Of course, Persi Diaconis was noted for being about to flip a coin so it would come up whichever way he wanted. And there’s the issue of the variability of human behavior: is there a difference between truly random (undetermined) and highly variable beyond our practical ability to predict?
On Feb 28, 2015, at 4:44 PM, Jeffry Ricker, Ph.D. <[email protected]> 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. Paul Brandon Emeritus Professor of Psychology Minnesota State University, Mankato [email protected] --- You are currently subscribed to tips as: [email protected]. To unsubscribe click here: http://fsulist.frostburg.edu/u?id=13090.68da6e6e5325aa33287ff385b70df5d5&n=T&l=tips&o=42374 or send a blank email to leave-42374-13090.68da6e6e5325aa33287ff385b70df...@fsulist.frostburg.edu
