Question

A die is rolled 10 times and the number of twos that come up is tallied. If this experiment is repeated find the standard deviation for number of two?

Answer 1

The standard deviation is `sqrt(10 * 1/6 * 5/6)= (5sqrt(2))/6`.

Explanation:

This experiment involves repeating identical independent trials (the rolling of the die), with the same condition for "success" each time (rolling a "2"). So, it will have a binomial distribution—this means the probability of rolling `k` 2's will be

`((n), (k))p^k(1-p)^(n-k)," "k=0,1,2,...,n`

Where:

• `n` is the number of trials in the experiment (`10`), and
• `p` is the probability of success in each trial (`1/6`).

Binomial distributions have a mean of `np`, and a standard deviation of `sqrt(np(1-p))`. Thus, the standard deviation `sigma` of this distribution is

`sigma = sqrt(np(1-p))=sqrt(10*1/6*(1-1/6))`
`color(white)(sigma = sqrt(np(1-p)))=sqrt(10*1/6*5/6)`
`color(white)(sigma = sqrt(np(1-p)))=sqrt(50/36)`
`color(white)(sigma = sqrt(np(1-p)))=sqrt((5^2*2)/6^2)`
`color(white)(sigma = sqrt(np(1-p)))=(5sqrt(2))/6`.