
You Can Load a Die, but You Can’t Bias a Coin (2002) [pdf] - leephillips
http://www.stat.columbia.edu/~gelman/research/published/diceRev2.pdf
======
BhavdeepSethi
Previous discussion:
[https://news.ycombinator.com/item?id=9363322](https://news.ycombinator.com/item?id=9363322)

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kevinpet
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.

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scott_s
This paper assumes that the coin is _caught_.

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Retric
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.

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YokoZar
Or decide whether or not to flip it by putting it on top of your other hand
depending on how it landed when you caught it.

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seanp2k2
Put something on one side of the coin so you can feel what side is up and
decide to flip it.

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JohnHaugeland
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...](https://izbicki.me/blog/how-to-create-an-unfair-coin-and-prove-it-
with-math.html)

~~~
kybernetikos
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)

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

~~~
ziyao_w
And some more magic by Persi:
[https://youtu.be/ZdA7TeoZD_g?t=2m10s](https://youtu.be/ZdA7TeoZD_g?t=2m10s)

~~~
davidb_
That's a neat trick. Based on the article and rewatching the video, he's not
actually flipping the coin.

> In Section Three we prove that the angle ψ between M and the normal to the
> coin stays constant. If this angle is less than 45 ◦ , the coin never turns
> over. It wobbles around and always comes up the way it started. Magicians
> and gamblers can carry out such controlled flips which appear visually
> indistinguishable from normal flips.

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steve19
He refers to it at the end of that video clip as tossing. looking closely
that's pretty much what he is doing. it looks like he might be flipping it
once during the toss for effect. Such a slow rotation allows him to easily
catch it at the right moment.

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tbabb
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.

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

