Binomial Formula

Binomial distributions are a class of frequency distributions that resemble certain real world distributions and have the fortunate property that they can be described with a simple equation. In this web page, we look at data from around the solar system to illustrate binomial distributions. Binomial distributions also form the basis of a simple test of statistical significance called the sign test. These notes contain an example of the sign test (see also Chapter 10 in Pagano) which you should look at.

One should expect any frequency distribution to approximate the binomial distribution if it fulfills the following criteria:

1. There is a series of n trials.
2. There are two possible outcomes each trial.(I called them win/lose or success/failure).
   We called probability(success)=p.
3. The outcomes are mutually exclusive
4. The outcomes are independent.
5. The probability of the outcomes remains constant across trials.

This is illustrated using the example of 3 coin throws. The example could be one coin tossed 3 times or 3 coins tossed once, it makes no difference (as long as the 5 criteria are not breached). If 3 fair coins are thrown, and the order of heads and tails counts, there are 8 equally probable outcomes. The probability of each of the outcomes is 0.5*0.5*0.5 = 0.125.

But a more general example is not coin tossing because that is a very special case where both outcomes are equally likely.

Suppose we repeat an experiment that has probability of succes $p$ $n$ times and count the number $k$ of successes, the distribution of $k$ is called the Binomial $B(n,p)$ random variable.

Probability of exactly $k$ success:

$$P(k)==\frac{n!}{k! \times (n-k)!}p^kq^{(n-k)} ={\binom{n}{k}} p^kq^{(n-k)}$$

A very good lecture, with examples

A page from Phil Starks' glossary, with calculator

Go to appendix of chapter 5