Coin flipping and waiting times paradoxes: Why fair coins are exceptional

TL;DR

This paper analyzes independent string races in coin and die tossing, revealing that fairness induces a total order on waiting times, while bias introduces complex, non-transitive patterns and incomparable waiting times.

Contribution

It provides compact formulas for waiting times and odds, characterizes when stochastic dominance induces a total order, and explores how bias affects transitivity and comparability.

Findings

Fair coins induce a total order on string waiting times.

Bias introduces non-transitive cycles and incomparable waiting times.

Explicit examples and computational classifications for short patterns are provided.

Abstract

Penney's Ante exhibits non-transitivity when two target strings race to appear in a shared stream of coin tosses. We study instead independent string races, where each player observes their own independent and identically distributed (i.i.d.) coin/die stream (possibly biased), and the winner is the player whose target appears first (under an explicit tie convention). We derive compact generating-function formulas for waiting times and a Hadamard-generating-function calculus for head-to-head odds. Our main theorem shows that for a fair -sided die, stochastic dominance induces a total pre-order on all strings, ordered by expected waiting time. For binary coins, we also prove a converse: total comparability under stochastic dominance characterises the fair coin (), and any bias yields patterns whose waiting times are incomparable under stochastic dominance. In contrast, bias allows both (i) reversals between mean waiting time and win probability and (ii) non-transitive cycles; we give explicit examples and certified computational classifications for short patterns.