Question

A die is rolled 45 times, and let x be the number of 3s obtained. What is the standard deviation of the probability distribution of x?

Answer 1

A die is rolled 45 times.

X is the number of 3's obtained.

We have to find the standard deviation of the probability distribution of X.

Now, we know that, on rolling a die, there could be 6 outcomes, vizually 1, 2, 3, 4, 5 and 6.

So, as these outcomes are equally, the probability of getting a 3, on any roll, is 1/6.

Now, if X is the number of 3's obtained in 45 rolls, then X follows binomial with parameters n=45 and p=1/6.

We know, for a binomial random variable with parameters n and p, the standard deviation is

Vnp(1 - p)

Here, n=45 and p=1/6. Putting these values

$=\sqrt{np(1-p)}$

1 5 45 * * 6 6

$=\sqrt{\frac{225}{36}}$

$=\frac{15}{6}$

$=2.5$

The answer is

If X is the number of 3's obtained in 45 rolls of a die, then the standard deviation of the probability distribution of X is 2.5.