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Old 04-04-2003, 01:58 AM
2ndGoat 2ndGoat is offline
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Join Date: Sep 2002
Location: DC Area
Posts: 147
Default creating a 2:1 shot with a fair coin

so here's a probability curiosity I've been pondering. I imagine there may be a very trivial way to do this, but I can't figure it:

You've got a *finite* number of fair coins (Pr(H) = Pr(T) = .5). Any number you like, but not infinite. How do you define a test, with any *finite* number of coin flips using any combination of the coins, that have a Pr(Success) of exactly 1/3?

Basically, I'm saying you and your buddy have a handful of pennies, and you want to create an even-money game based entirely on coin flips where one player lays the other 2:1 odds.

Obviously, to create an even-money game, one player could place two coins facing heads up, one tails, and have the other try to pick tails. But that wasn't a random coin flip. No using infinite series either, obviously one can create a situation with a Probability arbitrarily close to 1/3 by adding more coin flips and picking the binomial expiriment with the probability closest to 1/3. We're talking finite numbers here.

Is this possible? I can't get away from probabilities that are sums of powers of two, I'm starting to believe it can't be done.

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