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  • How to lose as little as possible
    Addona, Vittorio ; Wagon, Stan ; Wilf, Herbert S., 1931-2012
    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 ▫$N(q,p)$▫.
    Vir: Ars mathematica contemporanea. - ISSN 1855-3966 (Vol. 4, no. 1, 2011, str. 29-62)
    Vrsta gradiva - članek, sestavni del
    Leto - 2011
    Jezik - angleški
    COBISS.SI-ID - 16261977

    Povezava(-e):

    http://amc.imfm.si/index.php/amc/article/view/178
    Digitalna knjižnica Slovenije - dLib.si

vir: Ars mathematica contemporanea. - ISSN 1855-3966 (Vol. 4, no. 1, 2011, str. 29-62)

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