Unfair Coin Flip

[Tiberius]

Suppose you have a coin, and you suspect it is unfair (i.e. it lands on one side more often than the other). You don't know which side it lands on more, and for sake of argument assume you can't find out.

How can you simulate a fair coin flip using this coin only? By "fair" I mean that you should get a "heads" result exactly 50% of the time, and a "tails" result exactly 50% of the time.

[vorlon13]

drop coin on checkerboard

whatever color it most covers, red or black, corresponds to heads or tails

[Tiberius]

drop coin on checkerboard

whatever color it most covers, red or black, corresponds to heads or tails

Haha, points for thinking outside the box. Let's add the additional constraint that you can only use the coin and the action of flipping the coin to get a result.

[Edwardo Piet]

Suppose you have a coin, and you suspect it is unfair (i.e. it lands on one side more often than the other). You don't know which side it lands on more, and for sake of argument assume you can't find out.

How can you simulate a fair coin flip using this coin only? By "fair" I mean that you should get a "heads" result exactly 50% of the time, and a "tails" result exactly 50% of the time.

Do 100 flips and subtract the extra % that the unfair side would win from its total wins after 100 flips. So basically the unfair wins don't count as real wins.

If sample size is a problem then do it more than 100 times. But it's the same principle.

[pocaracas]

drop coin on checkerboard

whatever color it most covers, red or black, corresponds to heads or tails

Haha, points for thinking outside the box. Let's add the additional constraint that you can only use the coin and the action of flipping the coin to get a result.

Interesting problem, indeed...

Define flipping the coin... throwing it in the air, spinning, and catching it with another hand? or letting it fall to the ground (where the unfairness would be much more noticeable)?... I see a checkered floor is out of the question.

Can we throw it more than once for each result?

[Tiberius]

Suppose you have a coin, and you suspect it is unfair (i.e. it lands on one side more often than the other). You don't know which side it lands on more, and for sake of argument assume you can't find out.

How can you simulate a fair coin flip using this coin only? By "fair" I mean that you should get a "heads" result exactly 50% of the time, and a "tails" result exactly 50% of the time.

Do 100 flips and subtract the extra % that the unfair side would win from its total wins after 100 flips. So basically the unfair wins don't count as real wins.

If sample size is a problem then do it more than 100 times. But it's the same principle.

You don't know what the unfair side % is though, and your method doesn't give you a single result at the end (i.e. "heads" or "tails") unless I've misunderstood.

[Tiberius]

Haha, points for thinking outside the box. Let's add the additional constraint that you can only use the coin and the action of flipping the coin to get a result.

Interesting problem, indeed...

Define flipping the coin... throwing it in the air, spinning, and catching it with another hand? or letting it fall to the ground (where the unfairness would be much more noticeable)?... I see a checkered floor is out of the question.

Can we throw it more than once for each result?

Let's make it simple and say you have a machine which perfectly flips a coin with a degree of randomness and catches it perfectly, eliminating any unfairness based on the flipping action and the catch. So the unfairness is all within the coin and how it rotates in the air.

Yes you can flip it more than once for each result.

[Edwardo Piet]

Do 100 flips and subtract the extra % that the unfair side would win from its total wins after 100 flips. So basically the unfair wins don't count as real wins.

If sample size is a problem then do it more than 100 times. But it's the same principle.

You don't know what the unfair side % is though, and your method doesn't give you a single result at the end (i.e. "heads" or "tails") unless I've misunderstood.

Oh if we don't know what the unfair % is then I have no idea how to solve this problem unless it's some kind of trick question or something.

EDIT: Wait can't we just flip it so many times that we can find the unfair % advantage through statistics and a huge sample size?

By the way I do know that given a normal coin it's more likely to lend on heads than tails because of the indentation. But we're assuming here that it's 50/50 to begin with before the unfair advantage is added right?

[RoadRunner79]

Suppose you have a coin, and you suspect it is unfair (i.e. it lands on one side more often than the other). You don't know which side it lands on more, and for sake of argument assume you can't find out.

How can you simulate a fair coin flip using this coin only? By "fair" I mean that you should get a "heads" result exactly 50% of the time, and a "tails" result exactly 50% of the time.

Assuming only 1 flip of the coin, then if I don't know which side the coin lands on more, then it would be a fair coin flip, if I call it.  (since I have a 50/50 chance of getting the dominant side).

If you want more trials, you could spin the coin, and pick a feature on both sides of the coin, and determine the direction that feature is pointing.

[Tiberius]

You don't know what the unfair side % is though, and your method doesn't give you a single result at the end (i.e. "heads" or "tails") unless I've misunderstood.

Oh if we don't know what the unfair % is then I have no idea how to solve this problem unless it's some kind of trick question or something.

EDIT: Wait can't we just flip it so many times that we can find the unfair % advantage through statistics and a huge sample size?

It's not a trick question.

The problem is to get a perfect 50/50 probability you would have to flip the coin an infinite number of times to do the analysis. You don't know the unfairness % so you could flip the coin 100 time, get "heads" 60 times and "tails" 40 times, and think that it's more likely to come down "heads", but then you flip it 100 more times and it comes down "heads" 30 times and "tails" 70 times.

Increasing the sample size gets you closer to the actual value but you never touch it.