The binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials with the same probability of success. It is used when there are exactly two mutually exclusive outcomes of a trial often labelled as "success" and "failure". The distribution is defined by two parameters: the probability of success in a single trial and the number of trials.
Binominal Distribution in practice
Let's consider a binomial experiment where n is the number of trials, p is the probability of success on a single trial, and X is the number of successes. The probability of getting exactly k successes (where k can go from 0 to n) in n trials is given by the function:
P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))
where P(X=k) is the probability of k successes in n trials, C(n, k) is the binomial coefficient, p is the probability of success on a single trial, and (1-p) is the probability of failure on a single trial.
The mean and variance of a binomial distribution are np and np(1-p) respectively.
The binomial distribution can be used to answer questions like "If a fair coin is tossed 10 times, what is the probability of getting exactly 6 heads?" or "If 20% of phones produced by a factory are defective, what is the probability that a sample of 10 phones will include exactly 2 defective phones?"
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