Question

A box contains 5 balls. One is numbered 1, two are numbered 2, and two are numbered 3. The balls are mixed and one is selected at random. After a ball is selected, its number is recorded. Then it is replaced. If the experiment is repeated many times, find the standard deviation of the numbers on the balls.

Answer 1

$mean=E(x)=\sum xp(x)\\ E(x)=1\times\frac{1}{5}+2\times\frac{2}{5}+3\times\frac{2}{5}\\E(x)=2.2\\ variance=E(x^2)-(E(x))^2\\E(x^2)=1^2\times\frac{1}{5}+2^2\times\frac{2}{5}+3^2\times\frac{2}{5}\\ E(x^2)=5.4\\(E(x))^2=(2.2)^2=4.84\\variance=E(x^2)-(E(x))^2\\5.4-4.84=0.56\\ S.D=\sqrt{0.56}=0.7483$