On Jan 30, 2:29 am, William Elliot <ma...@panix.com> wrote: > There is a fair coin with a different integer on each side that you can't > see and you have no clue how these integers were selected. The coin is > flipped and you get to see what comes up. You must guess if that was the > larger of the two numbers or not. Can you do so with probability > 1/2?
I think it depends on how you interpret the question.
I guess we're supposed to think of this as a 2-person infinite game. The Opponent chooses two different integers, the coin is tossed, the Player guesses whether the visible integer is the larger of the two.
For a fixed integer n, let S_n be the following pure (deterministic) strategy for the Player: Guess that the visible number is bigger if it's greater than n, otherwise guess that the hidden number is bigger. (Other kinds of strategy are possible but I think we can ignore them.)
If the Player uses the strategy S_n, then he wins for sure if the number n + 1/2 lies between the two numbers on the coin; otherwise he wins half the time, according to the fall of the coin.
A mixed (randomized) strategy for the Player chooses randomly among the S_n's according to some specified probability distribution on the set Z of integers. It seems like a good idea to use a distribution which assigns a positive probability to each integer. The Opponent will naturally choose two consecutive integers. For any given choice by the Opponent, the Player's probability of winning exceeds 1/2. On the other hand, if the Opponent knows the Player's strategy, he can make the probability as close to 1/2 as he pleases.
I don't know if this counts as a yes or a no to the question, "Can you [guess right] with probability > 1/2?"