Play The Coin One Is Dealt
Three men in a room all covet a certain object. They agree to flip a coin for it in such a way that they all have equal chances of getting it.
Unfortunately, the only coin they can find is a biased one, rigged to land heads of the time and tails of the time. The men all know the exact odds for the biased coin.
Using the biased coin, what is the minimum number of flips in the best case needed to determine in a fair way which of the three gets the object?
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1 solution
Moderator note:
Nice idea!
To prove that is is indeed the minimum, can we show that it cannot be done with 2 flips?
What is the generalization of this result?
If all goes well, it can be done with 3 flips.
Man #1 takes the sequence tails-heads-heads.
Man #2 takes the sequence heads-tails-heads.
Man #3 takes the sequence heads-heads-tails.
Since all 3 of those sequences involve the same number of heads and tails, they are all equally likely to come up.
The biased coin is flipped 3 times, and if one of those sequences comes up, the man with the particular sequence gets the object.
(If not, they flip the coin 3 more times, etc...)