Question

A box contains 5 balls. Two are numbered 3, one is numbered 4,and two are numbered 5. The balls are mixed and one is selected at random. After a ball is selected,it’s number is recorded. Then it is replaced. If the experiment is repeated many times. Find the 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:

now, the mean is

$\mu=\sum_^X\text{ . P\lparen X})$

According to the data, this Mean would be:

$\mu=\sum_^X\text{ . P\lparen X})\text{ }=\text{ 3 . }\frac{2}{5}\text{ }+4\text{ . }\frac{1}{5}\text{ }+5\text{ . }\frac{2}{5}\text{ }=4$

So, we get that the Mean is:

$\mu=4$

Now, the variance is

$\sigma\text{ }=\text{ }\sum_^\lbrack X^2\text{ . P\lparen X})\rbrack-\mu^2$

According to the data of the problem, we get that the variance is:

$\sigma=\text{ }\lbrack\text{3}^2\text{ . }\frac{2}{5}\text{ }+4^2\text{ . }\frac{1}{5}\text{ }+5^2\text{ . }\frac{2}{5}\text{ }\rbrack\frac{}{}-4^2$

this is equivalent to:

$\sigma=\frac{4}{5}$

Thus, the standard deviation would be:

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

Then, we can conclude that the correct answer is:

Variance:

$\frac{4}{5}$

Standard deviation:

$0.894$