Solving Coffee Shop Customers Problem: Probability Mass Function

[cosmonova]

I've been working on this problem for quite a long time now but I'm not being able to solve it. It goes as follows :

A coffee shop opens his doors from 9.00 to 1.pm.
supposedly 8 customers will enter this shop and stay for a period of half an hour.
Let X be the number of customers present in this shop at a certain time t, what is the probability mass function of X ?

I would be very much grateful if you could please help me in solving this problem.
Thank you.

 

[HallsofIvy]

? All you say is that 8 customers will enter this shop and stay for a period of half an hour. Without more information, we can't say what the probability function is- it might well be that the 8 customers are guarenteed to come at 9 and leave and 9:30!

I could see two plausible probability functions here: the simpler (in concept, probably not in calculation) is that the time each customer enters the shop is uniform over the 3 1/2 hour period from 9 to 12:30.

However, it is more common for problems like this to obey a Poisson distribution. I suspect what you intend here is a Poisson distribution for the number of customers entering, with an average of 8 customers, each staying for 1/2 hour.

 

[cosmonova]

Thank you for your reply.
The problem here is that the number of customers that are present at the same time (which is X) could take the following values : 0,2,3,4,5,6,7 and 8.
Hence they could as you said be all present at the same time (hence, X=8), as well as they could enter each for half an hour and leave , without anyone meeting the other (Hence X=0).
The problem actually is to find the probability mass fuction of X for n customers over a period t of time, with each choosing a interval of time of length l , where sum of all intervals = t. But I just gave a simple example here.
Thank you again for your reply and hope I have made the problem clearer now.