Expected Number of Distinct Balls Drawn with Replacement
Problem Statement:A box contains 5 balls labeled from 1 to 5. If three balls are drawn with replacement, let X be the number of distinct balls that are drawn at least once in the three draws. Find E(X). Answer: \(P(X=3)=\frac{5}{5}\times\frac{4}{5}\times\frac{3}{5}=\frac{12}{25}\) \(P(X=2)=\frac{5}{5}\times\frac{1}{5}\times\frac{4}{5}\times\text{C}^3_2=\frac{12}{25}\) \(P(X=1)=\frac{5}{5}\times\frac{1}{5}\times\frac{1}{5}=\frac{1}{25}\) \(E(X)=3 \times P(X=3)+2 \times P(X=2)+1 \times P(X=1)\) \(=3\times\frac{12}{25}+2\times\frac{12}{25}+1\times\frac{1}{25}\) \(=\frac{61}{25}\) There are approximately 2.44... » read more