Probability - what distribution/ model should be used in this context

[SavvyAA3]

Homework Statement

Suppose we are given a table of conditional probabilities as follows (probabilities are in brackets):

Service Level
Demand: Poor/ Standard/ Exceptional
Low: Poor Demand: (0.1) Standard Demand: (0.6) Exceptional Demand:( 0.3)

High: Poor Demand: (0.5) Standard Demand: (0.4) Exceptional Demand: (0.1)


We are told the following: At a telephone call centre, the service levels during each hour (from 0000 hrs to 0100hrs, 0100hrs to 0200hrs and so on) are classified as Poor, Standard or Exceptional, and the classifications for successive hours can be regarded as independent.

The probabilities of the different classification levels being achieved depend on whether demand each hour is High or Low (which is also independently determined in different hourly periods).

Supposing demand levels are low from 1200 to 1800 hrs, find the probability that exactly four of those hours are classified as exceptional

Homework Equations

Can someone please tell me how to go about answering this? Should I use a poisson distribution? If so what is the parameter. I know the interval is of length '4 hours' but what is the mean?

Can I use conditional probability theory? if so How?

Thanks. I've spent a whilie trying to see what model to use but I can't igure it out.

The Attempt at a Solution

 

[EnumaElish]

I'd treat this as a "4 dice out of 6" kind of a problem. E.g., what is the probability that 4 dices out of 6 come up "two"?

In this example, each die has 3 faces (poor/std/exceptional), with the corresponding probabilities.

Since demand levels are low throughout, you know the set of probabilities you can assign to each "face" of each 3-sided die.

 

[SavvyAA3]

OK, so you suggest that I simply state, since we know their is Low demand we need not assign this a probability. I suppose this is a sensible assumption since the question states the two events (Demand and Service Level) are independent.

So from there I can just go about using these conditional probabilities (ignoring that they are conditional probabilities because the two events are independent):

Please tell me if my assumptions are correct:

A success in this scenario is obtaining an ‘exceptional service’

Fail is obtaining ‘not exceptional service’

So we have:

Ncr *(success)^r *failure^(1-r) [^ reps to the power of]

6c4*(.3)^4 * (0.7)^2

15 * (0.0081) * (0.49)

0.059535

I hope this is correct. Your example of the die scenario really helped me to see the logic behind this!

 

[SavvyAA3]

sorry the second line should read: ncr *(success)^r *(failure)^(n-r)