When we have to randomly but fairly choose between two outcomes, we instinctively reach for the coin toss. It is because it is an article of faith that the two outcomes of heads and tails are equally likely. But the two sides of the coin are not identical, and hence that slight difference may make a difference in outcome probabilities. In fact, there are four possible forms of bias that may exist. It is possible that either heads or tails may come out on top slightly more frequently or that there is a same-side bias (i.e., the side that is on top when flipped is more likely to be on top when it falls) or an opposite side bias.
Hence this is really an empirical question that can be tested experimentally by tossing coins and counting the outcomes. We know that due to statistical fluctuations, any limited number of tosses are highly unlikely to result in exactly equal number of heads and tails even if the coin is completely unbiased. Statistically, the equal number result for a fair coin is a limit that is approached as the number of tosses gets larger and larger.
The Stanford statistician Persi Diaconis (who is also an expert magician) developed a physics model based on human coin tossing that predicted a same-side bias of 51%. A new study has tested his theory by flipping coins an incredible 350,757 times, a number that is large enough to draw fairly robust conclusions. The result seems to be in agreement with Diaconis’s model, with a slight bias in favor of coin falling with the same side up as it was when flipped. The result was 50.8% with 95% confidence limits of being between 50.6% and 50.9%.
The standard model of coin flipping was extended by Persi Diaconis 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 Diaconis 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’).
In order to get such a large number of tosses, 48 of the co-authors of the paper tossed coins of 46 different currencies and denominations to get the huge number of data points. One of them did as many as 20,100 flips while another (clearly a slacker) did just 701. There was considerable variation among the flippers, with the same-side bias ranging from 0.487 to 0.601.
This variability is consistent with Diaconis’ 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’.
So what are the implications for this, especially when it comes to betting?
Could future coin tossers use the same-side bias to their advantage? The magnitude of the observed bias can be illustrated using a betting scenario. If you bet a dollar on the outcome of a coin toss (i.e., paying 1 dollar to enter, and winning either 0 or 2 dollars depending on the outcome) and repeat the bet 1,000 times, knowing the starting position of the coin toss would earn you 19 dollars on average. This is more than the casino advantage for 6 deck blackjack against an optimal-strategy player, where the casino would make 5 dollars on a comparable bet, but less than the casino advantage for single-zero roulette, where the casino would make 27 dollars on average. These considerations lead us to suggest that when coin flips are used for high-stakes decision-making, the starting position of the coin is best concealed.
So there you are. Science news you can use for profit!