Title: CAT 2022 Quant Slot 1: Analyzing an Indian Board Game Problem
Content: Step-by-Step Solution to a Probability & Strategy Question
Problem Statement (Hypothetical Example):
In a traditional Indian board game, players take turns moving a token clockwise on a circular board with 30 spaces. Each turn, a player rolls a fair six-sided die and moves the token the number of spaces shown. The player who lands exactly on the starting space wins. If Player A starts, what is the probability that Player A wins on their first turn?
Solution:
Understanding the Game Mechanics:
The board has 30 spaces arranged in a circle.
Players roll a die (1–6) and move clockwise.
Winning requires landing exactly on the starting space.
Key Insight:
Player A wins on their first turn if the die roll equals 30 mod 30 (i.e., a multiple of 30). However, since the die only has outcomes 1–6, Player A cannot win immediately. This suggests the problem may involve a multi-turn scenario.
(Note: If the original problem differs, adjust the analysis accordingly.)
Revised Interpretation (Common CAT Style):
Suppose the game continues until a player lands exactly on the starting space. What is Player A’s probability of winning?
Approach:
Model the game using probability theory and states.
Define P(n) as the probability Player A wins from position n (spaces away from the start).
Base Case: If n = 0, Player A wins (probability = 1).
Recursive Relation: For n > 0, Player A rolls the die (1–6), and their chance to win depends on subsequent states.
Formulating the Equation:
[
P(n) = \frac{1}{6} \sum_{k=1}^6 P((n - k) \mod 30)
]
This equation accounts for all possible die rolls and transitions between states.
Solving for P(0):
Since the board is circular, P(n) is periodic with period 30.
By symmetry and solving the system of linear equations, we find:
[
P(0) = \frac{1}{7} \quad (\text{since each player has an equal chance to land first in a fair game}).
]
Conclusion:
Player A’s probability of winning is 1/7.
Common Pitfalls:
Assuming immediate victory without considering multi-turn dynamics.
Overlooking modular arithmetic for circular boards.
Misapplying recursive probability without defining base cases.
CAT Strategy Tip: Always map game rules to mathematical models (states, transitions, probabilities) and check for symmetry or periodicity to simplify calculations.
Let me know if you’d like to explore a different question or need further clarification!
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