Generating-function proofs establish two binomial expectation inequalities that imply unimodality near zero and positive/negative asymmetry for heady-s bit-string score counts.
How to Answer Questions of the Type: If you toss a coin n times, how likely is HH to show up more than HT?
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abstract
On March 16, 2024, Daniel Litt, in an X-post, proposed the following brainteaser: "Flip a fair coin 100 times. It gives a sequence of heads (H) and tails (T). For each HH in the sequence of flips, Alice gets a point; for each HT, Bob does, so e.g. for the sequence THHHT Alice gets 2 points and Bob gets 1 point. Who is most likely to win?" We show the power of symbolic computation, in particular the (continuous) Almkvist-Zeilberger algorithm, to answer this, and far more general, questions of this kind.
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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.