Here’s a bet. I flip a coin a total number of times and you win £1000 if it comes up heads times, otherwise you lose £1000. What’s the probability of you winning for different values of and ?

The binomial distribution for different pairs of values of *n* and *k*. The graph plots the probability that *k* out of *n* trials are a success against the value of *k*.

The answer is given by the *binomial distribution*. Phrased in more general terms, the distribution gives the probability of achieving successes (eg getting heads) in a total number of trials of a process (eg flipping a coin).

To work out the distribution we need to know the probability of a success. In our coin example, this would be for a fair coin, but could be another value for a crooked coin. To capture the general case let’s just write for that probability.

The probability of observing a particular sequence of successes and failures in a particular order is equal to the product of the individual probabilities. For example, the probability of observing the sequence Success, Success, Failure is

But we don't mind what order our successes and failures come in, we just care about the total number of successes. That means we need to add up the probabilities of each sequence that results in the right number of successes. Going back to the example of three trials and two successes, we could have

- Success, Success, Failure
- Success, Failure, Success
- Failure, Success, Success.

Each of these has a probability so the probability of one of the three possible outcomes occurring is

Writing this more generally,

Here the variable can take values , , , up to , and the first term is the *binomial coefficient*

The exclamation mark denotes the *factorial* which is defined as

The binomial coefficient can be thought of as the number of ways of choosing an unordered list of outcomes from possibilities. You can find out more here.

The binomial distribution has*mean*and

*variance*. To see a worked example of the binomial distribution in action, read this article.