40,000 coin tosses yield ambiguous evidence for dynamical bias

Background

However, no experiment with actual coin-tosses has been done to investigate whether the predicted effect is empirically observed. Diaconis et al noted, correctly, that to estimate the probability with a S.E. of 0.1% would require 250,000 tosses, but this seems unnecessarily precise. Let's work with numbers of tosses rather than percents. With 40,000 tosses the S.E. for ``number landing same way" equals 100, and the means are 20,000 under the unbiased null and 20,320 under the "0.8% bias" alternative. So, if the alternative were true, it's quite likely one would see a highly statistically significant difference between the observed number and the 20,000 predicted by the null.

And 40,000 tosses works out to take about 1 hour per day for a semester .........

The experiment

of 20,000 Heads-up tosses (tossed by Janet) 10231 landed Heads

of 20,000 Tails-up tosses (tossed by Priscilla) 10014 landed Tails

Analysis

Applying textbook statistics:

testing the "unbiased" null hypothesis with the combined data, we get z = 2.45 and a (1-sided) p-value < 1%

assuming dynamical bias with possibly different individual biases, and testing the null hypothesis that these two individuals have the same bias, we get z = 2.17 and a (2-sided) p-value = 3 %

Finally, for anyone contemplating repeating the experiment, we suggest getting a larger group of people to each make 20,000 iconic tosses, for two reasons. Studying to what extent different people might have different biases is arguably a richer question that asking about overall existence of dynamical bias. And if the "few rotations bias" exists then we would see it operating in both directions for different people, whereas the predicted "precession bias' is always positive.

Iconic tosses and the few rotations bias

The obvious elementary analysis of coin tossing is that a coin lands "same way up" or "opposite way up" according to whether the number r of full rotations (r real, because a rotation may be incomplete) is in [n - 1/4, n+1/4] or in [n + 1/4, n+3/4] for some integer n. When the random r for a particular individual has large spread we expect these chances to average out to be very close to 1/2; but when r has small spread, in particular when its mean \mu is not large, one expects a "few rotations bias" toward "same way up" if \mu is close to an integer, or toward "opposite way up" if \mu is close to a half integer.

Detailed protocol

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