Richard Barker sent in this photograph and the following note:
Matt just pointed me to your article: You can load a die but you can’t bias a coin. You might be interested in the attached, a photo of a bent NZ 50c coin that I had pressed in the Physics lab here a few years ago because I got bored using flat coins in classroom demonstrations where everyone knows what Pr(heads) is. Fortunatley that particular style of coin is no longer legal tender so I am unlikely to be prosecuted for defacing her Majesty’s coinage.
In discussing this with Matt this afternoon we conjured up a counter example where the coin is completely pressed into a sphere. Then it has Pr(heads) = 1. If the pressing is not quite complete it will be a little less than one, so we claim the statement in the title of your article is not true. We think you can bias a coin.
At about 300 flips it looks as though Pr(Heads) is about 0.55.
When I first bent the coin I did some experiments letting the coin land on the ground. On soft carpet it was not obvioulsly biased but it was on a hard surface. On hard surfaces, most of the time it bounces up and starts spinning on its edge. When this happens it then always lands heads up.
Yeah, sure, he’s right. We were thinking of weighting a coin, but if you bend it enough, then it is no longer set to land “heads” for half of its rotation. And bouncing, sure, then anything can happen. We were always assuming you catch it in the air!
Finally, we were addressing the concept of the “biased coin,” which, by analogy to the “loaded die,” looks just like a regular die but actually has probabilities other than 50/50 when caught in the air. In that sense, the bent coin is not a full counterexample since it clearly looks funny.
Hi. I only recently discovered this wonderful blog.
I read the paper on how a biased coin is not possible, almost the same day I caught the book by John D Barrow ( 100 essential things [you didn't know]^2)that has an interesting piece on how you can *still* extract H / T with equal probability from a biased coin.
I apologise in advance for the amateur notation. The method is attributed to von Neumann. Assume a 'p' NotEqualTo 0.5, and P[H] is p. So P[T] is ( 1-p).
You toss twice, and ignore the H >H and T>T. The probability of getting H > T or T > H is the same regardless of 'p'. Call these pairs Heads / Tails respectively, and you still get an outcome as tho' it's a fair coin.
So, in the sense that a biased coin is a biased coin only if it always gives a biased outcome and one just cannot get a fair outcome …. a biased coin is indeed impossible!
With Best wishes
B. 'Nary' Narayanaswamy / New Delhi / India