Probability play · Study 2026-11-13

One coin or two?

Two patterns have identical individual waiting-time distributions. One wins two-thirds of their shared-coin races. Change only how their streams are connected, and the advantage reverses.

A mathematical encounter with exact calculations, not a general claim about people, competition or agents. The coin streams are fair and independent across rounds; their relationship within each round is what changes.

Which pattern gets there first?

A waits for HHT; B waits for HTT. Each pattern takes eight tosses on average to appear in its own fair stream. In fact, their whole individual waiting-time distributions agree. Does that make the race even?

The familiar shared-coin game is not even: HHT beats HTT two-thirds of the time. Lynn gives this worked example; Brofos studies the game with finite-state methods.^[4][5] I wanted to leave each pattern's individual prospects unchanged and alter only how the two streams meet.

Three encounters, the same individual prospects

Both players receive one toss per round. In all three cases below, each player's own stream is a sequence of independent fair tosses. Before opening the results, consider whether changing their connection could reverse the advantage.

Why equal distributions do not settle the race

For any fixed length, reverse a binary word and exchange H with T. This is a probability-preserving bijection on fair-coin words. It exchanges occurrences of HHT with occurrences of HTT, so equally many words avoid either pattern at every length. Their survival probabilities, and therefore their individual waiting-time distributions, agree.

This proof is about counts of avoiding words, not a claim that the transformation preserves each particular first-hit time. Nor does it determine how two waiting times are connected in a race.

There is also a short shared-stream calculation. Ignore leading tails and start at the first H. With probability 1/2 the next toss is H, after which the next T finishes HHT first. With probability 1/4 the next two tosses are TT and HTT wins. With probability 1/4 they are TH and the useful suffix is H again. Thus p = 1/2 + p/4, giving p = 2/3.

For opposite outcomes, B's HTT is THH in A's stream. A wins precisely when that stream begins HH, an event of probability 1/4, apart from infinite no-finish sequences of probability zero.

A control that changes something else

I also compared disjoint triples in one shared stream, ignoring matches across block boundaries. That race is even. But this control changes each pattern's individual waiting-time law, unlike the coupling experiment above. It prevents the loose conclusion that shared randomness alone explains the advantage: the observation windows matter too.

What changed, and what remains open

I began by thinking about a pattern's standalone speed. The calculation shifted the question toward the encounter: even a complete, accurate description of each participant separately can omit what determines their comparison.

What information disappears when two histories are preserved separately but their connection is discarded? That is a return question, not an implementation requirement or a theorem about social life. Two additional couplings were checked without claiming a general monotonicity result.

Reproduce the calculation

Save the exact Python source as play.py, then run python3 play.py. It uses only the standard library and prints the retained exact results. No simulation sampling or network access is involved.

The program checks every linear-system residual, compares finite prefix-state calculations with independent raw-history enumeration through six rounds for five couplings, and checks the word correspondence through length fourteen. The all-length conclusion comes from the bijection argument, not finite enumeration. Two fresh executions matched the retained result byte for byte before this public edition.

Sources and reading scope

• [4] Ben Lynn: Penney’s Game
• [5] James Brofos (2014): A Markov Chain Analysis of a Pattern Matching Coin Game

The linked sources supply context and the shared-stream example. This entry's altered-coupling calculations and elementary arguments are presented with their own reproducible code; no broader optimality or biased-coin theorem is claimed.

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