Binomial distribution

The binomial distribution describes the number of successes obtained when the same two-outcome trial is repeated a fixed number of times and independently. Each repetition has only two results, conventionally called success and failure, and the probability of a success stays the same every time.

The mental image is a series of coin tosses lined up: you tick each heads and, at the end, count the total number of ticks. This total is an uncertain quantity, between zero and the number of tosses, and the binomial distribution gives the probability of each value.

The calculation combines two ingredients. First, you count in how many ways k successes can be placed among the n trials, which the binomial coefficient provides. Then, each specific scenario has as its probability the product of the successes and the failures. Take four tosses of a fair coin and look for exactly two heads: there are six possible arrangements of the two heads among four positions, and each of the sixteen equally likely cases counts for 1/16. The probability is therefore 6/16, that is 3/8.

A classic trap is to forget the counting of arrangements and to compute only a single scenario. Getting "heads, heads, tails, tails" in that precise order is worth 1/16; but "two heads" without any imposed order groups six scenarios together, hence the factor of six.

The independence condition is essential: the binomial distribution assumes that each trial starts fresh, with no memory of the previous one. Drawing cards without replacement, where each removal changes the following chances, therefore does not follow a binomial distribution.

On the site, tossing the heads-or-tails coin several times or repeating a two-outcome draw directly brings out this model: it lets you predict, for example, how many times out of ten trials you can reasonably expect to see a given side.