Title: CAT 2022 Quant Slot 1 Solution (Indian Game Context)
Problem Statement (Hypothetical Example):
In a traditional Indian game, players pick tiles from a set of 30 tiles (each labeled 1–30). If a player picks a tile numbered ≥ 15, they win a prize. If they pick a tile numbered 1–10, they lose immediately. For tiles 11–14, they get another chance to pick again. What is the probability of winning a prize?
Solution:
Total Tiles: 30
Winning Tiles (≥15): 16 tiles (15–30 inclusive).
Losing Tiles (1–10): 10 tiles.
Neutral Tiles (11–14): 4 tiles → 2 more picks.
Step 1: Probability of Winning on First Pick
[ P(\text{Win}) = \frac{16}{30} = \frac{8}{15} ]
Step 2: Probability of Neutral Tiles (11–14)
[ P(\text{Neutral}) = \frac{4}{30} = \frac{2}{15} ]
Step 3: Probability of Winning After Neutral Pick
If a neutral tile is picked, the player gets another chance. The probability of winning in the second pick is again (\frac{16}{30}).
[ P(\text{Neutral} \rightarrow \text{Win}) = \frac{2}{15} \times \frac{16}{30} = \frac{32}{450} = \frac{16}{225} ]
Step 4: Total Winning Probability
[ P(\text{Total Win}) = P(\text{First Win}) + P(\text{Neutral} \rightarrow \text{Win}) ]
[ = \frac{8}{15} + \frac{16}{225} = \frac{120}{225} + \frac{16}{225} = \frac{136}{225} \approx 0.6044 ]
Answer: (\boxed{\dfrac{136}{225}})
Key Takeaways:
Use probability trees for multi-stage events.
Neutral tiles create recursive opportunities.
Simplify fractions by finding common denominators.
Let me know if you need further clarification or a different problem setup!
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