The Game

  • Two players, A and B, take turns flipping a fair coin.
  • Player A flips first, then B, and so on.
  • The game ends as soon as the pattern HT (a Head immediately followed by a Tail) appears.
  • The player who flipped the Tail in the HT wins.

Question

What is the probability that Player A wins the game?

Strategy and State Analysis

Let’s define some game states:

  • Start (S): No flips yet.
  • H_A: Last flip was H by Player A.
  • H_B: Last flip was H by Player B.
  • A_Wins, B_Wins: terminal winning states.

From the start:

  • A flips:
    • H with probability \( \frac{1}{2} \) → state H_A
    • T with probability \( \frac{1}{2} \) → back to S (no pattern HT)

From H_A:

  • B flips:
    • T → B wins (completes HT)
    • H → state H_B

From H_B:

  • A flips:
    • T → A wins (completes HT)
    • H → back to H_A

Recursive Probabilities

Let:

  • \( P_A \): probability that A wins starting from the initial state

Using the recursive transitions, we derive:

\[ P_A = \frac{1}{2}P_{H_A} + \frac{1}{2}P_A \]

Solve for \( P_{H_A} \) via:

\[ P_{H_A} = \frac{1}{2} \cdot 0 + \frac{1}{2} \cdot P_{H_B} = \frac{1}{2} P_{H_B} \]

\[ P_{H_B} = \frac{1}{2} \cdot 1 + \frac{1}{2} \cdot P_{H_A} \]

Substitute backwards and solve the system to get:

\[ P_A = \frac{4}{9} \]

Final Answer

The probability that Player A wins is:

\[ \boxed{\frac{4}{9}} \]

Reference