The Poisson Variation of the Binomial

My recent post about Why the Other Line Moves Faster reminded me of another variation of the binomial distribution that is not covered in AP Statistics, the Poisson Distributions. (Ok, the Poisson is lots more than just a special case of the binomial, more about that later, but in the days before hand-held calculators it was in that sense that we met it.)

The distribution is named for the French Mathematician Simeon Denis Poisson (hey, his name is on the Eiffel Tower.. 2nd from the right on the South East side).




Suppose you had a very rare binomial event, let's say something that happened only one in 200 trials (p=.005) and we wanted to know how probable it was to happen four times (x=4) in three-hundred trials (n=300). The calculation of the binomial probability is now as easy to do with modern calculators as with the Poisson approximation to it, but in my youth a calculation of looked nearly impossible.

The Poisson frequently uses the Greek letter lambda, , for the mean or expected value np. In this case NP=300(.005) = 3/2 indicates that, on average, we would expect only 1.5 successes in 300 trials. We want to calculate the probability of getting four successes. The Poisson probability is given by . The difference between the two calculations is less than .0002 on my Ti-84 calculator.

The Poisson is not limited to binomial events. More often it is applied to events which are distributed randomly across time. The same calculation above could be used to calculate the probability of four people entering a bank in a 15 minute period when only and average of 1.5 visitors would be expected. If a manufacturing process produces an average of 1.5 failures per day and you wanted to calculate the probability of four failures, you use the same calculation again.

And if you wanted to know how many cashiers to keep open at the market, how many mutations in a string of DNA exposed to radiation, the number of deaths from a rare side effect of a drug, or other similar situations which involve a very rare event with a very large number of possible incidences, the Poisson will often suffice as long as conditions of independence are met.

The Poisson distribution is sometimes called the Law of Small Numbers after a book by the same name by Ladislaus Bortkiewicz published in 1898 on the Poisson distribution.

For AP Statistics Students, it is often possible to approximate the Poisson with the Normal Distribution as long as the value of lambda is about ten or more. For the Poisson, the mean and variance are both the same, so if you approximate it with a normal, use a mean of lambda, and a standard deviation that is the square root of lambda. Do remember to do the continuity correction for the fact that the Poisson is discrete and the Normal is continuous.

More "Almost Binomial" Distributions

In my recent post I illustrated the extension of the binomial to a Multinomial Distribution. In a similar way, the geometric distribution and the Pascal (aka the Negative Binomial) Distribution are very much like special cases of the binomial.

I will illustrate each with a simple probability example. In a limited version of the game of "greedy pig" you roll a die as many times as you wish each turn and you add the points on the top of the die to your score for that turn...but... if you roll a one, your turn ends and you lose all the points you have earned for that round. One might inquire, what is the probability that you could roll the die n times without rolling a one. Since the probability on each roll is the same, we could handle this using the binomial (or multinomial) distribution with n trials, p=5/6, and the number of successes also equal to n. For n=5 for example, we get (using the notation established in that blog) which simplifies to just (5/6)⁵ .

But a slightly different question might be, what is the probability that our first failure (rolling a one) would occur on the sixth roll. This is asking for the probability that the first five rolls succeed, and then the final roll is a failure. This is the general model for a geometric distribution. The reason it is called a geometric distribution is clear if you calculate the probability of the first failure happening on the first, second, etc rolls.

Roll...1....2.......3........4...

Prob...1/6..5/36...25/216 ..5^3/6^4..

notice that each probability is the previous probability multiplied by a constant ratio of 5/6. The terms for a geometric sequence (which must sum to one to be a probability distribution......check)

In general, if the probability of failure is q = 1-p.. then the probability of the first failure occurring on the n^th trial is given by (p)^n-1(q)

It often surprises students that the mean for such a distribution is 1/q where q is the probability of a failure. OK before I confuse someone.. the geometric distribution is sometimes described as the number of trials to the first success, so you may see the expected or mean value as 1/p. In any event, if the probability of an event happening (whether you call it success or failure) is p, the expected number of trials before it happens is 1/p.

Now if you are really clever you can figure out how to do the next problem without me, but let's walk through it anyway, (hey...it's MY blog).
Suppose instead, you could keep rolling until you had three rolls of one..... sort of "three strikes and you're out." Now what is the probability that the third strike comes on the tenth roll.
The idea of course, a collection of 9 rolls with 2 failures anywhere in the string, and then a third failure on the tenth roll. To get the probability of all the possible ways to get 7 successes and 2 failures in the first nine rolls is a straight binomial (multinomial) probability problem.

We just multiply this by a failure on the tenth roll and we have the probability we seek. Since we have a couple of "failures" in that (1/6)^2, we might as well just up it to a three and be done. The final probability is
If you would like to experiment with these distributions, I came across a nice experimental applet here

This experiment uses the trials to k successes instead of failures, and so p and q are switched here (and it seems I could only adjust these in .05 increments). This is a nice routine and you can simulate trials by clicking on the "step" button to see how many trials it took to get three successes.
This applet is part of a nice virtual laboratory created by Kyle Siegrist of the Department of Mathematical Sciences at the University of Alabama in Huntsville. There is lots of nice stuff. See the home page here.

When we deal with integer numbers of failures this is called a Pascal Distribution after Blaise Pascal. It can be extended to any real and is then called a Polya Distribution, after George Polya. This has application for events which are very rare, but related to each other, such as hurricanes. Both are special cases of the general Negative Binomial Distribution.