From a fun little article by Andrew Gelman and Deborah Nolan:

The law of conservation of angular momentum tells us that once the coin is in the air, it spins at a nearly constant rate (slowing down very slightly due to air resistance). At any rate of spin, it spends half the time with heads facing up and half the time with heads facing down, so when it lands, the two sides are equally likely (with minor corrections due to the nonzero thickness of the edge of the coin); see Figure 3. Jaynes (1996) explained why weighting the coin has no effect here (unless, of course, the coin is so light that it floats like a feather): a lopsided coin spins around an axis that passes through its center of gravity, and although the axis does not go through the geometrical center of the coin, there is no difference in the way the biased and symmetric coins spin about their axes.

On the other hand, a weighted coin spun on a table will show a bias for the weighted side. The article describes some experiments and statistical tests to use in the classroom. There are some entertaining stories too. Like how the King of Norway avoided losing the entire Island of Hising to the King of Sweden by rolling a 13 with a pair of dice (“One die landed six, and the other split in half landing with both a six and a one showing.”)

Visor volley: Toomas Hinnosaar.

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March 2, 2011 at 1:49 pm

Jeff SAbout a year ago, I had a longer than would be expected debate with a friend of mine as a result of this paper. The rejoinder comes from some Stanford researchers:

Click to access headswithJ.pdf

My reply at the time:

A lot of fancy math obscures the fact that their [the Stanford] findings are completely trivial. What they call “procession”, Gelman calls “tossing like a frisbee”. Gelman already points out that it is not only possible to have a small amount of bias if you allow for procession, you can have as much bias as you want! But, Gelman assumes from the outset of his paper that the coin toss is “fast, with lots of spin”. This is an imprecise way of saying that it is a toss without procession, which, according to the Stanford study, is the only case where bias does not exist.

March 2, 2011 at 2:09 pm

wellplacedadjective…i always thought you were supposed to bend the coin..?

March 4, 2011 at 10:34 am

twickerNo such thing as a biased coin, but you can learn to flip a coin the same way each time so that you get whatever result you like (assuming you catch it in the air). With about 15 min. of practice, I can reliably end up with the result I want around 80% – 90% of the time (“reliably” = “every time I’ve bothered to try it, which is about 8 times so far”). With a bit more practice (holding it the same way, launching it with the same force, catching it at the same position, etc.), I’d likely be able to get as close to 100% as makes no difference.

No biased coins, but, unless you’re the one flipping, you always want it to fall on the ground (surfaces differ in their elasticity; thus, the bounces also differ in generally unpredictable ways).

June 10, 2013 at 5:47 am

Sylvianejohn byrdWell, I must confess that I do not ssrbcbiue to any of the various Bayesian approaches. Several years ago I became very intrigued and excited about what biological anthropologists were trying to do with Bayesian statistics. However, I have not read a single paper in my field where the use of priors did not make me cringe. Further, as Bayesian models have become more popular in forensics, I have begun to appreciate the danger inherent in the flippant application of these models, which tend to yield posteriors that speak to central questions. You might recall the article I referenced awhile ago that reported on the rejection of a Bayesian analysis from a UK court.So, I am uncomfortable with how priors are derived in practice, and even more disturbed by how we tend to ignore that issue after analyses are done and conclusions drawn. I worry about scientists relaxing their focus on the appropriateness of data as they employ more sophisticated models. I look askance at the fashion trend.All that confessed, I appreciate Prof Berger`s concerns about loose thinking about priors. I also find it interesting to see how Bayesians might seek objectivity in their model building. What I most want to know, however, is how the Bayesian proponents will answer the critique of Birnbaum`s thesis. If the strong likelihood principle does not hold, what happens to the Bayesian house?

June 10, 2013 at 10:39 pm

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November 3, 2013 at 7:45 pm

Andreijohn byrdIt seems the problem is that wiouhtt the likelihood principle, and no means of dealing with sampling error, and no explicit concern for how the data were generated, then I should have no reason to think the likelihood component of a Bayesian model is informative. And that is supposed to be the interesting part.

February 8, 2012 at 10:45 am

Alex TabarrokNOT TRUE!

http://www.amazon.com/Double-Sided-Coin-Nickel-ALWAYS/dp/B006H6L0D2