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

In summary, the problem asks for the probability that exactly four out of six hours with low demand will be classified as exceptional in a telephone call centre, given a table of conditional probabilities for different demand and service level scenarios. The solution involves treating the problem as a "4 out of 6 dice" scenario and using the corresponding probabilities for each face of the die. The final answer is 0.059535, assuming that a success is obtaining an exceptional service and a failure is obtaining any other service level.
  • #1
SavvyAA3
23
0

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




 
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  • #2
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.
 
Last edited:
  • #3
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!
 
  • #4
sorry the second line should read: ncr *(success)^r *(failure)^(n-r)
 

1. What is the difference between a normal distribution and a binomial distribution?

A normal distribution is a continuous probability distribution that is symmetrical and bell-shaped, with the mean, median, and mode all being equal. It is often used to model real-world data that is normally distributed, such as heights or test scores. On the other hand, a binomial distribution is a discrete probability distribution that is used to model events with only two possible outcomes, such as heads or tails in a coin toss. It is characterized by a fixed number of trials and a probability of success for each trial.

2. When should I use a Poisson distribution instead of a normal distribution?

A Poisson distribution is used to model the number of occurrences of a rare event within a specific time or space interval. It is typically used when the mean and variance of the data are equal, and when the data is discrete rather than continuous. In contrast, a normal distribution is used for continuous data and is not suitable for modeling rare events. If the data is skewed or has a different mean and variance, a Poisson distribution may not be appropriate.

3. What is the difference between a discrete and a continuous distribution?

A discrete distribution is used to model data that can only take on specific values, such as whole numbers. It is represented by a probability mass function, which gives the probability of each possible outcome. On the other hand, a continuous distribution is used to model data that can take on any value within a given range, such as height or weight. It is represented by a probability density function, which gives the probability of a value falling within a certain range.

4. How do I determine which distribution to use for my data?

Choosing the appropriate distribution for your data depends on several factors, including the type of data (continuous or discrete), the shape of the data (symmetrical or skewed), and the underlying process that generated the data. It is important to understand the characteristics and assumptions of different distributions and to consider the context and purpose of your analysis. In some cases, it may be necessary to test multiple distributions and choose the best fit for your data.

5. Can I use a single distribution to model all types of data?

No, different types of data require different types of distributions. For example, continuous data is best modeled by a continuous distribution such as the normal distribution, while discrete data is best modeled by a discrete distribution such as the binomial or Poisson distribution. Attempting to use a single distribution to model all types of data may lead to inaccurate results and incorrect conclusions.

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