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Data Interview Question

Coin Flip Outcomes

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Solution & Explanation

To solve the problem of determining the probability that Player A wins when the sequence {HHT} appears before {HTT}, we need to analyze the possible outcomes of consecutive coin flips.

Understanding the Problem

  • Sequences:
    • HHT: Player A wins.
    • HTT: Player B wins.
  • Both sequences start with an H, so the first flip must be a head (H) for any player to win.

Probability Analysis

  1. First Flip (H):

    • The first flip must be H for either sequence to start.
    • Probability of H = 1/2.

  2. Second Flip:

    • HHT: The second flip must be H (probability = 1/2), leading to a win for Player A if the third flip is T.
    • HTT: The second flip must be T (probability = 1/2), leading to a win for Player B if the third flip is T.

  3. Third Flip:

    • HHT: If the second flip is H, the third must be T for A to win (probability = 1/2).
    • HTT: If the second flip is T, the third must be T for B to win (probability = 1/2).

Calculating Probabilities

  • Probability of HHT (Player A wins):

    • Probability of first flip being H = 1/2.
    • Probability of second flip being H = 1/2.
    • Probability of third flip being T = 1/2.
    • Total probability for HHT = (1/2) * (1/2) * (1/2) = 1/8.

  • Probability of HTT (Player B wins):

    • Probability of first flip being H = 1/2.
    • Probability of second flip being T = 1/2.
    • Probability of third flip being T = 1/2.
    • Total probability for HTT = (1/2) * (1/2) * (1/2) = 1/8.

Solving the Problem

  • Probability of A winning vs. B winning:

    • Since both sequences have equal probability of occurring (1/8 each), we need to consider the recursive nature of the game.

  • Recursive Calculation:

    • After the first head (H), the game can be viewed recursively:
      • If H appears, the probability of A winning next is higher due to the sequence length.
    • Using recursion and considering that the game continues until one sequence appears, the probability of A winning is 2/3.

Conclusion

The probability that Player A wins by obtaining the sequence {HHT} before {HTT} is 2/3.