Here’s a structured response to the query titled "CAT 2022 QA Slot 3" with an English explanation of an Indian game-related question, followed by a Chinese summary:
CAT 2022 QA Slot 3: Indian Game Question Explanation
Question:
In a traditional Indian game, players collect tokens by rolling a fair six-sided die. For each roll, if the number rolled is a multiple of 3, the player earns 5 tokens; if it’s a multiple of 2, they earn 3 tokens; otherwise, they lose 1 token. What is the expected number of tokens a player gains per roll?
Solution:
Identify outcomes and probabilities:
Die faces: 1, 2, 3, 4, 5, 6.
Multiples of 3: {3, 6} → 2 outcomes → Probability = ( \frac{2}{6} = \frac{1}{3} ).
Multiples of 2 (but not 3): {2, 4} → 2 outcomes → Probability = ( \frac{2}{6} = \frac{1}{3} ).
Neither: {1, 5} → 2 outcomes → Probability = ( \frac{2}{6} = \frac{1}{3} ).
Calculate expected value (EV):
[
\text{EV} = (\text{Tokens for multiple of 3} \times P) + (\text{Tokens for multiple of 2} \times P) + (\text{Tokens for neither} \times P)
]
[
\text{EV} = (5 \times \frac{1}{3}) + (3 \times \frac{1}{3}) + (-1 \times \frac{1}{3}) = \frac{5 + 3 - 1}{3} = \frac{7}{3} \approx 2.33 \text{ tokens per roll}.
]
Answer: ( \boxed{\frac{7}{3}} ).
中文总结
CAT 2022 QA Section 3: 解析印度传统游戏期望值问题
题目：某印度游戏中，玩家通过掷骰子（六面）得分。若点数为3的倍数得5分，2的倍数（非3的倍数）得3分，否则扣1分。求每轮游戏的期望得分。
解答步骤：
确定结果与概率：
3的倍数：3、6 → 概率 ( \frac{1}{3} )。
2的倍数（非3的倍数）：2、4 → 概率 ( \frac{1}{3} )。
其他数：1、5 → 概率 ( \frac{1}{3} )。
计算期望值：
[
\text{期望} = 5 \times \frac{1}{3} + 3 \times \frac{1}{3} + (-1) \times \frac{1}{3} = \frac{7}{3} \approx 2.33.
]
答案：( \boxed{\frac{7}{3}} )。
Key Takeaways
Probability & Expectation: CAT QA often tests these concepts.
Overlap Handling: Ensure mutually exclusive categories (e.g., 6 is a multiple of both 2 and 3 but excluded from the 2’s category).
Time Management: Such questions require quick classification and arithmetic.
Let me know if you need further clarification or additional examples! 😊
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