The generalized Alice HH vs Bob HT problem
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915618975907840 |
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| author | Janson, Svante Nica, Mihai Segert, Simon |
| author_facet | Janson, Svante Nica, Mihai Segert, Simon |
| contents | In 2024, Daniel Litt posed a simple coinflip game pitting Alice's "Heads-Heads" vs Bob's "Heads-Tails": who is more likely to win if they score 1 point per occurrence of their substring in a sequence of n fair coinflips? This attracted over 1 million views on X and quickly spawned several articles explaining the counterintuitive solution. We study the generalized game, where the set of coin outcomes, {Heads, Tails}, is generalized to an arbitrary finite alphabet A, and where Alice's and Bob's substrings are any finite A-strings of the same length. We find that the winner of Litt's game can be determined by a single quantity which measures the amount of prefix/suffix self-overlaps in each string; whoever's string has more overlaps loses. For example, "Heads-Tails" beats "Heads-Heads" in the original problem because "Heads-Heads" has a prefix/suffix overlap of length 1 while "Heads-Tails" has none. The method of proof is to develop a precise Edgeworth expansion for discreteMarkov chains, and apply this to calculate Alice's and Bob's probability to win the game correct to order O(1/n). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_19035 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The generalized Alice HH vs Bob HT problem Janson, Svante Nica, Mihai Segert, Simon Probability Combinatorics In 2024, Daniel Litt posed a simple coinflip game pitting Alice's "Heads-Heads" vs Bob's "Heads-Tails": who is more likely to win if they score 1 point per occurrence of their substring in a sequence of n fair coinflips? This attracted over 1 million views on X and quickly spawned several articles explaining the counterintuitive solution. We study the generalized game, where the set of coin outcomes, {Heads, Tails}, is generalized to an arbitrary finite alphabet A, and where Alice's and Bob's substrings are any finite A-strings of the same length. We find that the winner of Litt's game can be determined by a single quantity which measures the amount of prefix/suffix self-overlaps in each string; whoever's string has more overlaps loses. For example, "Heads-Tails" beats "Heads-Heads" in the original problem because "Heads-Heads" has a prefix/suffix overlap of length 1 while "Heads-Tails" has none. The method of proof is to develop a precise Edgeworth expansion for discreteMarkov chains, and apply this to calculate Alice's and Bob's probability to win the game correct to order O(1/n). |
| title | The generalized Alice HH vs Bob HT problem |
| topic | Probability Combinatorics |
| url | https://arxiv.org/abs/2503.19035 |