โ Scribed by Helge Tverberg
No coin nor oath required. For personal study only.
Tverberg, H., On a coin tossing problem by G. Bennett, Discrete Mathematics 115 (1993) 293-294.
For a certain class of games, Bennett proved that player B never has a smaller chance of winning than player A. Here we give a proof which keeps strictly to the original problem environment.
Bennett [ 11, discusses the following problem: A and B are two players, playing with coins P and Q. A target set T in N x N is given, and A plays as follows. He tosses P until he gets heads in, say, the mth tossing, and then he tosses Q until he gets heads in, say, the nth tossing. He wins if (m, n)~ T.
B plays differently. He tosses P and Q simultaneously, and stops when he gets heads on both in, say, the kth tossing. If k<l TI, he wins. Bennett [l] proves that the probability of winning is at least as large for B as for A, and asks for a more 'probabilistic' proof of this. This is what we give here.
We first modify the target Tin a way which improves the situation for A, without affecting that of B. If, say, r 3 1 and (r, s)4 T, while (r + 1, S)E T, then A is better off if one adds (r, s) to T and removes (r + 1, s). This follows from the fact that A's probability of winning is
x(1 -P)~-~P (~-~~-'cL (j, WET, where p(q) denotes the probability of heads with P (Q).
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