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8 members have voted

Wizard
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Wizard 
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January 17th, 2017 at 5:56:09 PM permalink
Suppose you have a coin where the probability of heads equals 1/pi =~ 31.8309886%. Next, suppose you and a friend can't agree on which movie to see so you use this coin to choose. To be fair, each of you should have a 50% chance of winning. Multiple flips are allowed.

Question 1: Name any method which will result in a probability of success of 50%.

Question 2: Name a method which will equal or beat a method I know of that requires an average of 4.61 flips to have an outcome.

First person to answer question 2 wins a beer.

As usual, please put answers in spoiler tags.

The question for the poll is which statements do you agree with (multiple votes allowed).
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
MathExtremist
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charliepatrick
January 17th, 2017 at 6:22:23 PM permalink
The method you know is Von Neumann's method: flip pairs of coins until there's a difference, then pick the first. For 1/pi the average number of flips is 4.61.
But here's a paper about more efficient methods; rather than typing it out:
http://www.eecs.harvard.edu/~michaelm/coinflipext.pdf


Edit: for the record, I think my hands are invisible to those bathroom faucets.

Also, I saw a TV ad for those Delta kitchen faucets where you tap the faucet anywhere, say with your wrist or forearm, and the water turns on. What happens if you're not at home and a fly lands on it?
"In my own case, when it seemed to me after a long illness that death was close at hand, I found no little solace in playing constantly at dice." -- Girolamo Cardano, 1563
megapixels
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January 17th, 2017 at 6:29:24 PM permalink
Dunno about #1... Dunno about #2 but my answer for a beer:

One person writes down their choice with a head or tail next to it, and the other person points to a side of the coin. 0 flips.
TomG
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January 17th, 2017 at 6:43:57 PM permalink
Question one:
Flip the coin twice. If you get the same result on both flips, keep going until your two flips are different. Once you have two flips with different results, go see RoboCop if the first flip was heads, or go see Steel Magnolias if the first flip was tails
MathExtremist
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January 17th, 2017 at 6:55:53 PM permalink
Question 3: What is the shape of a coin that has probability of heads equal to 1/pi?

I don't actually know the answer. I figure it'd be a frustum with a specific ratio of bases but I have no idea how to figure it out...
"In my own case, when it seemed to me after a long illness that death was close at hand, I found no little solace in playing constantly at dice." -- Girolamo Cardano, 1563
Wizard
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January 17th, 2017 at 7:10:08 PM permalink
Quote: MathExtremist

The method you know is Von Neumann's method: flip pairs of coins until there's a difference, then pick the first. For 1/pi the average number of flips is 4.61.
But here's a paper about more efficient methods; rather than typing it out:
http://www.eecs.harvard.edu/~michaelm/coinflipext.pdf



This was the answer I was looking for. I didn't know it was called what you called it. That paper explains a way to slightly decrease the expected flips.

(Beers I owe you)++;

I also totally agree about those touch-less faucets. To me it is like playing a slot machine, hoping water will come out. For me, the hit frequency is about 25%. There must be some trick to it that those in the know aren't sharing.

Okay, somebody else submit a good math or logic puzzle for a change.
Last edited by: Wizard on Jan 17, 2017
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
RS
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January 18th, 2017 at 1:30:45 AM permalink
We each pick either heads or tails. If we both pick the same, I win. If we both pick differently, you win. No need to flip the coin.

Another method, not sure if lower than 4.61 expected flips -- but we both flip the coin once. If one person gets heads and the other tails, the one who got heads wins. If both get heads or both get tails, then flip again -- continue until one gets tails and the other heads.
jsm
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January 20th, 2017 at 4:19:35 PM permalink
Draw parallel lines with a pen separated by a distance equal to its length. Throw the pen, and if it lands on the line, toss the coin. If it comes up heads, it's a winner. This happens 50% of the time. No need for floating-point arithmetic!
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