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 aggregrate 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

 

[tacman]

Based on the given information, the probability of one of the three preferred time slots being available with at least one dentist can be calculated as follows:

First, we need to find the probability of a specific time slot being booked by one of the three dentists through the traditional call and book method. This can be calculated as 6/8 = 0.75, since on average, each dentist has 6 bookings out of 8 daily appointments.

Next, we need to find the probability of a specific time slot being available for booking through the web-based self-booking system. This can be calculated as 5/8 = 0.625, since there are 8 daily appointment slots and 3 of them are already booked by the dentists, leaving 5 available slots.

To find the probability of at least one of the three preferred time slots being available with at least one dentist, we can use the complement rule. This means that the probability of at least one time slot being available is equal to 1 - the probability of all three time slots being booked. This can be expressed as:

P(at least one slot available) = 1 - (0.625)^3 = 0.9844

Therefore, there is a 98.44% probability that at least one of the three preferred time slots will be available with at least one dentist.

To find the probability of two of the three preferred time slots being available with at least one dentist, we can use the same approach. The only difference is that we need to consider the probability of two specific time slots being booked and one being available, which can be expressed as:

P(two slots available) = (0.75)^2 * (0.625) = 0.3516

Therefore, there is a 35.16% probability that two of the three preferred time slots will be available with at least one dentist.

One way to solve similar problems is by using basic probability principles such as the complement rule and the multiplication rule. It is also important to clearly define the events and their probabilities before calculating the final probability. Additionally, using a simulation or a mathematical model can help in obtaining more accurate results.