JohnF
Posts:
218
Registered:
5/27/08


Re: Beating the Odds?
Posted:
Jan 31, 2013 6:46 AM


Butch Malahide <fred.galvin@gmail.com> wrote: > 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.
Yeah, that's how it looked to me, too. It says, "...you have no clue how these integers were selected." But, as you point out below, that doesn't say anything about how they >>actually<< were selected. Suppose they were selected randomly (without replacement) from a jar containing 1000 paper slips with 11000 written on them. Then your S_500 is certainly the best strategy. You won't >>know<< it's the best strategy (physics might call that ~ epistemic uncertainty), but the question doesn't ask whether you know the strategy, just whether or not one exists ( ~ ontological uncertainty). But that's more of a trickydicky word question. Either the op meant to be tricky, or he needs to phrase it more precisely.
> I guess we're supposed to think of this as a 2person 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?"
 John Forkosh ( mailto: j@f.com where j=john and f=forkosh )

