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.