Title: CAT 2024 Quant Slot 1: Solving "Lucky Draw" Probability Game
Problem Statement (Hypothetical Example):
In the "Lucky Draw" game, players randomly pick 3 numbers from 1 to 10 (with replacement). A prize is awarded if at least two numbers are the same. What is the probability of winning?
Solution:
Total Possible Outcomes:
Since numbers are picked with replacement, each selection has 10 choices. Total outcomes = (10 \times 10 \times 10 = 1000).
Unfavorable Outcomes (No Repeats):
Calculate the probability of all three numbers being distinct:
[
\frac{10}{10} \times \frac{9}{10} \times \frac{8}{10} = \frac{720}{1000} = 0.72
]
Winning Probability (Complementary Event):
Subtract the unfavorable probability from 1:
[
P(\text{Win}) = 1 - 0.72 = 0.28 \quad \text{(or 28%)}
]
Key Takeaways for CAT 2024 Quant Slot 1:
Probability Shortcuts: Use complementary probability to simplify calculations (e.g., "at least two same" → (1 - P(\text{all distinct}))).
With/Without Replacement: Always clarify if selections are independent (with replacement) or dependent (without replacement).
Combinatorics: For distinct selections, use permutations ((P(n, k))) or combinations ((C(n, k))) appropriately.
Common Pitfalls:
Misapplying combinations instead of permutations (or vice versa).
Forgetting to account for replacement in probability trees.
Final Answer:
The probability of winning the "Lucky Draw" game is 28%.
This structured approach ensures clarity and aligns with CAT 2024’s emphasis on efficient problem-solving in Quantitative Slot 1.
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