Random Variables

Solution

Answer: 0.221

A binomial random variable \(X \sim B(12,p)\) has probability mass function \[ P(X=k) = \binom{12}{k} p^k (1-p)^{12-k}. \] For \(k=0\), \[ P(X=0) = \binom{12}{0} p^0 (1-p)^{12}. \] Since \(\binom{12}{0}=1\) and \(p^0=1\), \[ P(X=0) = (1-p)^{12}. \] We are given \[ (1-p)^{12} = 0.05. \] Taking the \(12\)th root of both sides, \[ 1-p = 0.05^{1/12}. \] \[ p = 1 - 0.779. \] \[ p \approx 0.221. \]