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The binomial distribution is a discrete probability distribution which describes the number of successes in a sequence of n independent experiments, each of which yielding success with probability p. Such a success/failure experiment is also called a Bernoulli experiment.
A typically example is the following: 5% of the population are HIV-positive. You pick 500 people randomly. How likely is it that you get 30 or more HIV-positives? The number of HIV-positives you pick is a random variable X which follows a binomial distribution with n = 500 and p = .05. We are interested in the probability Pr[X ≥ 30].
In general, if the random variable X follows the binomial distribution with parameters n and p, we write X ~ B(n, p). The probability of getting exactly k successes is given by
If X ~ B(n, p), then the expected value of X is
If X ~ B(n, p) and Y ~ B(m, p) are independent binomial variables, then X + Y is again a binomial variable; its distribution is B(n+m, p).
Two other important distributions arise as approximations of binomial distributions:
The formula for Bézier curves was inspired by the binomial distribution.
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