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convergent series on
A familiar example of this is the expansion
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
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
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?
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.