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In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: success (with probability p) or failure (with probability q = 1 − p).
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, the power (+) expands into a polynomial with terms of the form , where the exponents and are nonnegative integers satisfying + = and the coefficient of each term is a specific positive integer ...
However, as the example below shows, the binomial test is not restricted to this case. When there are more than two categories, and an exact test is required, the multinomial test, based on the multinomial distribution, must be used instead of the binomial test. [1] Most common measures of effect size for Binomial tests are Cohen's h or Cohen's g.
X is a beta-binomial random variable with parameters (n, α, β). Let p = α/(α + β) and suppose α + β is large, then X approximately has a binomial(n, p) distribution. If X is a binomial (n, p) random variable and if n is large and np is small then X approximately has a Poisson(np) distribution.
The probability density function (PDF) for the Wilson score interval, plus PDF s at interval bounds. Tail areas are equal. Since the interval is derived by solving from the normal approximation to the binomial, the Wilson score interval ( , + ) has the property of being guaranteed to obtain the same result as the equivalent z-test or chi-squared test.
Graphs of probability P of not observing independent events each of probability p after n Bernoulli trials vs np for various p.Three examples are shown: Blue curve: Throwing a 6-sided die 6 times gives a 33.5% chance that 6 (or any other given number) never turns up; it can be observed that as n increases, the probability of a 1/n-chance event never appearing after n tries rapidly converges to 0.
Within a system whose bins are filled according to the binomial distribution (such as Galton's "bean machine", shown here), given a sufficient number of trials (here the rows of pins, each of which causes a dropped "bean" to fall toward the left or right), a shape representing the probability distribution of k successes in n trials (see bottom of Fig. 7) matches approximately the Gaussian ...
There are many other combinatorial interpretations of binomial coefficients (counting problems for which the answer is given by a binomial coefficient expression), for instance the number of words formed of n bits (digits 0 or 1) whose sum is k is given by (), while the number of ways to write = + + + where every a i is a nonnegative integer is ...