Binomial Distribution

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Consider $N$ independent trials of an experiment (like flipping a coin), where each trial produces one of two possible results --- we'll call them success and failure --- with some probability of success that does not change between trials. The Binomial distribution, and the related Bernoulli distribution, give the probability that some number $n$ of those $N$ trials are successes.

Let

p

be the probability of success, which implies

1 - p

is the probability of failure. The probability of any particular sequence of trials containing

n

successes, and

N - n

failures, is simply

p n (1 - p) N - n

regardless of the order in which the successes occur. The number of distinct sequences of

N

trials containing

n

successes is given by the binomial coefficient ${N\choose n}= \frac {N!}{n!(N-n)!}.$ Putting these together, we find the binomial distribution

P B i n (n; N)

for the probability of

n

successes in

N

trials $P_{Bin}(n;N) = {N\choose n} p^n(1-p)^{N-n}.$

If we define $γ$ as the ratio between the probability of success and the probability of failure, $\gamma = \frac{p }{ 1-p }$ , we can write $p = \frac \gamma { 1 + \gamma }.$ Substituting into the Binomial distribution, we find the Bernoulli distribution,