0 KB667 KB62 KB200820252011Addona, VittorioWagon, StanWilf, Herbert S.številka:1letnik:4str. 29-62COBISSID:16261977ISSN:1855-3966URN:URN:NBN:SI:doc-G7RWY3S3enDruštvo matematikov, fizikov in astronomov SlovenijeArs mathematica contemporaneagame theoryLegendre polynomialsLegendrovi polinomiprobabilitysimbolno seštevanjesymbolic summationteorija igerverjetnostHow to lose as little as possible|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 ?$\left\lfloor \frac{1}{2(p-q)} - \frac{1}{2} \right\rfloor \leq N(q, p) \leq \left\lceil \frac{\max(1-p,q)}{p-q} \right\rceil$?. 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 ?$N(q,p)$?TEXTznanstveno časopisjejournalsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije