Binomial theorem

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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?

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

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 Poisson distribution is often used to get an approximate result, rather than using the binomial theorem directly.

References

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