Answer to Question #240124 in Statistics and Probability for ice

Question

A box contains 5 balls. Two are numbered 1, one is 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 mean, variance, and standard deviation of the numbers on the balls.

Answer 1

Solution:

Let X be the number on each ball. The probability distribution is

The mean is

 $\mu=\Sigma X \cdot P(X)=1 \cdot \frac{2}{5}+2 \cdot \frac{1}{5}+3 \cdot \frac{2}{5}=2$  

The variance is

$\begin{aligned} \sigma &=\Sigma\left[X^{2} \cdot P(X)\right]-\mu^{2} \\ &=1^{2} \cdot \frac{2}{5}+2^{2} \cdot \frac{1}{5}+3^{2} \cdot \frac{2}{5}-2^{2} \\ &= \frac{24}{5}-4 \\ &=\frac{4}{5} \end{aligned}$  

The standard deviation is

 $\sigma=\sqrt{\frac{4}{5}}=\sqrt{0.8}=0.894$