Unfair Coin Flip

[vorlon13]

cheaters never prosper

[Brian37]

cheaters never prosper

If that were true 45 would not be in office.

[Alex K]

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.

Invert the result of every other flip?

Probably too easy - it depends on what you want to accomplish. It'll give you exactly 50:50, but with strong correlation between subsequent flips if the coin is unfair.

[bennyboy]

Yep, that will definitely give 50%, which satisfies the OP.

The next step is how to remove that alternating correlation if someone figures out the pattern.

How about before each "for keeps" flip, you flip twice, and do the following:
1) both heads (after inversion adjustment): don't invert the "for keeps" flip
2) both tails (after inversion adjustment): invert the "for keeps" flip
3) one of each: do another pair of flips until you don't get one of each

Of course, at the time of the "for keeps" flip, you'd still have information about how it was likely to come out, so all betting or whatever would have to be done before the pre-flips. But this should break the alternating correlation at least.

[ErGingerbreadMandude]

If it's that simple then why not toss 2 times, if the result is same then flip the result if the result is different make another coin toss to select which result to choose 🤙

[Tiberius]

How would the percentage change in a relevant way if the ratio is the same?

Because the two larger percentages, when combined, will result in a large value. Likewise, the two smaller percentages, when combined, will result in a smaller value. The smaller combined with the larger, when combined, produce a middle value, and because you can combine them in two different ways to get the same result, you can use that to determine the result of the simulated flip.

It seems more to me like a math problem with the heads and tails thing only used to make it more palatable to the layman. As grouping together multiple flips and calling that 'simulating a flip' seems like cheating to me. Cheating insofar as it's equivocating and what I meant by the trick question thing.

I am very impressed by the math. Not impressed by the redefinition. I can't math well anyhow.

EDIT: Wait how is disregarding HH and TT results allowed?

lol

It is a math problem but the original problem involved flipping an unfair coin. My words were "simulate a fair coin flip", not just "simulating a flip". I then went on to clarify that: 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.

So I don't see how the fuck it is a trick question. There's no redefinition. You're not actually performing a single coin flip with a fair coin, you are simulating one by flipping an unfair coin twice (or more).

"simulation" - the imitative representation of the functioning of one system or process by means of the functioning of another.

Also, disregarding HH and TT results is allowed because why wouldn't it be? The goal is to get the results of a fair coin flip using an unfair coin. You do that by eliminating flip combinations that occur too often or not enough, and only look at the ones which occur equally likely (TH and HT).

[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.

Invert the result of every other flip?

Probably too easy - it depends on what you want to accomplish. It'll give you exactly 50:50, but with strong correlation between subsequent flips if the coin is unfair.

Can you show the math on that, I'm not seeing how inverting every other flip means you get a 50:50 result regardless of the unfairness of the coin. If anything, doesn't it make the coin slightly more biased towards the less likely result?

Also your method can't be used to generate a single fair coin flip, because the first flip you get will always be un-inverted and will therefore have the probability of the unfair coin, so you've achieved nothing. Note my question said to "simulate a fair coin flip", not requiring multiple flips to average out 50/50. The first simulated flip (and therefore all subsequent simulated flips) should have a 50/50 chance of being heads or tails.

[ErGingerbreadMandude]

Ohh so that's the problem.

In that case, I want to show an example.

So you make 12 coin toss. You get 3 heads and 9 tails. So your question is how can we make this fair as in instead of getting 3 heads and 9 tails how can we get 6 heads and 6 tails.

Isn't that a bit easy then?

I made 12 tosses so first I divide it by 2. Now I have 6.

Then I do two subtract operation: 6-3=3 and 6-9=-3

So that means I have to flip 3 tails to heads because there is less heads so that becomes 3+3=6 heads or I can flip 3 tails to heads because there is more tails so that becomes 9-3=6 tails.

So now I have 50% heads and 50% tails...

[Tiberius]

Ohh so that's the problem.

In that case, I want to show an example.

So you make 12 coin toss. You get 3 heads and 9 tails. So your question is how can we make this fair as in instead of getting 3 heads and 9 tails how can we get 6 heads and 6 tails.

Isn't that a bit easy then?

I made 12 tosses so first I divide it by 2. Now I have 6.

Then I do two subtract operation: 6-3=3 and 6-9=-3

So that means I have to flip 3 tails to heads because there is less heads so that becomes 3+3=6 heads or I can flip 3 tails to heads because there is more tails so that becomes 9-3=6 tails.

So now I have 50% heads and 50% tails...

I think you've misunderstood the problem. You want to simulate a single fair coin toss, not 12. Besides, how do you know that tails will always come up 9 times out of 12. Maybe if you'd tossed 12 more coins, you would end up with 12 heads in a row. The point is you don't know the probability of flipping heads or tails.

[Losty]

Somebody quote me when the answer is revealed xD