How to lose as little as possible - Addona, Vittorio; Wagon, Stan; Wilf, Herb (2010)
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? The problem of determining the optimal$N$first appeared in wa. 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$\frac{1}{2(p-q)}-\frac12\le N(q,p)\le \frac{\max{(1-p,q)}}{p-q}$ . 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$N(q,p)$ .