Chapter 15.9, Problem 1.5E

To determine

To find: The average of all scores after rolling a die many times will approximately.

Answer to Problem 1.5E

The average score of the score getting after $n$ rolls is approximately is $3.5$ .

Explanation of Solution

Given:

The given information is as given:

The die rolls many times with faces numbered $1\text{to}6$ .

The possible outcome of a die rolls are given as:

  $\left\{1,2,3,4,5,6\right\}$

The probability of getting any face of a die is $\frac{1}{6}$ .

The number you get when averaging all the outcomes of the $n$ rolls is therefore roughly equal to :

  $\begin{array}{c}\text{Average}=\frac{\left(\frac{n}{6}\cdot 1+\frac{n}{6}\cdot 2+\frac{n}{6}\cdot 3+\frac{n}{6}\cdot 4+\frac{n}{6}\cdot 5+\frac{n}{6}\cdot 6\right)}{n}\\ =\frac{n\cdot \left(\frac{1}{6}\cdot 1+\frac{1}{6}\cdot 2+\frac{1}{6}\cdot 3+\frac{1}{6}\cdot 4+\frac{1}{6}\cdot 5+\frac{1}{6}\cdot 6\right)}{n}\\ =3.5\left[\text{as expected score of rolling a die is}3.5\right]\end{array}$

Therefore the average score of the score getting after $n$ rolls is approximately is $3.5$ .