Formula of the binomial distribution

Theorem — Formula of the binomial distribution

Under the hypotheses of the Bernoulli scheme, the probability of $k$ successes in $n$ trials is: $P_n(k) = \binom{n}{k}\, p^k\, (1-p)^{n-k}, \qquad k=0,1,\ldots,n.$

Proof

A specific sequence of $n$ trials with $k$ successes and $n-k$ failures (for example $SSI\ldots SIS$) has probability $p^k(1-p)^{n-k}$: the outcomes are independent, so the probabilities multiply. The distinct sequences that have exactly $k$ successes and $n-k$ failures number $\binom{n}{k}$: choosing the $k$ positions where the success sits is equivalent to choosing a subset of $k$ elements out of $n$. Since these outcomes are mutually exclusive, their probabilities add up; being all equal, the total is $\binom{n}{k}\,p^k(1-p)^{n-k}$. ∎

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Topics: Probability
Concepts: Simple combinations · Binomial distribution · Bernoulli scheme
Methods: Binomial
Skills: Probability calculation · Proving · Using formulae