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Answer by sigoldberg1 for How is it that you can guess if one of a pair of random numbers is larger with probability > 1/2?

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As previously explained above, the success of the proposed contestant strategy is entirely dependent upon the assumption made by the contestant about the host's method of choosing x and y.

An example where a fixed contestant strategy almost always fails. Suppose the host always chooses an "unexpectedly" large negative number for x, say x = -n, where n has >1000 digits, and r is more "ordinary", say having <100 digits. The method depends on the contestant betting that x < r < y. If the host's uniform strategy has always chosen y = x-1, then the contestant will almost certainly lose round after round of play. (If, after numerous rounds of this, the contestant then realizes this strategy of the host and tries to change his/her betting strategy, the host can adapt as well, choosing y = -x = n for later rounds)


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