Fair coins tend to land on the same side they started: Evidence from 350,757 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 Montgomery (DHM; 2007). The model asserts that when people flip an ordinary coin, it tends to land on the same side it started -- DHM 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, $\text{Pr}(\text{same side}) = 0.508$, 95% credible interval (CI) [$0.506$, $0.509$], $\text{BF}_{\text{same-side bias}} = 2359$. 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: $\text{Pr}(\text{heads}) = 0.500$, 95% CI [$0.498$, $0.502$], $\text{BF}_{\text{heads-tails bias}} = 0.182$. Furthermore, this lack of heads-tails bias does not appear to vary across coins. Additional exploratory analyses revealed that the within-people same-side bias decreased as more coins were flipped, an effect that is consistent with the possibility that practice makes people flip coins in a less wobbly fashion. 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 the DHM physics model of coin tossing.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2023
- DOI:
- 10.48550/arXiv.2310.04153
- arXiv:
- arXiv:2310.04153
- Bibcode:
- 2023arXiv231004153B
- Keywords:
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- Mathematics - History and Overview;
- Physics - Data Analysis;
- Statistics and Probability;
- Statistics - Other Statistics