coin flips

FAIR COINS TEND TO LAND ON THE SAME SIDE THEY STARTED:
EVIDENCE FROM 350,757 FLIPS
František Bartoš1∗, Alexandra Sarafoglou1, Henrik R. Godmann1, Amir Sahrani1, David Klein Leunk2,
Pierre Y. Gui2, David Voss2, Kaleem Ullah2, Malte J. Zoubek3, Franziska Nippold,
Frederik Aust1, Felipe F. Vieira4, Chris-Gabriel Islam5,6, Anton J. Zoubek7, Sara Shabani8,
Jonas Petter1, Ingeborg B. Roos9, Adam Finnemann1,10, Aaron B. Lob2,11, Madlen F. Hoffstadt1,
Jason Nak2, Jill de Ron2, Koen Derks12, Karoline Huth2,13, Sjoerd Terpstra14,
Thomas Bastelica15,16, Magda Matetovici2,17, Vincent L. Ott2, Andreea S. Zetea2, Katharina Karnbach2,
Michelle C. Donzallaz1, Arne John2, Roy M. Moore2, Franziska Assion1,8, Riet van Bork2,
Theresa E. Leidinger2, Xiaochang Zhao2, Adrian Karami Motaghi2, Ting Pan1,9, Hannah Armstrong2,
Tianqi Peng2, Mara Bialas20, Joyce Y.-C. Pang2, Bohan Fu2, Shujun Yang2,
Xiaoyi Lin2, Dana Sleiffer2, Miklos Bognar21, Balazs Aczel22, and Eric-Jan Wagenmakers1
1Department of Psychological Methods, University of Amsterdam,
2Department of Psychology, University of Amsterdam,
3Institute of Economic Law, University of Kassel,
4Faculty of Psychology and Educational Sciences, KU Leuven,
5Chairs of Statistics and Econometrics, Georg August University of Göttingen,
6Centre for Environmental Sciences, Hasselt University,
7Gutenberg School Wiesbaden,
8Department of Psychology, Justus Liebig University,
9Department of Adult Psychiatry, Amsterdam University Medical Centre,
10Centre for Urban Mental Health, University of Amsterdam,
11Cognitive and Behavioral Decision Research, University of Zurich,
12Center of Accounting, Auditing & Control, Nyenrode Business University,
13Department of Psychiatry, Amsterdam UMC, University of Amsterdam,
14Institute for Marine and Atmospheric Research Utrecht, Utrecht University,
15Centre Hospitalier Le Vinatier,
16INSERM, U1028; CNRS, UMR5292, Lyon Neuroscience Research Center,
Psychiatric Disorders: from Resistance to Response Team,
17Research Institute of Child Development and Education, University of Amsterdam,
18Faculty of Behavioral and Movement Sciences, Vrije Universiteit Amsterdam,
19Department of Develomental Psychology, University of Amsterdam,
20Behavioural Science Institute, Radboud University,
21Doctoral School of Psychology, ELTE Eotvos Lorand University,
22Institute of Psychology, ELTE Eotvos Lorand University
*Correspondence concerning this article should be addressed to František Bartoš at [email protected].
For all but the last four positions, the authorship order aligns with the number of coin flips contributed.
arXiv:2310.04153v2 [math.HO] 10 Oct 2023

350,757 Coin Flips
ABSTRACT
Many people have flipped coins but few have stopped to ponder the statistical and physical intricacies
of the process. In a preregistered study we collected 350,757 coin flips to test the counterintuitive
prediction from a physics model of human coin tossing developed by Diaconis, Holmes, and Mont-
gomery (D-H-M; 2007). The model asserts that when people flip an ordinary coin, it tends to land on
the same side it started—D-H-M estimated the probability of a same-side outcome to be about 51%.
Our data lend strong support to this precise prediction: the coins landed on the same side more often
than not, Pr(same side)=0.508 , 95% credible interval (CI) [ 0.506 , 0.509 ], BFsame-side bias = 2364 .
Furthermore, the data revealed considerable between-people variation in the degree of this same-side
bias. Our data also confirmed the generic prediction that when people flip an ordinary coin—with the
initial side-up randomly determined—it is equally likely to land heads or tails: Pr(heads)=0.500 ,
95% CI [ 0.498 , 0.502 ], BFheads-tails bias = 0.183 . Furthermore, this lack of heads-tails bias does not
appear to vary across coins. Our data therefore provide strong evidence that when some (but not
all) people flip a fair coin, it tends to land on the same side it started. Our data provide compelling
statistical support for D-H-M physics model of coin tossing.
Introduction
A coin flip—the act of spinning a coin into the air with your thumb and then catching it in your hand—is often
considered the epitome of a chance event. It features as a ubiquitous example in textbooks on probability theory and
statistics [ 1 , 2 , 3 , 4 , 5 ] and constituted a game of chance (‘capita aut navia’ – ‘heads or ships’) already in Roman times
([6], 1.7:22).
The simplicity and perceived fairness of a coin flip, coupled with the widespread availability of coins, may explain why
it is often used to make even high-stakes decisions. For example, in 1903 a coin flip was used to determine which of the
Wright brothers would attempt the first flight; in 1959, a coin flip decided who would get the last plane seat for the
tour of rock star Buddy Holly (which crashed and left no survivors); in 1968, a coin flip determined the winner of the
European Championship semi-final soccer match between Italy and the Soviet Union (an event which Italy went on to
win); in 2003, a coin toss decided which of two companies would be awarded a public project in Toronto; and in 2004
and 2013, a coin flip was used to break the tie in local political elections in the Philippines.
Despite the widespread popularity of coin flipping, few people pause to reflect on the notion that the outcome of a
coin flip is anything but random: a coin flip obeys the laws of Newtonian physics in a relatively transparent manner
[ 3 ]. According to the standard model of coin flipping [ 7 , 8 , 9 , 10 ], the flip is a deterministic process and the perceived
randomness originates from small fluctuations in the initial conditions (regarding starting position, configuration,
upward force, and angular momentum) combined with narrow boundaries on the outcome space. Therefore the standard
model predicts that when people flip a fair coin, the probability of it landing heads is 50% (i.e., there is no ‘heads-tails
bias’; conversely, if one side of a coin would land on one side more often than the other, we would say there is a
‘heads-tails bias’).1
The standard model of coin flipping was extended by Diaconis, Holmes, and Montgomery (D-H-M; [ 12 ]) who proposed
that when people flip a ordinary coin, they introduce a small degree of ‘precession’ or wobble—a change in the direction
of the axis of rotation throughout the coin’s trajectory. According to the D-H-M model, precession causes the coin to
spend more time in the air with the initial side facing up. Consequently, the coin has a higher chance of landing on the
same side as it started (i.e., ‘same-side bias’). Based on a modest number of empirical observations (featuring coins
with ribbons attached and high-frame-rate video recordings) D-H-M [ 12 ] measured the off-axis rotations in typical
human flips. Based on these observations, the D-H-M model predicted that a coin flip should land on the same side as it
started with a probability of approximately 51%, just a fraction higher than chance.
Throughout history, several researchers have collected thousands of coin flips. In the 18
th
century, the famed naturalist
Count de Buffon collected 2,048 uninterrupted sequences of ‘heads’ in what is possibly the first statistical experiment
ever conducted [ 13 ]. In the 19
th
century, the statistician Karl Pearson flipped a coin 24,000 times to obtain 12,012 tails
[ 14 ]. And in the 20
th
century, the mathematician John Kerrich flipped a coin 10,000 times for a total of 5,067 heads
while interned in Nazi-occupied Denmark [ 15 ]. These experiments do not allow a test of the D-H-M model, however,
mostly because it was not recorded whether the coin landed on the same side that it started. A notable exception is
a sequence of 40,000 coin flips collected by Janet Larwood and Priscilla Ku in 2009 [ 16 ]: Larwood always started
1
Some even assert that a biased coin is a statistical unicorn—everyone talks about it but no one has actually encountered one [ 5 ].
Physics models support this assertion as long as the coin is not bent [11] or allowed to spin on the ground [3, 4].
2

350,757 Coin Flips
the flips heads-up, and Ku always tails-up. Unfortunately, the results (i.e., 10,231/20,000 heads by Larwood and
10,014/20,000 tails by Ku) do not provide compelling evidence for or against the D-H-M hypothesis.
In order to provide a diagnostic empirical test of the same-side bias hypothesized by D-H-M, we collected a total
of 350,757 coin flips, a number that –to the best of our knowledge– dwarfs all previous efforts. To anticipate our
main results, the data reveal overwhelming statistical evidence for the presence of same-side bias (and for individual
differences in the extent of this bias). Furthermore, the data suggested moderate evidence for the absence of a heads-tails
bias.
Results
A group of 48 people (i.e., all but three of the co-authors) tossed coins of 46 different currencies × denominations
and obtained a total number of 350,757 coin flips. Raw data, video recordings of the coin flips, and the preregistered
analysis plan can be found at https://osf.io/pxu6r/ . The data confirm the prediction from the D-H-M model:
the coins landed how they started more often than 50%. Specifically, the data feature 178,078 same-side landings
from 350,757 tosses, Pr(same side) = 0.508 , 95% CI [ 0.506 , 0.509 ] (see Table 1 for a by-person summary), which is
remarkably close to D-H-M prediction of (approximately) 51%.
In addition, the data show no trace of a heads-tails bias. Specifically, we obtained 175,420 heads out of 350,757 tosses,
Pr(heads) = 0.500, 95% CI [0.498,0.502] (see Table 2 in the methods section for a by-coin summary).
A preregistered Bayesian informed binomial hypothesis test
2
indicates extreme evidence in favor of the same-side bias
predicted by the D-H-M model, BFsame-side bias = 1.71 ×1017 . A similar (not-preregistered) analysis yields moderate
evidence against the presence of a heads-tails bias, BFheads-tails bias = 0.168.
With the data in hand we realized that the same-side bias was possibly subject to considerable between-people
heterogeneity. Therefore we specified a more complex Bayesian hierarchical model that includes both heterogeneity
in same-side bias between people and heterogeneity in heads-tails bias between coins; this hierarchical model was
then used to estimate the parameters and to test the hypotheses using Bayesian model-averaging and inclusion Bayes
factors (for details see the methods section). These analyses were not preregistered. The posterior distribution of the
same-side bias is slightly wider than in the simple preregistered analysis, Pr(same side)=0.5098 , 95% CI [ 0.5049 ,
0.5147 ], which is caused by the substantial between-people heterogeneity in the probability of the coin landing on the
same side, sdpeople(psame-side) = 0.0156 , 95% CI [ 0.0119 , 0.0200 ]. The additional uncertainty lowers the evidence for
the same-side bias to BFsame- side bias = 2364 , which however remains extreme (e.g., when the hypothesis of a same-side
bias has a prior probability of 0.50 , a Bayes factor of about 2364 results in a posterior probability of 0.9996 ). Consistent
with the visual impression from Figure 1, the hierarchical model reveals overwhelming evidence for the presence of
between-people heterogeneity in same-side bias, BFpeople heterogeneity = 3.04 ×1024.
The posterior estimates of the overall probability of heads remains practically unchanged, Pr(heads)=0.5005 , 95%
CI [ 0.4986 , 0.5026 ] with virtually no between-coin heterogeneity, sdcoins(Pr(heads)) = 0.0019 , 95% CI [ 0.0002 ,
0.0048 ]. The evidence against the presence of heads-tails bias also remains practically unchanged, BFheads-tails bias =
0.183 . The model shows moderate evidence against the presence of between-coin heterogeneity in heads-tails bias,
BFcoin heterogeneity = 0.179.
We repeated the statistical analyses after excluding four potential outliers with same-side sample proportions larger
than 53% (i.e., the four largest and right-most estimates in the top panel of Figure 1) but this does not qualitatively
affect the conclusion. The posterior distribution of the same-side bias and the between-people heterogeneity in
the probability of the coin landing on the same-side bias decreases, Pr(same side) = 0.5060 , 95% CI [ 0.5031 ,
0.5089 ] and sdpeople(psame side) = 0.0072 , 95% CI [ 0.0049 , 0.0099 ], whereas the posterior distribution of the overall
probability of heads and the between-coin heterogeneity in the probability of heads remains practically unchanged,
Pr(heads)=0.5008 , 95% CI [ 0.4988 , 0.5031 ] and sdcoins(Pr(heads)) = 0.0021 , 95% CI [ 0.0002 , 0.0051 ]. The
evidence for the same-side bias and the between-people heterogeneity in the same-side bias decreases but remains
extreme, BFsame-side bias = 793 and BFpeople heterogeneity = 2.84 ×107 , whereas the evidence against the heads-tails bias
and against between-coin heterogeneity in heads-tails bias remains practically unchanged, BFheads-tails bias = 0.213 and
BFcoin heterogeneity = 0.222.
2The preregistration can be found at https://osf.io/cf6nw.
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350,757 Coin Flips
Table 1: By-person summary of the same-side bias.
Person Same side Flips Coins Proportion [95% CI] BF10
XiaoyiL 780 1600 2 0.487[0.463,0.512] 0.361
JoyceYCP 1126 2300 3 0.490[0.469,0.510] 0.292
AndreeaSZ 2204 4477 4 0.492[0.478,0.507] 0.179
KaleemU 7056 14324 8 0.493[0.484,0.501] 0.042
FelipeFV 4957 10015 3 0.495[0.485,0.505] 0.095
ArneJ 1937 3900 4 0.497[0.481,0.512] 0.325
AmirS 7458 15012 6 0.497[0.489,0.505] 0.081
ChrisGI 4971 10005 5 0.497[0.487,0.507] 0.130
FrederikA 5219 10500 5 0.497[0.487,0.507] 0.128
FranziskaN 5368 10757 3 0.499[0.490,0.509] 0.186
RietvanB 1801 3600 4 0.500[0.484,0.517] 0.510
JasonN 3352 6700 7 0.500[0.488,0.512] 0.351
PierreYG 7506 15000 9 0.500[0.492,0.508] 0.196
KarolineH 2761 5500 5 0.502[0.489,0.515] 0.525
SjoerdT 2510 5000 5 0.502[0.488,0.516] 0.547
SaraS 5022 10000 3 0.502[0.492,0.512] 0.415
HenrikRG 8649 17182 8 0.503[0.496,0.511] 0.513
IrmaT 353 701 1 0.504[0.466,0.541] 0.931
KatharinaK 2220 4400 5 0.505[0.490,0.519] 0.833
JillR 3261 6463 2 0.505[0.492,0.517] 0.810
FrantisekB 10148 20100 11 0.505[0.498,0.512] 1.068
IngeborgBR 4340 8596 1 0.505[0.494,0.516] 0.872
VincentLO 2475 4900 5 0.505[0.491,0.519] 0.905
EricJW 2071 4100 5 0.505[0.490,0.521] 0.911
MalteZ 5559 11000 7 0.505[0.496,0.515] 1.046
TheresaEL 1769 3500 4 0.505[0.489,0.522] 0.952
DavidV 7586 14999 5 0.506[0.498,0.514] 1.402
AntonJZ 5069 10004 2 0.507[0.497,0.517] 1.577
MagdaM 2510 4944 6 0.508[0.494,0.522] 1.432
ThomasB 2540 5000 5 0.508[0.494,0.522] 1.526
BohanF 1118 2200 3 0.508[0.487,0.529] 1.209
JonasP 5080 9996 5 0.508[0.498,0.518] 2.677
HannahA 1525 3000 4 0.508[0.490,0.526] 1.325
AdrianKM 1749 3400 3 0.514[0.497,0.531] 3.272
AaronBL 3815 7400 5 0.516[0.504,0.527] 21.31
KoenD 3309 6400 7 0.517[0.505,0.529] 22.26
MichelleCD 2224 4300 5 0.517[0.502,0.532] 7.982
RoyMM 2020 3900 4 0.518[0.502,0.534] 7.501
TingP 1658 3200 4 0.518[0.501,0.536] 5.280
MaraB 1426 2750 3 0.519[0.500,0.537] 4.370
AdamF 4334 8328 2 0.520[0.510,0.531] 291.9
AlexandraS 9080 17434 8 0.521[0.513,0.528] 5.14 ×105
MadlenFH 3705 7098 1 0.522[0.510,0.534] 226.3
DavidKL 7895 15000 1 0.526[0.518,0.534] 2.87 ×107
XiaochangZ 1869 3481 4 0.537[0.520,0.554] 276.2
FranziskaA 2055 3800 4 0.541[0.525,0.557] 1443
JanY 956 1691 2 0.565[0.541,0.589] 249.9
TianqiP 1682 2800 3 0.601[0.582,0.619] 9.89 ×108
Combined 178078 350757 46 0.508 [0.506,0.509]1.71 ×1017
Note. ‘Proportion’ refers to the observed proportion of coin flips that landed same-side, BF10 corresponds to a Bayesian
binomial test of the same side proportion with H1:π∼Beta(5100,4900)[0.5,1] vs. H0:π= 0.5.
4

350,757 Coin Flips
0.40
0.45
0.50
0.55
0.60
Pr(landing on same side)
chance
D−H−M
Observed probability
Hierarchical mean estimate and 95% CI
0.46 0.48 0.50 0.52 0.54
0
50
100
150
200
250
Density
Pr(landing on same side)
chance D−H−M
0.00 0.01 0.02 0.03
0
50
100
150
200
250
Between tosser standard deviation
Prior
Posterior
Figure 1: Coins have a tendency to land on the same side they started, confirming the predictions from the Diaconis,
Holmes, and Montgomery (D-H-M) model of coin flipping. Top panel: posterior estimates of the same-side bias
separately for each coin tosser, as obtained from the hierarchical Bayesian model with weakly informative, estimation-
tailored prior distributions described in the methods section; Bottom-left panel: prior and posterior distributions for the
overall same-side bias; Bottom-right panel: prior and posterior distributions for the between-tosser heterogeneity in
same-side bias.
Discussion
We collected 350,757 coin flips and found strong empirical confirmation of the counterintuitive and precise prediction
from D-H-M model of human coin tossing: when people flip a coin, it tends to land on the same side as it started.
Moreover, the data revealed a substantial degree of between-people variability in same-side bias: as can be seen from
Figure 1, some people appear to have little or no same-side bias, whereas others do display a same-side bias, albeit
to a varying degree. This variability is consistent with D-H-M model, in which the same-side bias originates from
off-axis rotations (i.e., precession or wobbliness), which can reasonably be assumed to vary between people. Future
work may attempt to verify whether ‘wobbly tossers’ show a more pronounced same-side bias than ‘stable tossers’. The
effort required to test this more detailed hypothesis appears to be excessive, as it would involve detailed analyses of
high-speed camera recordings for individual flips (cf. [12]).
In order to ensure the quality of the data, we videotaped and audited the data collection procedure (see the method
section for details). However, there remains a legitimate concern: at the time when people were flipping the coins they
were aware of the main hypothesis under test. Therefore it cannot be excluded that some of the participants were able
to manipulate the coin flip outcomes in order to produce the same-side bias. In light of the nature of the coin tossing
5