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Old 11-24-2010, 08:02 PM   #1
bachfan
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Fun puzzle: How to make an unfair coin fair

I read about this old, elegant puzzle, having forgotten about it nearly a decade ago and thought some here might enjoy it. It's trivial to google for the answer so I urge you to resist that temptation. PLEASE DON'T POST THE ANSWER (not like I can stop you), but feel free to brag if you figured it out.

On to the problem:

Let's assume you are handed a coin, and told that it is biased, but you are told neither in which direction the coin is biased (heads or tails) nor how strong the bias is (70%/30%, 51%/49%...). Your task - devise a procedure, using only the biased coin, that results in either HEADS or TAILS with a probability of exactly 50% for either outcome. You may use as many coin tosses as you want to generate a HEADS or TAILS.

Cheers,
bachfan

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Old 11-25-2010, 09:46 AM   #2
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Re: Fun puzzle: How to make an unfair coin fair

Spoiler:
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Old 11-25-2010, 09:50 AM   #3
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Re: Fun puzzle: How to make an unfair coin fair

Solution (I think):

Spoiler:
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Old 11-25-2010, 12:39 PM   #4
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Re: Fun puzzle: How to make an unfair coin fair

von neumann corrector.
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Old 11-25-2010, 12:50 PM   #5
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Re: Fun puzzle: How to make an unfair coin fair

Nice job hep!
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Old 11-25-2010, 12:52 PM   #6
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Re: Fun puzzle: How to make an unfair coin fair

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Originally Posted by Pyromantha View Post
von neumann corrector.
Indeed. Old tricks are the best tricks.
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Old 11-25-2010, 04:12 PM   #7
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Re: Fun puzzle: How to make an unfair coin fair

I thought of that and thought there must be a more elegant solution.
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Old 11-25-2010, 04:18 PM   #8
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Re: Fun puzzle: How to make an unfair coin fair

Try the one I asked spadebidder yesterday, it's more interesting:

You have a run calculator for which you can enter 3 numbers m,n,p and it will tell you the probability of at least 1 run of length at least m of events with probability p in n trials. For example, if n=100, m=5, p=0.5 it will tell you the probability that you will get at least 5 consecutive heads in 100 flips of a fair coin. What numbers would you enter to compute the probability of 5 consecutive heads OR tails in 100 coin flips? Note that these are not independent events.
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Old 11-25-2010, 04:25 PM   #9
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Re: Fun puzzle: How to make an unfair coin fair

Quote:
Originally Posted by hepzebah View Post
Spoiler:
That's how you generate an arbitrary probability given a fair coin.
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Old 11-26-2010, 10:03 AM   #10
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Re: Fun puzzle: How to make an unfair coin fair

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Originally Posted by BruceZ View Post
Try the one I asked spadebidder yesterday, it's more interesting:

You have a run calculator for which you can enter 3 numbers m,n,p and it will tell you the probability of at least 1 run of length at least m of events with probability p in n trials. For example, if n=100, m=5, p=0.5 it will tell you the probability that you will get at least 5 consecutive heads in 100 flips of a fair coin. What numbers would you enter to compute the probability of 5 consecutive heads OR tails in 100 coin flips? Note that these are not independent events.
My first 3 guesses are, without any checking or real thinking....
> Double the flips, otherwise unchanged
> 1 minus the probability squared, otherwise unchanged
> m = 4, otherwise unchanged

...will have a think now.
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Old 11-27-2010, 11:27 AM   #11
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Re: Fun puzzle: How to make an unfair coin fair

Spoiler:
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Old 11-27-2010, 04:57 PM   #12
BruceZ
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Re: Fun puzzle: How to make an unfair coin fair

Quote:
Originally Posted by hepzebah View Post
My first 3 guesses are, without any checking or real thinking....
> Double the flips, otherwise unchanged
> 1 minus the probability squared, otherwise unchanged
> m = 4, otherwise unchanged

...will have a think now.
One of these guesses is very close.
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Old 12-01-2010, 05:22 PM   #13
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Re: Fun puzzle: How to make an unfair coin fair

Seriously, nobody?

Answer:
Spoiler:
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Old 12-02-2010, 05:45 AM   #14
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Re: Fun puzzle: How to make an unfair coin fair

Yeah I got that Bruce, with a little help from your post. I used the inspection method (i.e. guesswork) to solve.
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Old 12-03-2010, 04:16 AM   #15
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Re: Fun puzzle: How to make an unfair coin fair

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Originally Posted by BruceZ View Post
Seriously, nobody?

Answer:
Nice puzzle. Wish I'd been on 2+2 to mull it over before you spoiled it (I have little self-control sometimes!)
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Old 12-04-2010, 06:26 AM   #16
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Re: Fun puzzle: How to make an unfair coin fair

Sooooo Eeezeeeeee, do I get to put another star on my wizard hat? Oh forgot my wizard hat has dollar signs...
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