An eccentric uncle has left a fortune to you and your brother - with a very strange condition.

He's bequeathed to you his special fake coin that lands heads more often than it lands tails. And all his money will go either to you or to your brother depending on who wins a sequence of tosses of his special coin.

What method can you and your brother devise so that you'll both agree that a fair winner has been determined by repeated tosses of the biased coin? It's unlikely that more than 26 tosses would be required.

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TopNewestThere should be a limitation on what method is allowed, otherwise I can do this without any toss:

Or easier:

On the other hand, if the method can't be just:

Because the second player can just pick the one with all heads and have large probability of winning if the probability of heads is big, like, 99.9999% or something. – Ivan Koswara · 9 months ago

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I'll give a helpful hint in a couple of days. – Martin Coles · 9 months ago

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– Ivan Koswara · 9 months ago

With that edit, the answer is obvious. (Toss the coin twice. If it's HT, first player wins; if it's TH, second player wins; if it's anything else, toss again.) I'm very sure the 26-toss mark can be easily passed, though; just try seeing what happens if the probability of heads is 99% ("much more often").Log in to reply

They can count how many times there was a heads on even tosses and odd tosses and see which one has more. Before they toss the 26 times, one of then will pick even and one will pick odd. – Josh Kari · 9 months ago

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– Martin Coles · 9 months ago

This would work. However, there's a more elegant solution. I'll give you a hint. With this solution, the winner could be decided with as few as two tosses of the biased coin.Log in to reply

– Ivan Koswara · 9 months ago

I don't see how it works, since there is a possibility of a tie.Log in to reply

– Martin Coles · 9 months ago

The decision might be made after two tosses, but it might take more tosses. It's extremely unlikely that as many as 26 tosses would be required to determine the winner.Log in to reply

– Ivan Koswara · 9 months ago

"Extremely unlikely" is not "never". Your problem requires you to have the winner within 26 tosses; you're not allowed to go over, even if the probability that will occur is tiny.Log in to reply

– Martin Coles · 9 months ago

OK. Problem is hereby rewritten. Once the correct strategy has been adopted by the two brothers, it's extremely unlikely that more than 26 tosses would be required to determine the winner.Log in to reply

– Ivan Koswara · 9 months ago

What happens if there is a tie?Log in to reply

Is the probability of landing on heads known or unknown?

Interesting problem! Can we make a fair outcome from biased outcomes? – Calvin Lin Staff · 9 months ago

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– Martin Coles · 9 months ago

The probability of heads is not known.Log in to reply