Question

Suppose that you roll 117 fair six-sided dice. Find the probability that the sum of the dice is less than 400. (Round your answers to four decimal places.)

Answer 1

The probability that the sum of the dice is less than 400 is 0.3050.

Explanation:

Let X = outcomes of a single roll of a dice.

The possible outcomes of X are

S = {1, 2, 3, 4, 5, 6}

The probability of the random variable X is:

`P(X)=p=(1)/(6)=0.167`

Compute the mean and variance of the random variable X as follows:

`E(X)=\sum x.P(X=x)=3.5\\V(X)=E(X^(2))-(E(X))^(2)=2.917`

The dice was rolled n = 117 times.

The sum of the values of X in these 117 rolls follows a Normal distribution with mean 3.5 and variance 2.917.

Compute the probability that the sum of the dice is less than 400 as follows:

`P(\sum X<400)=P((\sum X)/(n)<(400)/(117))\\=P(\bar X<3.42)\\=P((\bar X-\mu)/(\sigma/√(n))<(3.42-3.5)/(√(2.917/117)))\\=P(Z<-0.51)\\=1-P(Z<0.51)\\=1-0.695\\=0.305`

*Use the z-table for the probability.

Thus, the probability that the sum of the dice is less than 400 is 0.3050.