Question

charlie is playing a game in which a 10 - sided die, numbered from 0 to 9, is rolled to determine the number of spaces moved. what is the expected number of spaces a player will move on each turn? a. 5 b. 5.5 c. 9

Explanation:

Step1: Recall expected - value formula

The expected value $E(X)$ of a discrete - uniform distribution with values $x_1,x_2,\cdots,x_n$ is given by $E(X)=\sum_{i = 1}^{n}x_ip_i$. For a fair 10 - sided die numbered from 0 to 9, $n = 10$, and $p_i=\frac{1}{10}$ for $i = 0,1,\cdots,9$.

Step2: Calculate the sum

$E(X)=\sum_{x = 0}^{9}x\times\frac{1}{10}=\frac{0 + 1+2+\cdots+9}{10}$.
We know that the sum of the first $n$ non - negative integers is given by $\sum_{k = 1}^{n}k=\frac{n(n + 1)}{2}$. Here, $\sum_{x = 0}^{9}x=\frac{9\times(9 + 1)}{2}=45$.

Step3: Find the expected value

$E(X)=\frac{45}{10}=4.5$.

Answer:

A. 4.5