You Can Load a Die, but You Can’t Bias a Coin (2002) [pdf]

http://www.stat.columbia.edu/~gelman/research/published/diceRev2.pdf

by leephillips • 11 years ago

34 18 11 years ago

stat.columbia.edu

It is very easy to bias a coin. Ask any stage magician. You can bend them, you can taper the outer edge, you can cover one side in a thin layer of something bounce-deadening (like rubber cement), et cetera.

There's a big difference between "you can't bias a coin using weight" and "you can't bias a coin."

https://izbicki.me/blog/how-to-create-an-unfair-coin-and-pro...

kybernetikos2• 11 years ago

The article says that the flip should be caught and not allowed to bounce or spin on the ground.

It still seems to me possible to bias it with weight, based on angular momentum. I think a cavity internally that went from the center of the coin towards one side, with a small weight that moved up and down freely would cause one part of the spin to go faster (when the weight was in the center) than the other part, and would cause a bias if the spin were slow enough not to keep the weight locked at the outside (maybe a spring would be needed? Turns out I don't have a good intuition for what happens when you mix gravity with fast rotation)

GFK_of_xmaspast• 11 years ago

Persi Diaconis, 2004: http://statweb.stanford.edu/~susan/papers/headswithJ.pdf and http://news.stanford.edu/pr/2004/diaconis-69.html

mrestko• 11 years ago

Your concern is addressed within the first three sentences of the article.

kevinpet• 11 years ago

Consider a coin which consists of a strip along one side made of a dense metal, and the remainder made of a light metal. This would move the center of rotation to the side of the coin. Now place the coin on your forefinger with heads up and the heavy band facing away from you. Assuming the rate of rotation was high relative to the falling rate of the coin, then it will tend to first impact the surface with the leading edge of the light two thirds, face down.

The article makes assumes that conservation of momentum guaranteeing a constant rotation rate guarantees fairness. But this relies on the assumption that biasing the coin cannot affect the timing of when it lands.

Of course, this doesn't bias the coin in either direction, it just lets the flipper choose which way to bias a particular flip.

scott_s• 11 years ago

This paper assumes that the coin is caught.

Retric• 11 years ago

In that case you don't need to bias the coin, you can flat out learn to do a normal looking coin flip with catch and get a biased result.

cmsmith• 11 years ago

kevin_pet's coin would still be biased if it were caught. The article contends that you can bias the bouncing part of a coin flip, but not the falling part - GP proposes a coin with a biased falling motion.

BhavdeepSethi• 11 years ago

Previous discussion: https://news.ycombinator.com/item?id=9363322

tbabb• 11 years ago

I wonder if this result holds if the coin is given an "unusual" moment of inertia such that it does not spin about a single axis.

fixxer• 11 years ago

From the title alone, I knew it was going to be Gelman. I guess I have strong priors.

You Can Load a Die, but You Can’t Bias a Coin (2002) [pdf]

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