I recently experienced three consecutive days that are major anniversaries for me: June 12 is my baptismal anniversary, June 13 is my birthday, and June 14 is my wedding anniversary (also my daughter turned 15 months old on this day). These events put to the forefront of my mind a famous problem in probability, the birthday problem.
— The Uniform Probability Model for a Finite Sample Space —
There are many different probability models. We’ll start with the simplest of these models, the uniform probability model for a finite sample space.
Let be a finite sample space. The uniform probability model for is defined by
as defined satisfies the axioms for a probability measure. By adopting this model, we are predicting that in a sequence of experiments with sample space the observed relative frequency of events in the sequence of experiments will be equal to the relative frequency, , of events in the set . In particular, for each outcome , we have , or as one says, “All outcomes in are equally likely (i.e., equally probable)." Incidentally, for events consisting of a single outcome are sometimes called elementary events.
assuming all tuples are equally likely. Below are some values of :
Remarkably, in a room with just people, the probability is greater than that two or more people have the same birthday!
— Binomial Random Variables and Poisson Approximations —
In the case of a binomial experiment there are independent trials, with the results of each trial labeled “success" or “failure." On each trial the probability of success is and (hence) the probability of failure is . The sample space consists of the set of words in the alphabet and the random variable on is defined by (each word) the number of ‘s in that word. The probablity density function (pdf) for is given by
We say that such a random variable is a binomial random variable with parameters and , abbreviating this with the notation . We also say that X has a binomial distribution with parameters and , and that is a binomial density function with parameters and . Note, in general, if , then .
If , where is “large" and is “small," then for
where . This is called the Poisson approximation (Denis Poisson, 1781-1840) to the binomial distribution.
The Poisson approximation of this quantity is