Here’s a structured analysis and solution approach for CAT 2024 Slot 3 Quantitative Ability (QA) questions, assuming hypothetical "game-related" problems (as official 2024 questions are not publicly available). This guide focuses on common QA patterns in CAT exams, including probability, permutations/combinations, and logical reasoning.
Sample Problem 1: Game Probability
Question:
A board game involves rolling two fair six-sided dice. If the sum is a prime number, the player wins. What is the probability that a player wins in a single round?
Solution:
Total outcomes: (6 \times 6 = 36).
Prime sums: Possible primes between 2–12 are 2, 3, 5, 7, 11.
Sum = 2: (1,1) → 1 way
Sum = 3: (1,2), (2,1) → 2 ways
Sum = 5: (1,4), (2,3), (3,2), (4,1) → 4 ways
Sum = 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 ways
Sum = 11: (5,6), (6,5) → 2 ways
Favorable outcomes: (1 + 2 + 4 + 6 + 2 = 15).
Probability: (\frac{15}{36} = \frac{5}{12}).
Answer: (\boxed{\dfrac{5}{12}}).
Sample Problem 2: Game Strategy (Optimization)
Question:
In a card game, players draw 3 cards from a deck of 12 cards (values 1–12). The score is the sum of the two highest cards. What is the minimum possible score for a player?
Solution:
To minimize the sum of the two highest cards, the player should have the two lowest possible values in the two highest positions.
Worst-case scenario: The three cards drawn are the three smallest values: 1, 2, 3.
Two highest cards: 2 and 3 → Sum = 5.
Verification: Any other combination (e.g., 1,2,4) would result in a higher sum (2+4=6).
Answer: (\boxed{5}).
Sample Problem 3: Game Theory (Nash Equilibrium)
Question:
Two players, A and B, play a game where they each choose a number between 1 and 5. The player with the higher number wins. If they choose the same number, it’s a tie. What is the Nash equilibrium strategy for each player?
Solution:
Nash equilibrium occurs when neither player can benefit by unilaterally changing their strategy.
Symmetric equilibrium: Both players randomize uniformly over all numbers (1–5).
Probability distribution: Each number has a (\frac{1}{5}) chance.
Outcome: No player gains an advantage by deviating.
Answer: Both players choose numbers uniformly at random (1–5).
Key CAT QA Strategies for Game Problems
Break down complex rules into mathematical terms (e.g., probability, combinatorics).
Visualize scenarios using tables or probability trees.
Optimization problems: Use extreme cases (minimize/maximize sums/differences).
Game theory: Identify symmetry or mixed strategies for equilibrium.
Let me know if you need further clarification or additional problem types!
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