Question

Suppose that we roll a die 166 times. What is the approximate probability that the sum of the numbers obtained is between 544 and 606, inclusive.

(if possible, could you explain the steps/ state theorems used). Thank you :)

Answer 1

Let define random variables X1,X2,...,X166 represents number on 1st roll, 2nd roll,..., 166th roll.

When we roll a die the possible outcomes are

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

X1~U[1,6] (X1 follow discrete uniform distribution)

X2~U[1,6] (X2 follow discrete uniform distribution)

.

.

.

X166~U[1,6] (X166 follow discrete uniform distribution)

So, the mean and variance of Xi are given by

Let define a random variable Y that represent sum of the numbers on 166 roll of die.

Y=X1+X2+...+X166

The mean and variance of Y is given by

By central limit theorem, the approximate distribution of Y is ,

We need to find the P(544<Y<606)

The probability is 0.82572674 of sum lies between 544 and 606.