# How to design a deliberately biased coin?

For demonstrating basic probability concepts, it would be nice to have a coin-like object that lands heads/tails not in 50/50% ratio, but biased in a way that can be revealed in a short experiment. What I'd like is to make an object satisfying:

• Thin disk shape, say thickness around 1/10 to 1/20 of diameter.
• Lands heads/tails with some given lopsided ratio such as 60/40.
• Feels evenly balanced to the students' hands.
• Can be made by the average machinist, hobbyist woodworker, or 3D printing designer.
• Size not important, but maybe 10-20cm diameter. It's meant to be a theatrical prop visible to a small audience, not an actual coin.

Would a disk with an interior hollow zone closer to one side than the other do the job? I doubt it, since the coin will still rotate uniformly in the air, exposing both sides equally to any direction, including the floor. Making both sides "heads" is too obvious a cheat, and I don't want 100/0% probabilities anyway.

Note I'm not asking for any practical how-to workshop details, just the physical principle for designing such an object.

-
These don't seem like the best tags. Feel free to improve them, anyone smarter than I. – DarenW May 26 '13 at 6:36
The cavity closer to one side will surely impact the odds. In the air, it may spend the same time on both sides but when it touches the surface, it will tend to fall to one side and not the other. – Luboš Motl May 26 '13 at 6:52
I hope you mean closer to one face, because side could be thought as half the periphery. – anna v May 26 '13 at 7:23
Glue on one side of the coin. – Elements in Space May 26 '13 at 12:52
You could perhaps try gluing two coins of different material but same diameter to each other. That ought to shift the odds. As for getting specific odds like 60/40, it's not gonna be easy. – Bolt64 May 26 '13 at 13:18

I would suggest making a magnetic coin. Then you can alter its results by having an electric magnet behind the table you are flipping the coin on it.

However, if the magnetic field is two strong it would be easily distinguished by bare eyes. So you have to balance its strength, by flipping the coin numerous times(e.g. few thousands) and calibrating the magnet with the probability of heads/tails.

This way you will not only have an unfair coin, but a variable unfair coin which can be used for many demonstrations.

Anyway, note that even a fair coin has a lot of uncertainty, which means by flipping a fair coin a hundred times; getting 40/60 is not highly unprobable.

-

Maybe this answer won't be useful to you at all, but still.

Since you mentioned that all this is for a demostration, you might as well use a loaded dice than put in all the effort to make an unbiased coin. Even if the audience is completely new to concepts of probability, a dice will keep concepts as simple as in the case of a coin. If you are interested in the mechanics, then you can read http://en.wikipedia.org/wiki/Dice#Loaded_dice

If it is required that you have to construct it yourself then "shaved die" will be the best option: get a wooden or metallic cube made and file one side to make it asymmetric. See http://en.wikipedia.org/wiki/Dice#Shaved_dice You will have to see for yourself how much filing goes unnoticed. In any case it will need a few hundreds of throws atleast to reveal its bias.

-
Dang, wikipedia has everything! I hadn't thought of looking up "loaded dice" there. – DarenW May 26 '13 at 21:12
I wonder by how much dice are shaved. Dishonest people doing that to cheat casinos, not physics teachers doing demos, don't want anyone to notice, but then by how much are the probabilities tweaked? – DarenW May 26 '13 at 21:13
If the dimensions of the shaved die are a,a and a-h, then out of the three possible configurations of the sitting dice, the anomalous one will be more stable by a fraction of h/a. Take a = 1 cm and h = 1 ~ 1.5 mm (That will be imperceptible) will give you 10~15 % extra stability. This difference will definitely come up in 1000 throws. – Man May 27 '13 at 11:00