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Topic: Beating the Odds?
Replies: 35   Last Post: Feb 6, 2013 3:44 PM

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Posts: 219
Registered: 5/27/08
Re: Beating the Odds?
Posted: Jan 31, 2013 6:46 AM
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Butch Malahide <> wrote:
> On Jan 30, 2:29?am, William Elliot <> 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, " 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 1-1000 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 tricky-dicky
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 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?"

John Forkosh ( mailto: where j=john and f=forkosh )

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