What to do if you're stuck in one place for a long while? Flip a coin...10,000 times. An early study of coin tosses...and a byproduct of WWII.
http://en.m.wikipedia.org/wiki/John_Edmund_Kerrich Hugh Sent from my iPad On Feb 28, 2015, at 5:44 PM, Jeffry Ricker, Ph.D. <jeff.ric...@scottsdalecc.edu<mailto:jeff.ric...@scottsdalecc.edu>> wrote: I was surprised to learn today that physicists have been studying coin tosses since the mid—1980s. The question they usually are trying to answer is: ‘do the results of coin tosses reflect a stochastic process?’ The answer may surprise you. For example, here is the abstract from a paper published by Diaconis, Holmes, and Montgomery (2007): We analyze the natural process of flipping a coin which is caught in the hand. We prove that vigorously-flipped coins are biased to come up the same way they started. The amount of bias depends on a single parameter, the angle between the normal to the coin and the angular momentum vector. Measurements of this parameter based on high-speed photography are reported. [I’ve omitted the final sentence because it would have spoiled the Shyamalan–esque ending of this post.] And here is the abstract from a report by Strzałko, Grabski, Stefański, Perlikowski, and Kapitaniak (2008): The dynamics of the tossed coin can be described by deterministic equations of motion, but on the other hand it is commonly taken for granted that the toss of a coin is random. A realistic mechanical model of coin tossing is constructed to examine whether the initial states leading to heads or tails are distributed uniformly in phase space. We give arguments supporting the statement that the outcome of the coin tossing is fully determined by the initial conditions, i.e. no dynamical uncertainties due to the exponential divergence of initial conditions or fractal basin boundaries occur. [Again, I’ve omitted the final sentence.] I cannot follow the math in either article at all; but it’s truly impressive, which leads me to conclude that such smart people cannot possibly be wrong (and please don’t confuse me by pointing to the many, many examples of brilliant physicists who were wrong, OK? Thank you very much). There’s lotsa’ stuff filling up the space between the abstract and the conclusion in each paper. I barely glanced at any of it. I recommend that you follow my lead. Now to the Shyamalan–esque ending. The final sentence of Diaconis, Holmes, and Montgomery’s (2007) abstract is: “For natural flips, the chance of coming up as started is about .51.” Whaaaa…? Strzałko, et al. (2008) make a similar conclusion, but in a much less “user friendly” way: In practice although heads and tails boundaries are smooth, the distance of a typical initial condition from a basin boundary is so small that practically any finite uncertainty in initial conditions can lead to the uncertainty of the result of tossing…. One can consider the tossing of a coin as an approximately random process. Why the flip—flop (surprisingly, no pun was intended)? The Diaconis, Holmes, and Montgomery (2007) paper spells this out more clearly than the other paper. The researchers’ assumptions, as well as the experimental conditions, made it difficult to generalize their results to real life: * The coin was flipped with a known side facing upwards. * There was no air resistance. * There was no variation in “flight time” across tosses. * The side of the coin facing up was positioned perfectly (i.e., there is no tilt). * The coin didn’t bounce when landing. * And there were various technical limitations in the experiment. They concluded: “For tossed coins, the classical assumptions of independence with probability 1/2 are pretty solid.” Case closed? Perhaps not. I noticed that the literature on coin tossing is pretty extensive. I’ll need to look further. My reason for posting this discussion is related to the following point made by Diaconis, Holmes, and Montgomery (2007): The discussion … highlights the true difficulty of carefully studying random phenomena. If we can find this much trouble analyzing a common coin toss, the reader can imagine the difficulty we have with interpreting typical stochastic assumptions in an econometric analysis. For me, the discussion highlights the difficulty of designing, conducting, analyzing, and interpreting research studies, in general. These experiments on the physics of coin tossing—a phenomenon that, on the surface, might seem to be relatively simple and straightforward—illustrate many of the points we try to make in our classes. I want to elaborate on this, and perhaps I will tomorrow. But I am out of time now. Best, Jeff REFERENCES Diaconis, P., Holmes, S., & Montgomery, R. (2007). Dynamical bias in the coin toss. SIAM review, 49(2), 211-235. doi:10.1137/S0036144504446436 PDF here: https://statistics.stanford.edu/sites/default/files/2004-32.pdf Strzałko, J., Grabski, J., Stefański, A., Perlikowski, P., & Kapitaniak, T. (2008). Dynamics of coin tossing is predictable. Physics reports, 469(2), 59-92. doi:10.1016/j.physrep.2008.08.003 PDF here: http://www.math.hu-berlin.de/~synchron/web/publications/papers/PR2008.pdf -- --------------------------------------------------------------------------------- Jeffry Ricker, Ph.D. Professor of Psychology --------------------------------------------------------------------------------- Scottsdale Community College 9000 E. Chaparral Road Scottsdale, AZ 85256-2626 Office: SB-123 Phone: (480) 423-6213 Fax: (480) 423-6298 --- You are currently subscribed to tips as: hfo...@skidmore.edu<mailto:hfo...@skidmore.edu>. 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