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Re: Beating the Odds?
Posted:
Jan 31, 2013 5:10 AM
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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?"
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