Question

two students created a game of dice. they determined that each roll would add the value of the die if it was an odd number. if the roll was even, it would subtract the value of the die. (for example, a roll of 2 would subtract two points, but a roll of 3 would add three points.) what is the expected value for this game?
○ -3
○ -0.5
○ 0
○ 0.5

Explanation:

Step1: List outcomes & probabilities

A die has 6 faces, each with probability $\frac{1}{6}$.
Odd rolls (1,3,5): +value; Even rolls (2,4,6): -value.
Outcomes: $+1, -2, +3, -4, +5, -6$

Step2: Calculate expected value

Multiply each outcome by its probability, sum all terms.

$$\begin{align*} E(X) &= \frac{1}{6}(1) + \frac{1}{6}(-2) + \frac{1}{6}(3) + \frac{1}{6}(-4) + \frac{1}{6}(5) + \frac{1}{6}(-6)\\ &= \frac{1 - 2 + 3 - 4 + 5 - 6}{6}\\ &= \frac{(1+3+5) - (2+4+6)}{6}\\ &= \frac{9 - 12}{6}\\ &= \frac{-3}{6} \end{align*}$$

Step3: Simplify the fraction

Reduce the resulting fraction to lowest terms.
$\frac{-3}{6} = -0.5$

Answer:

B. -0.5