### How to lose as little as possible

#### Abstract

Suppose Alice has a coin with heads probability

*q*and Bob has one with heads probability*p*>*q*. Now each of them will toss their coin*n*times, and Alice will win iff she gets more heads than Bob does. Evidently the game favors Bob, but for the given*p*,*q*, what is the choice of*n*that maximizes Alice's chances of winning? We show that there is an essentially unique value*N*(*q*,*p*) of*n*that maximizes the probability*f*(*n*) that the weak coin will win, and it satisfies ⌊1/2(*p*−*q*) − 1/2⌋ ≤*N*(*q*,*p*) ≤ ⌈max(1 −*p*,*q*)/(*p*−*q*)⌉. The analysis uses the multivariate form of Zeilberger's algorithm to find an indicator function*J*_{n}(*q*,*p*) such that*J*> 0 iff*n*<*N*(*q*,*p*) followed by a close study of this function, which is a linear combination of two Legendre polynomials. An integration-based algorithm is given for computing*N*(*q*,*p*).#### Keywords

Legendre polynomials, symbolic summation, probability

DOI: https://doi.org/10.26493/1855-3974.178.12b

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications