Do you know your binomial from your Poisson?
Many business situations are subject to significant uncertainty. What are the chances of one or more serious accidents in our worldwide operations in a given period? Or two or more key business process interruptions over the same time? Or three or more sales from 16 appointments? Or two heads in two flips of a fair coin?
Understanding these probabilities, and how to manage or improve them, could be life-saving. Misunderstanding, or applying the wrong model, could court disaster.
Some uncertain situations can be analysed using a simple probability model. For example, the coin flip. Practical business situations are usually much more complex, but probability models can still be useful tools to start our analysis.
BINOMIAL AND POISSON DISTRIBUTIONS
We need to understand them both. But we also need to know when to use each of them, or neither of them, and how we can change the odds in our favour.
BINOMIAL LIMITED
The ‘binomial’ part of the binomial distribution means it is built up from a number of ‘success’ or ‘fail’ trials. It is a good model for a limited series of simple, independent trials, such as coin flips. Let’s consider two flips of a fair coin and the probability of scoring two heads, a head being defined as a success. There is a limited number of independent trials, two, and a constant probability of success (p) in each trial, 0.5.
The probability of failure, a tail in this case, is 1 – p. This is also 0.5 for a fair coin flip.
The probability of two heads from two flips modelled by the binomial is:
0.5 x 0.5 = 0.25, or 25%.
Similarly, the probability of three heads in three flips would be:
0.5 x 0.5 x 0.5 = 0.125, or 12.5%.
A key feature of binomial models is that the number of binomial trials is limited, and the maximum number of successes is also limited, namely to the total number of trials.
POISSON UNLIMITED
Most business risks aren’t as simple as that. Key business process interruptions or even serious accidents can happen at any time. Examples include apparently random interruptions, such as computers going down, phone calls or other types of faults.
These situations are better modelled by the Poisson distribution, rather than the binomial.
The time period is sometimes known as a ‘continuum’. This means:
The event can occur at any time during the total time period.
The total number of times the event could occur is unlimited.
COMPARE AND CONTRAST
In the meantime, let’s summarise and contrast both the binomial and Poisson models.
|
|
Binomial |
Poisson |
|
1 |
A process consists of a limited whole number of identical trials or situations (n). |
Continuous observation is needed, rather than a finite number of trials. |
|
2 |
Each trial results in just one of only two possible outcomes, for example, success or failure. |
The variable takes a positive whole number value, with no upper limit. |
|
3 |
The probability of success (p) remains constant for each independent trial. |
The average number of occurrences is known or can be estimated. |
|
4 |
Interest is in the number of successes or failures in the n trials. |
Interest is in the number of times the event occurs in a given time period. |
LET’S LOOK AT AN EXAMPLE
A car salesperson has 16 appointments with potential buyers. However, there is only a one-in-five chance that an appointment will result in a car being bought.
The salesperson will receive a bonus if at least three cars are bought as a result of the 16 appointments.
Is the binomial or the Poisson the better model here? Applying the characteristics we have identified, the number of sales appointments (trials) is limited, namely to the 16 appointments. For this reason, the number of successful appointments can never exceed 16. So the binomial distribution is relevant, and not the Poisson.
____________________
Author: Doug Williamson
Source: The Treasurer magazine