown up in the 2 minutes since the previous student?

Solutions

Expert Solution

ANSWER::

X is the random variable which gives us the time between two students arriving at a bar in minutes.

It follows exponential distribution that is

f( X= k) =

where is the mean of no. of students coming to the bar in a minute.

Thus, = 30/60 = 0.5

(X has a mean rate of 30 students per hour so 0.5 mean rate for a minute).

a) In the first part, we want to find the Probability such that the next student arrives after 2 minutes that is P ( X > 2) which is given by

b) Here, we want to find the probability that the next student arrives in third minute that is X < 3 given that X >2.

i.e. P( X >2 | X >3) = P ( 2 < X < 3)

which is given by

NOTE:: I HOPE YOUR HAPPY WITH MY ANSWER....***PLEASE SUPPORT ME WITH YOUR RATING...

***PLEASE GIVE ME "LIKE"...ITS VERY IMPORTANT FOR ME NOW....PLEASE SUPPORT ME ....THANK YOU


Related Solutions

Students arrive at a local bar at a mean rate of 30 students per hour. Assume...
Students arrive at a local bar at a mean rate of 30 students per hour. Assume that the bouncer waits X (minutes) to card the next student. That is, X is the time between two students arriving at the bar. Then we know that X has approximately an exponential distribution. What is the probability that nobody shows up within the 2 minutes after the previous customer? What is the probability that the next student arrives in the third minute, knowing...
Customers arrive at a local ATM at an average rate of 14 per hour. Assume the...
Customers arrive at a local ATM at an average rate of 14 per hour. Assume the time between arrivals follows the exponential probability distribution. Determine the probability that the next customer will arrive in the following time frames. ​a) What is the probability that the next customer will arrive within the next 2 ​minutes? ​b) What is the probability that the next customer will arrive in more than 15 ​minutes? ​c) What is the probability that the next customer will...
Cars arrive at a parking lot at a rate of 20 per hour. Assume that a...
Cars arrive at a parking lot at a rate of 20 per hour. Assume that a Poisson process model is appropriate. Answer the following questions. No derivations are needed but justification of your answers are necessary. What assumptions are necessary to model the arrival of cars as a Poisson process? What is the expected number of cars that arrive between 10:00 a.m and 11:45 a. m? Suppose you walk into the parking lot at 10:15 a.m.; how long, on average,...
Suppose that customers arrive at a bank at a rate of 10 per hour. Assume that...
Suppose that customers arrive at a bank at a rate of 10 per hour. Assume that the number of customer arrivals X follows a Poisson distribution. A. Find the probability of more than 25 people arriving within the next two hours using the Poisson mass function. B. Find the probability of more than 25 people arriving within the next two hours using the normal approximation to the Poisson. C. Compute the percent relative difference between the exact probability computed in...
People arrive at a party according to a Poisson process of rate 30 per hour and...
People arrive at a party according to a Poisson process of rate 30 per hour and remain for an independent exponential time of mean 2 hours. Let X(t) be the number of people at the party at time t (in hours) after it started. Compute E[X(t)] and determine how long it takes to have on average more than 40 people at the party.
Customers arrive at the rate of 100 per hour. The ticket seller averages 30 seconds per...
Customers arrive at the rate of 100 per hour. The ticket seller averages 30 seconds per customer. What is the average customer time in the system?
Phone calls arrive at the rate of 48 per hour at the reservation desk for Regional...
Phone calls arrive at the rate of 48 per hour at the reservation desk for Regional Airways. a. Compute the probability of receiving one call in a 5-minute interval of time.   (to 4 decimals) b. Compute the probability of receiving exactly 13 calls in 15 minutes.   (to 4 decimals) c. Suppose no calls are currently on hold. If the agent takes 10 minutes to complete the current call, how many callers do you expect to be waiting by that time?...
Phone calls arrive at the rate of 48 per hour at the reservation desk for Regional...
Phone calls arrive at the rate of 48 per hour at the reservation desk for Regional Airways. a. Compute the probability of receiving four calls in a 5-minute interval of time. b. Compute the probability of receiving exactly 9 calls in 15 minutes. c. Suppose, no calls are currently on hold. If the agent takes 5 minutes to complete the current call, how many callers do you expect to be waiting by that time? d. Suppose, no calls are currently...
The population mean wage rate for workers at General Motors is $17.26 per hour. Assume the...
The population mean wage rate for workers at General Motors is $17.26 per hour. Assume the population standard deviation is $2.00. Suppose we have a random sample of 50 workers from the population of workers at General Motors. In the sample of 50 workers, the mean wage was $15 per hour. 1) Describe the sampling distribution of the sample mean. (Include the 3 parts: center, dispersion, and shape). 2) What is the probability that in a new sample of 50...
An average of 10 cars per hour arrive at a single-server drive-in teller. Assume that the...
An average of 10 cars per hour arrive at a single-server drive-in teller. Assume that the average service time for each customer is 4 minutes, and both interarrival times and service times are exponential. What is the arrival rate per minute? What is the servicing rate per minute? What is the servicing rate per hour? What is the traffic intensity? What is the probability that the teller is idle? What is the average number of cars waiting in line for...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT