We repeatedly flip a fair coin until either the pattern HT appears (A wins) or the pattern HH appears (B wins), whichever occurs first. Ties cannot occur. 1) Compute P(A wins) and P(B wins) exactly. 2) For m ≥ 2, derive the probability mass function of the stopping time: P(A wins at flip m) and P(B wins at flip m). 3) Compute the expected stopping time E[T]. 4) Extend part (1) to a biased coin with P(H)=p and provide closed-form expressions for P(A wins) and P(B wins). 5) Validate your results by simulation (≥ 10^6 trials) and compare empirical PMFs to theory.

This question evaluates understanding of discrete probability, stopping-time analysis, pattern occurrence in Bernoulli sequences, derivation of PMFs and expectations, and the ability to generalize results to biased coins.

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Analyze HT vs HH stopping-time probabilities

Coin-Flip Stopping Game: HT vs HH

You repeatedly flip a coin until either the pattern HT appears (Player A wins) or the pattern HH appears (Player B wins), whichever occurs first. These are evaluated on the last two consecutive flips; ties cannot occur because only one ordered pair can appear at a time.

Assume the coin is fair unless otherwise stated.

Compute P(A wins) and P(B wins) exactly for a fair coin.
For m ≥ 2, derive the probability mass function (PMF) of the stopping time by outcome: P(A wins at flip m) and P(B wins at flip m).
Compute the expected stopping time E[T].
Generalize part (1) to a biased coin with P(H) = p (so P(T) = 1 − p = q). Provide closed-form expressions for P(A wins) and P(B wins).
Validate by simulation with at least 10^6 trials and compare empirical PMFs to the theoretical results.

Coin-Flip Stopping Game: HT vs HH

Assume the coin is fair unless otherwise stated.

Compute P(A wins) and P(B wins) exactly for a fair coin.
For m ≥ 2, derive the probability mass function (PMF) of the stopping time by outcome: P(A wins at flip m) and P(B wins at flip m).
Compute the expected stopping time E[T].
Generalize part (1) to a biased coin with P(H) = p (so P(T) = 1 − p = q). Provide closed-form expressions for P(A wins) and P(B wins).
Validate by simulation with at least 10^6 trials and compare empirical PMFs to the theoretical results.

Analyze HT vs HH stopping-time probabilities

Quick Overview

Coin-Flip Stopping Game: HT vs HH

Solution

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Analyze HT vs HH stopping-time probabilities

Quick Overview

Coin-Flip Stopping Game: HT vs HH

Solution

Comments (0)