The expected value of a game involving rolling two dice and taking the positive dif...
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Problem
The expected value of a game involving rolling two dice and taking the positive difference of the outcomes. Yazmin states the average points per turn will be about 2.14. Hannah states that over 1,000 turns, the total points will be about 214. Determine whose statement is correct based on the expected value.
Answer
C
Step-by-step solution
- Understand the Game: The game involves rolling a 7-sided die and a 5-sided die. The points awarded in a turn are the positive difference between the numbers rolled. Let $X$ be the random variable representing the points in a given turn.
- Analyze Yazmin's Statement: Yazmin states that the average number of points per turn will be about $2.14$. This directly refers to the expected value of $X$, denoted as $E(X)$. The problem statement itself provides that $E(X) \approx 2.14$ points. Therefore, Yazmin's statement is correct.
- Analyze Hannah's Statement: Hannah states that over $1,000$ turns, the total points can be expected to be about $214$. The expected total points over $N$ turns is $N \times E(X)$. In this case, $N = 1000$ and $E(X) \approx 2.14$. So, the expected total points are $1000 \times 2.14 = 2140$. Hannah's statement claims the total will be about $214$, which is significantly different from $2140$. Therefore, Hannah's statement is incorrect.
- Conclusion: Based on the analysis, only Yazmin's statement is correct.