Stone Swap Showdown

Solution & Explanation

To solve the problem of determining the probability that Player B emerges victorious in 1, 2, ..., n rounds, we need to analyze the outcomes of each round and the conditions under which Player B wins or ties.

Initial Setup

• Player A starts with 8 stones.
• Player B starts with 6 stones.

Game Dynamics

1. Round 1:
   • Player A rolls a die and takes stones from Player B equal to the die roll.
   • Player B rolls a die and takes stones from Player A equal to the die roll.
   • The player with more stones wins the round.

Conditions for Winning

• Player B Wins Round 1:

  • After both players roll, Player B wins if:

    $8 + x_a - x_b < 6 + x_b - x_a$

    Simplifying, we get:

    $x_b > x_a + 1$

  • Possible outcomes for $(x_b, x_a)$ are:

    • (3,1), (4,1), (5,1), (6,1)
    • (4,2), (5,2), (6,2)
    • (5,3), (6,3)
    • (6,4)

  • Total favorable outcomes: 10 out of 36 possible combinations.

  • Probability of Player B winning the first round:

    $P(B \text{ wins in round 1}) = \frac{10}{36}$

• Game Ties in Round 1:

  • A tie occurs if:

    $x_b = x_a + 1$

  • Possible outcomes for $(x_b, x_a)$ are:

    • (2,1), (3,2), (4,3), (5,4), (6,5)

  • Total tie outcomes: 5 out of 36.

  • Probability of a tie in the first round:

    $P(\text{tie in round 1}) = \frac{5}{36}$

Subsequent Rounds

• Round 2 and Beyond:

  • If the game ties in round 1, both players have 7 stones each.

  • For Player B to win in subsequent rounds:

    $x_a < x_b$

  • Possible outcomes for $(x_b, x_a)$: 15 out of 36.

  • Probability of Player B winning in a subsequent round:

    $P(B \text{ wins in round } n) = \frac{15}{36}$

  • Probability of a tie in subsequent rounds:

    $P(\text{tie in round } n) = \frac{6}{36}$

General Probability for B Winning in n Rounds

• Probability Formula:

  • For Player B to win in n rounds (where n > 1):

    $P(B \text{ wins in } n \text{ rounds}) = \frac{5}{36} \times \left(\frac{6}{36}\right)^{n-2} \times \frac{15}{36}$

• For n = 1:

  • Simply the probability of winning in the first round:

    $P(B \text{ wins in 1 round}) = \frac{10}{36}$

This approach provides a detailed breakdown of how the probabilities are computed for each round and generalizes the solution for any number of rounds, ensuring a comprehensive understanding of the game's dynamics and outcomes.