Question

In a game of luck, a turn consists of a player rolling 121212 fair 666-sided dice. Let X=X=X, equals the number of dice that land showing "111" in a turn. Find the mean and standard deviation of XXX. You may round your answers to the nearest tenth.

Answer 1

The mean of X is 2 and the standard deviation is 1.3.

Explanation:

For each dice rolled, there are only two possible outcomes. Either it lands on one, or it does not. The probability of a dice rolled landing on one is independent of any other dice. This means that the binomial probability distribution is used to solve this question.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

The expected value of the binomial distribution is:

`E(X) = np`

The standard deviation of the binomial distribution is:

`√(V(X)) = √(np(1-p))`

Probability of rolling a 1.

Six sides, so:

`p = (1)/(6) = 0.1667`

12 dices are rolled:

This means that
`n = 12`

Find the mean and standard deviation of X

`E(X) = np = 12*0.1667 = 2`

`√(V(X)) = √(np(1-p)) = √(12*0.1667*0.8333) = 1.3`

The mean of X is 2 and the standard deviation is 1.3.