Binomial theorem

The binomial theorem is an important statement in mathematics, of value in probability theory and calculus. It states that the expansion of a binomial, x+y, raised to the power n is given by:

$$(x+y)^n = \sum_{k=0}^n {n \choose k}x^{n-k}y^k$$

A familiar example of this is the expansion

$$(x+y)^2 = x^2 + 2xy + y^2$$

Importance to probability
The binomial theorem allows calculation of the probability of an event with probability p happening k times given n chances. In this instance, the formula is

$$P(E) = {n \choose k}p^k(1-p)^{n-k}$$

This can be very handy in demonstrating how improbable things happen. For instance, assume the probability of a particular mutation is about 10-3. "Impossible," you might say, "such a mutation could never occur!" However, another key aspect is the number of opportunities for that mutation to develop. Assume that, in a particular timeframe, 500 (a remarkably small number) offspring are produced. In this situation, the chance of the mutation occurring at least once is given by

$$1 - (1-10^{-3})^{500} = 1-0.999^{500} = 0.3936$$

Despite the low chance, the probability of the mutation arising with only 500 new members in the population is almost 2 in 5, or 40%.

What if we say that the number of new members added to the population is 1000, but require the mutation to occur at least twice?

$$\begin{align} P(E) &= \sum_{k=2}^{1000} {1000 \choose k} (10^{-3})^k (1-10^{-3})^{1000-k} \\ &= 0.2642 \end{align}$$

As you can see, the chance of the mutation occurring is still greater than 1 in 4, despite the chance of any individual member receiving it being 1 in 1000. Given more time or a larger population, the mutation will almost definitely occur, even if its chance is decreased.

In practice, when calculating the probability of a given number of occurrences of a rare event in a large population, the is often used to get an approximate result, rather than using the binomial theorem directly.