Fair coins tend to land on the side they started (2023)

Hi, I'm the first author of the manuscript, so I thought I could answer some of the questions and clarify some issues (all details are in the manuscript, but who has the time to read it ;)

Low RPM tosses: Most of the recordings are on crapy webcams with ~ 30FPS. The coin spin usually much faster than the sensor can record which results in often non-spinning-looking flips. Why did we take the videos in the first place? To check that everyone collected the data and to audit the results.

Building a flipping matching: The study is concerned with human coin flips. Diaconis, Holmes, and Montgomery's (DHM, 2007) paper theorize that the imperfection of human flips causes the same-side bias. Building a machine completely defeats the purpose of the experiment.

Many authors and wasted public funding: We did the experiment in our free time and we had no funding for the study = no money was wasted. Also, I don't understand why are so many people angry that students who contributed their free time and spent the whole day flipping coins with us were rewarded with co-authorship. The experiment would be impossible to do without them.

Improper tosses: Not everyone flips coin perfectly and some people are much worse at flipping than others. We instructed everyone to flip the coin as if they were to settle a bet and that the coin has to flip at least once (at least one flip would create bias for the opposite side). We find that for most people, the bias decreased over time which suggests that people might get better at flipping by practice = decrease the bias and it also discredits the theory that they learned how to be biased on purpose. From my own experience - I flipped coins more than 20,000 times and I have no clue how to bias it. Also, we did a couple of sensitivity analyses excluding outliers - the effect decreased a bit but we still found plentiful evidence for DHM.

If you doubt my stats background, you are more than welcome to re-analyze the data on your own. They are available on OSF: https://osf.io/mhvp7/ (including cleaning scripts etc).

Frantisek Bartos

Hi, thanks for replying. I have no complaints about your analysis, and agree that your results strongly support the D-H-M model (that there is a slight bias in coin-flipping over all and that it is caused by precession). However, it looks like about a third of your volunteers had little or no bias, presumably due to flipping end-over-end with no precession, and about a third had a lot of precession and a lot of bias.

Your paper draws the conclusion that coin-flipping inherently has a small-but-significant bias, but looking at table 2 it seems like an equally valid conclusion would be that some people flip a coin with no bias and others don't. Did you investigate this at all? In particular, I'd expect that if you took the biggest outliers, explained what precession is and asked them to intentionally minimize it, that the bias would shrink or disappear.

The first thing I looked for was how high was the flip and did it land on a hard or soft surface. Neither seemed to be mentioned in the paper.

From the one video I looked at, the flip seems to be a few feet high at most, and land back in the hand.

> In each sequence, people randomly (or according to an algorithm) selected a starting position (heads-up or tails-up) of the first coin flip, flipped the coin, caught it in their hand, recorded the landing position of the coin (heads-up or tails-up), and proceeded with flipping the coin starting from the same side it landed in the previous trial (we decided for this “autocorrelated” procedure as it simplified recording of the outcomes). (p.3)

Wrt to the height, that naturaly varied among people and flips and we did not measure it.

Yes, there is indeed a lot of heterogeneity in the bias between flippers and we are going to put more emphasis on it in an upcoming revision. However, it's hard to tell whether there are two groups or a continuous scale of increasing bias. From our examination of the data, and continuum seem to be the more likely case, but we would need many many more people flipping a lot of coins to test this properly.

Yes, training the most wobbly flippers sounds like a very interesting idea. It might indeed answer additional questions but it's not really something I wanna run more studies on :)

The NFL still flips coins professionally. I wonder if they have better-than-webcam footage of each flip. Somewhere out there a bookie might be very interested in any potential bias.

That makes me wonder whether any bookmakers or sports betting arbitration shops have ever internally ran a study like this.

With how much money there is in sports betting, it could potentially be somewhat lucrative, though I wouldn't be surprised if the bias doesn't actually end up mattering that much in practice.

IIRC, past studies have suggested that letting the coin land, rather than catching it, reduces or eliminates the bias.

Understandable, but I guess it's hard to put much weight on this data given how easy it is to introduce the effect being studied intentionally. Were the subjects aware of D-H-M beforehand? I wasn't before today, but I've been able to fake a coin flip with precession for many years (a very useful skill for parents of two small children) and if I was participating in a study like this I would be pretty hyper-aware of how much "sideways" I was giving it.

Re: Low FPS webcam - here's an approach that attempts to analyze coin tossing data from the _sound_ rather than the _video_, since sound is typically recorded at a much higher sampling rate (high enough to "hear" the spinning of the coin). https://cs.stanford.edu/~kach/can-one-hear-the-fate-of-a-coi...

Some hi-res footage of an NFL toss: https://youtu.be/-sjAyQcv0oM

How do you control against the prospect of your coin flippers being biased in terms of the videos people choose to upload?

I have a question about the ethics of this study.

Were you not concerned that a study that shows a bias in coin flipping would undermine the trust people have in this simple method settling arguments, leading to even more arguments between people, possibly fights and injuries, in situations where a coin flip would have settled an existing argument?

Thank you.

PS: This isn't supposed to to be a serious question, if anyone has doubts. :)

We did not. However, we find it highly unlikely since everyone was incentivised to upload as much as possible, and the number of coin flips determined the order of the manuscript. Also, we did some basic analyses to check irregularities in the uploaded sequences, and we did not find any issues.

Couldn't a bit of a Benford's Law curve be at work with the lesser flippers? Assuming a minimum full flip, results begin with:

1.0 flip, lands on side it started

1.5 flips, lands on opposite side

2.0 flips, lands on side it started

etc