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 11-25-2010, 09:46 AM #2 hepzebah adept   Join Date: Aug 2008 Posts: 714 Re: Fun puzzle: How to make an unfair coin fair Spoiler: I've tried to figure this one out... it must be something to do with a sequence of tosses coming up a certain way. I'm mucking about with binary numbers at the moment to try and see if that gives a solution.
 11-25-2010, 09:50 AM #3 hepzebah adept   Join Date: Aug 2008 Posts: 714 Re: Fun puzzle: How to make an unfair coin fair Solution (I think): Spoiler: Hang on I think I have it. Flip the coin twice. If it comes up the same twice then spin again... if it comes up different then the second toss counts. I think that this is equally likely. Lets assume the probablity of head is h and tail is 1-h > Head Head has probability h^2 - spin again > Tail Tail has probability (1-h)^2 - spind again > Head Tail has probability h(1-h) - counts as TAIL with probability h-h^2 > Tail Head has probability (1-h)h - counts as HEAD with probability h-h^2 h-h^2 is always positive where h isn't 1 or zero. ...that's it. Nice problem.
 11-25-2010, 12:39 PM #4 Pyromantha veteran   Join Date: Dec 2007 Posts: 2,228 Re: Fun puzzle: How to make an unfair coin fair von neumann corrector.
 11-25-2010, 12:50 PM #5 bachfan adept   Join Date: Nov 2005 Location: Colorado Posts: 888 Re: Fun puzzle: How to make an unfair coin fair Nice job hep!
11-25-2010, 12:52 PM   #6
bachfan

Join Date: Nov 2005
Posts: 888
Re: Fun puzzle: How to make an unfair coin fair

Quote:
 Originally Posted by Pyromantha von neumann corrector.
Indeed. Old tricks are the best tricks.

 11-25-2010, 04:12 PM #7 BruceZ Carpal \'Tunnel     Join Date: Sep 2002 Posts: 11,877 Re: Fun puzzle: How to make an unfair coin fair I thought of that and thought there must be a more elegant solution.
 11-25-2010, 04:18 PM #8 BruceZ Carpal \'Tunnel     Join Date: Sep 2002 Posts: 11,877 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.
11-25-2010, 04:25 PM   #9
BruceZ
Carpal \'Tunnel

Join Date: Sep 2002
Posts: 11,877
Re: Fun puzzle: How to make an unfair coin fair

Quote:
 Originally Posted by hepzebah Spoiler: I've tried to figure this one out... it must be something to do with a sequence of tosses coming up a certain way. I'm mucking about with binary numbers at the moment to try and see if that gives a solution.
That's how you generate an arbitrary probability given a fair coin.

11-26-2010, 10:03 AM   #10
hepzebah

Join Date: Aug 2008
Posts: 714
Re: Fun puzzle: How to make an unfair coin fair

Quote:
 Originally Posted by BruceZ 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.

 11-27-2010, 11:27 AM #11 wull journeyman     Join Date: Jul 2007 Posts: 216 Re: Fun puzzle: How to make an unfair coin fair Spoiler: Ignorance is key! You throw twice. If it comes heads heads or tails tails you ignore it. 1.heads 2.tails we call heads 1.tails 2.heads we call tails since they are equally likely it's 50%. Nice one bachfan, took me a while.
11-27-2010, 04:57 PM   #12
BruceZ
Carpal \'Tunnel

Join Date: Sep 2002
Posts: 11,877
Re: Fun puzzle: How to make an unfair coin fair

Quote:
 Originally Posted by hepzebah 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.

 12-01-2010, 05:22 PM #13 BruceZ Carpal \'Tunnel     Join Date: Sep 2002 Posts: 11,877 Re: Fun puzzle: How to make an unfair coin fair Seriously, nobody? Answer: Spoiler: Instead of heads/tails, consider same/different from the previous flip which also have probability 0.5. For 5 consecutive heads or tails in 100 flips, we need 4 consecutive "sames" in 99 flips while the 1st flip can be anything. So m=4, n=99, p=0.5.
 12-02-2010, 05:45 AM #14 hepzebah adept   Join Date: Aug 2008 Posts: 714 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.
12-03-2010, 04:16 AM   #15
wobuffet

Join Date: Dec 2007
Location: USA
Posts: 1,084
Re: Fun puzzle: How to make an unfair coin fair

Quote:
 Originally Posted by BruceZ 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!)

 12-04-2010, 06:26 AM #16 PsiCloneB journeyman     Join Date: Oct 2010 Location: Under The Dome Posts: 340 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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