To determine the expected value per roll that Chris will receive in the game, we need to analyze each possible outcome on the six-sided die and the corresponding reward based on whether the rolled number is even or odd.
A six-sided die has the faces: {1, 2, 3, 4, 5, 6}. Each face has an equal probability of 1/6 of being rolled.
Let's break it down face-by-face:
1. Face 1: It's an odd number.
- Reward: [tex]$1 1 = $[/tex]1
2. Face 2: It's an even number.
- Reward: [tex]$2 2 = $[/tex]4
3. Face 3: It's an odd number.
- Reward: [tex]$1 3 = $[/tex]3
4. Face 4: It's an even number.
- Reward: [tex]$2 4 = $[/tex]8
5. Face 5: It's an odd number.
- Reward: [tex]$1 5 = $[/tex]5
6. Face 6: It's an even number.
- Reward: [tex]$2 6 = $[/tex]12
Next, calculate the expected value by summing the rewards, each multiplied by the probability of rolling that face (1/6):
[tex]\[
\text{Expected Value} = \left(\frac{1}{6} \times 1\right) + \left(\frac{1}{6} \times 4\right) + \left(\frac{1}{6} \times 3\right) + \left(\frac{1}{6} \times 8\right) + \left(\frac{1}{6} \times 5\right) + \left(\frac{1}{6} \times 12\right)
\][/tex]
[tex]\[
\text{Expected Value} = \frac{1}{6}(1 + 4 + 3 + 8 + 5 + 12)
\][/tex]
[tex]\[
\text{Expected Value} = \frac{1}{6}(33)
\][/tex]
[tex]\[
\text{Expected Value} = 5.5
\][/tex]
Therefore, the expected value per roll that Chris will get is [tex]$5.50. Thus, the correct answer is:
C. $[/tex]5.50
