If the mean and variance of a binomial distribution are 18 and 12 respectively, then n = ______.

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If the mean and variance of a binomial distribution are 18 and 12 respectively, then n = 54. Explanation: Mean = np = 18 ...(i) Variance = npq = 12 ...(ii) $\therefore$ (npq)/(np) = $\frac{12}{18}$ $\therefore$ q = $\frac{2}{3}$ $\therefore$ p = 1 − q = 1 - $\frac{2}{3}$ = $\frac{1}{3}$ np = 18 ...[From eqn. (i)] $\therefore$ n($\frac{1}{3}$) = 18 $\therefore$ n = 18 $\times$ 3 $\therefore$ n = 54

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