Fair coins tend to land on the same side they started: Evidence from 350,757 flips

2023-10-06

AI Summary

This study empirically tests the Diaconis–Holmes–Montgomery physical model’s prediction of the “same-side bias”—the tendency for a tossed coin to land with its initial upward-facing side up. Method: Analyzing 350,757 manually executed coin tosses across multiple participants, the study employs Bayesian inference, hierarchical modeling of inter- and intra-subject variability, and large-sample frequentist analysis. Contribution/Results: It provides the first ultra-large-sample confirmation of a robust same-side bias: same-side probability = 0.508 (95% CI [0.506, 0.509]; Bayes factor = 2359). The study further reveals significant individual differences and a practice effect—bias diminishes with repeated tossing—and falsifies the hypothesis of inherent coin asymmetry (i.e., Pr(heads) ≠ 0.500 under fair conditions). These findings advance stochastic modeling of human motor behavior, offering both novel empirical evidence and a methodological framework integrating Bayesian and frequentist approaches for studying action-induced randomness.

Abstract

Many people have flipped coins but few have stopped to ponder the statistical and physical intricacies of the process. 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, $ ext{Pr}( ext{same side}) = 0.508$, 95% credible interval (CI) [$0.506$, $0.509$], $ ext{BF}_{ ext{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: $ ext{Pr}( ext{heads}) = 0.500$, 95% CI [$0.498$, $0.502$], $ ext{BF}_{ ext{heads-tails bias}} = 0.182$. Furthermore, this lack of heads-tails bias does not appear to vary across coins. Additional 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.

Problem

• Tests if coin flips land same-side up more often
• Investigates human variation in same-side bias
• Examines practice effect on reducing same-side bias

Innovation

• Large-scale coin flip data collection
• Physics model validation via statistics
• Analysis of same-side bias variation