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Math Help - Net effect of more draws with lower "accuracy"...?

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    Question Net effect of more draws with lower "accuracy"...?

    Dear all,
    I am struggling with the following problem:

    We take n + 1 independent draws (n even), where each draw has a “success probability” gamma > 1/2. Let S be the probability of a majority (i.e. more than 50%) “successes.”

    What happens to S if we increase the number of draws by 2 (to keep n even) but , at the same time, decrease gamma in such a way that probability of a tie one draw before the end -- i.e., after n draws, resp., after n+2 draws -- remains constant?

    Numerical calculation show quite convincingly that S falls. But I cannot prove it!

    I have written out the problem in the attachment. Here, Prob(Pivot) denotes the probability of a tie after n draws (respectively, n+2 draws). Such a tie one draw before the end means that all draws are "pivotal," i.e., decisive.

    Very much looking forward to any suggestions that people may have.

    Best,

    Felix
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