The binomial distribution models the number of successes in a sample of size n drawn with replacement from a population of size N. If the sampling is carried out without replacement, as is usually the case in research programmes, the draws are not independent. However, where the population N is much (at least 10 times) larger than n, the binomial distribution is a good approximation.

If the random variable X follows the binomial distribution with parameters n ∈ N and p ∈ [0,1], we write X ~ B(n, p). The probability of getting exactly k successes in n trials is given by the probability mass function (Exhibit 32.8):

$$ P(X=k) = \binom{n}{k}p^k(1-p)^{n-k} $$where

$$\binom{n}{k}=\frac{n!}{k!(n-k)!}$$The mean and variance of this distribution is:

$$mean,\, E(X)= \bar{p} = µ_X = np $$ $$variance,\, Var(X)=σ_X^2= np(1-p)$$*Note: In Excel you may use the BINOM.DIST (k, n, p, cumulative) function to compute P(k). *

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