# An unbiased choice

Anthem
Posers and Puzzles 27 Jul '11 21:16
1. Anthem
The Ferocious Camel
27 Jul '11 21:16
You and your roommate are trying to decide who gets the last piece of pizza. You decide to flip a coin to choose who gets the slice, but the only coin you have is biased (that is, there is a x% chance of the coin coming up heads on each flip where x is between 0 and 100, but is not 50).

How can you use the results of flipping this coin to fairly decide who gets the slice?
2. Palynka
Upward Spiral
27 Jul '11 21:402 edits
Originally posted by Anthem
You and your roommate are trying to decide who gets the last piece of pizza. You decide to flip a coin to choose who gets the slice, but the only coin you have is biased (that is, there is a x% chance of the coin coming up heads on each flip where x is between 0 and 100, but is not 50).

How can you use the results of flipping this coin to fairly decide who gets the slice?
Flip it twice with each player winning once with heads and once with tails. If one wins both, he gets the slice but if it&#039;s one each then repeat.
3. 27 Jul '11 22:43
Originally posted by Anthem
You and your roommate are trying to decide who gets the last piece of pizza. You decide to flip a coin to choose who gets the slice, but the only coin you have is biased (that is, there is a x% chance of the coin coming up heads on each flip where x is between 0 and 100, but is not 50).

How can you use the results of flipping this coin to fairly decide who gets the slice?
I would reply but was looking into a question I had for you and saw the (or an) answer. Good one!
4. AThousandYoung
All My Soldiers...
28 Jul '11 00:10
Each player flips until one gets heads and one gets tails
5. sonhouse
Fast and Curious
28 Jul '11 00:351 edit
Originally posted by AThousandYoung
[hidden]Each player flips until one gets heads and one gets tails[/hidden]
That could still be by chance. You need to correct for the coin bias.
I would think that would only be the beginning, since now (assuming you captured the bias correctly) you would throw more tosses, say one side gets 3 tosses, the other 5 to compensate for the flip bias. I would assume you average the numbers then.
6. 28 Jul '11 00:36
does the coin bias change?
7. sonhouse
Fast and Curious
28 Jul '11 00:37
Originally posted by Banana King
does the coin bias change?
That would be difficult unless it was coated with ice.....
8. AThousandYoung
All My Soldiers...
28 Jul '11 02:28
Originally posted by sonhouse
That could still be by chance. You need to correct for the coin bias.
I would think that would only be the beginning, since now (assuming you captured the bias correctly) you would throw more tosses, say one side gets 3 tosses, the other 5 to compensate for the flip bias. I would assume you average the numbers then.
Both sides get the same number of flips. The bias will apply evenly to both sides because both sides win on heads. They flip seperately.
9. coquette
28 Jul '11 04:341 edit
if neither player knows the coin bias, then the coin can be flipped and either player can select heads or tails. it doesn't matter. the guess of the coin is a 50:50 guess that you pick the favored bias side. then the flip uses the coin and the bias has already been neutralized by the random choice. the conditions for knowing or not knowing were not stated in the problem.

if, however, both sides know the coin is biased and the decision is still to use the biased coin, the problem becomes interesting. So, let's say that tails is favored over heads to some degree and both sides know it. the use of the biased coin should be for the tip. the last piece of pizza should go to the winner of rock, scissors and paper
10. Palynka
Upward Spiral
28 Jul '11 08:30
Strange, why doesn't my answer appear when I hover over the spoiler? Well, if someone is interested click reply and quote.
11. 28 Jul '11 11:34
Originally posted by sonhouse
That would be difficult unless it was coated with ice.....
how is the coin biased in the first place?
12. Anthem
The Ferocious Camel
28 Jul '11 14:391 edit
A solution:
If I am understanding his post correctly, AThousandYoung got the solution that I was thinking of. In more detail: You each flip the coin once. If one person gets heads and the other gets tails the one who got heads wins. Repeat until someone wins. Palynka&#039;s solution also works.

sonhouse - You do not know what the bias is. If you try to figure it out experimentally (e.g. by flipping the coin a bunch of times), you can only approximate the bias and thus cannot assure that the toss is completely fair.
13. Palynka
Upward Spiral
28 Jul '11 15:03
Your solution (and ATY's) is more economical than mine. I like it.
14. roma45
st johnstone
28 Jul '11 16:16
Originally posted by Anthem
You and your roommate are trying to decide who gets the last piece of pizza. You decide to flip a coin to choose who gets the slice, but the only coin you have is biased (that is, there is a x% chance of the coin coming up heads on each flip where x is between 0 and 100, but is not 50).

How can you use the results of flipping this coin to fairly decide who gets the slice?
cut it in half, a bit each ðŸ˜€ðŸ˜€
15. AThousandYoung
All My Soldiers...
28 Jul '11 16:40
Originally posted by Palynka
[hidden]Flip it twice with each player winning once with heads and once with tails. If one wins both, he gets the slice but if it's one each then repeat.[/hidden]
Probably too long, or perhaps you're not allowed to use punctuation.