The binomial distribution is a fundamental probability distribution in statistics that describes the number of successes in a fixed number of independent trials, each with the same probability of success. It is used when there are exactly two possible outcomes in each trial, often termed "success" and "failure." The distribution is characterized by two parameters: the number of trials nnn and the probability of success ppp in each trial. The binomial distribution calculates the probability of obtaining exactly kkk successes out of nnn trials, using the formula:
P(X=k)=(nk)pk(1−p)n−kP(X = k) = \binom{n}{k} p^k (1-p)^{n-k}P(X=k)=(kn)pk(1−p)n−k
where (nk)\binom{n}{k}(kn) is the binomial coefficient representing the number of ways to choose kkk successes from nnn trials. This distribution is widely used in fields such as finance, biology, and quality control to model scenarios like pass/fail tests, yes/no surveys, or defective/non-defective products.
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