https://www.stat.berkeley.edu/~aldous/Real-World/coin_tosses.html

   40,000 coin tosses yield ambiguous evidence for dynamical bias

Background

The 2007 Diaconis - Holmes - Montgomery paper Dynamical bias in the
coin toss suggests that in coin-tossing there is a particular
``dynamical bias" that causes a coin to be slightly more likely to
land the same way up as it started. In brief, whether the coin lands
the same way up as it started depends deterministically on the
initial parameters of motion imparted at the instant of tossing. Each
person's individual "tossing style" gives some probability
distribution on the initial parameters, but (unless the spread is
unrealistically small) it turns out from a careful analysis of the
physics that the resulting overall probability always works out to be
1/2 or greater, though it would presumably vary from person to
person. The basic reason is that, instead of rotating around a
horizontal axis as one might imagine, a typical tossed coin is
rotating around a tilted axis which is precessing in 3-space, and
this entails a certain degree of ``memory" of the initial parameters.
Combining theory with data on initial parameters from a small number
of tosses obtained via high speed photography, Diaconis et al gave a
rough estimate of a 0.8% bias (i.e. a 50.8% chance of landing same
way up as started) for a typical tosser, and discuss a number of
possible caveats to the theory. It is important to distinguish this
subtle 3-dimensional effect ("precession bias"), which persists when
the number of rotations is large, from a more obvious 2-dimensional
bias when the number of rotations is small ("few rotations bias" -
see below).

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

Over the Spring 2009 semester two Berkeley undergraduates, Priscilla
Ku and Janet Larwood, undertook to do the required 40,000 tosses.
After preliminary experimentation with practical issues, there was
formulated a specific protocol, described in detail below. Cutting to
the chase, here is the complete data-set as a .xlsx spreadsheet (see
sheet 2). This constitutes a potentially interesting data-set in many
ways -- one could compare numerous theoretical predictions about pure
randomness (lengths of runs, for instance) with this empirical data.
For the specific question of dynamical bias, the relevant data can be
stated very concisely:

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

A first comment is that it would have been better for each individual
to have done both "Heads up"and "Tails up" tosses (which was part of
the intended protocol, but on this aspect of the protocol there was a
miscommunication); this would separate the effect of individual
tossing style from any possible effect arising from the physical
difference between Heads and Tails. But it is very hard to imagine
any such physical effect, so we presume the observed difference (if
real rather than just chance variation) is due to some aspect of
different individual tossing style.

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 %

We leave the statistically literate reader to draw their own
conclusions. A caveat is that the experiment did not use "iconic
tosses" (see below), and we can't really distinguish the possible
precession bias from the possible "few rotations" bias, even though
there was no visual indication of systematic difference between the
two tossing styles.

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

We visualize an "iconic toss" done standing; the coin moves roughly
vertically up, rising a height of 2 or 3 feet, spinning rapidly, and
is caught in the open hand at around the level it was tossed.

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

To avoid tiredness when tossing standing up, the participants sat on
the floor. One person did a long sequence of tosses (all starting the
same way up) while the other recorded the result directly onto the
spreadsheet. Tosses where the coin was dropped were disregarded.
Dates, times and person tossing were also recorded on the
spreadsheet. The coin used was an ordinary dime. Visually, the tosses
were typically rather low (maybe 18 inches high), rotating moderately
fast, and angled rather than purely vertical.

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