Probability related to an Appointment Scheduling Simulation

[wolfego]

I am trying to simulate the performance of a web booking/scheduling system and although there are many features of this problem that intrigue me the one currently giving me fits is demonstrated below.

Assume that a dentist has 8 appointments per day and that each appointment time is equally desireable from the perspective of the patient.

Assume that 3 dentists working together as a group each have appointments via the traditional call and book method averaging 6 of the 8 daily appointment slots. In other words, some days they have more or less than 6 bookings but in the aggregate they each have 6 bookings per day.

Then using a web based (last-minute/day-before) self-booking system where a prospective client picks 3 preferred time slots what is the probability that one of those 3 time slots would be available with at least 1 dentist? With 2 dentists? How can one solve this and similar problems?

Thanks,
Bernie

 

[balakrishnan_v]

Since the distribution is not given,a fair assumption is to take exactly 6 bookings that day
Let us say he books 1,2,3 as his time slot(of 1,2,3,4,5,6,7,8)
We first compute the probability that all the 3 time-slots are filled with all the dentists
Pr(of the 6 appointments,3 appointments fall in 1,2,3 for 1 dentist)=5C3/8C3=P(slots are filled for dentist 1)=5/28
So the probability that the slots are filled for all the dentists=(5/28)^3=
So the probability that atleast 1 slot is empty = 1-(5/28)^3
0.9943

For it to be available with 2 dentists=1-(5/28)^3-3C1*(5/28)^2*(23/28)
=0.9157