Generating-function proofs establish two binomial expectation inequalities that imply unimodality near zero and positive/negative asymmetry for heady-s bit-string score counts.
Segert, A proof that HT is more likely to outnumber HH than vice versa in a string of n coin flips, arXiv:2405.16660 [math.CO]
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abstract
Consider the following probability puzzle: A fair coin is flipped n times. For each HT in the resulting sequence, Bob gets a point, and for each HH Alice gets a point. Who is more likely to win? We provide a proof that Bob wins more often for every n>=3. As a byproduct, we derive the asymptotic form of the difference in win probabilities, and obtain an efficient algorithms for their calculation.
fields
math.CO 2years
2026 2representative citing papers
Derives recursive and closed formulas for moments of waiting times for prescribed words in coin flips and die rolls using one-parameter Eulerian number extensions, Goulden-Jackson cluster method, and Faà di Bruno's formula.
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On Two Combinatorial Inequalities That Explain the Blimpy Shape of Heady-s and Taily-s Bit Strings
Generating-function proofs establish two binomial expectation inequalities that imply unimodality near zero and positive/negative asymmetry for heady-s bit-string score counts.
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Moments for generalizations of a coin flip game
Derives recursive and closed formulas for moments of waiting times for prescribed words in coin flips and die rolls using one-parameter Eulerian number extensions, Goulden-Jackson cluster method, and Faà di Bruno's formula.