fbartos

The coins are as likely to fall on heads as on tails, so they are not biased for any of the sides.

People flipping them however introduce bias due to the wobble of their tosses.

Paper showing that "Fair Coins Tend to Land on the Same Side They Started: Evidence from 350,757 Flips" circulating here a couple of years back (https://news.ycombinator.com/item?id=42181345) is now published in the Journal of the American Statistical Association.

The main difference from the original version is that we now document a decrease in the same-side bias over time. Meaning the more people flip, the less biased they are. We guess that it's a practice effect -- they might be getting better at coin flipping over time.

(I'm the main author of the paper)

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.

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 :)

> 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.

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

Yes, I mentioned that in the original paper but it got eddited for some reason.

It's not about the number of rotations at all. I doubt that you can control it at all even after dozens of hours coin flipping (I did more than 20h and I can't eveb guess how many rotations the coin made) Diaconis, Holmes, and Montgomery (2007) proposed a physical model of coin flipping that introduces the bias as a result of wobblines (i.e., off-axis rotation in the flips).

Diaconis, P., Holmes, S., & Montgomery, R. (2007). Dynamical bias in the coin toss. SIAM Review, 49(2), 211-235. https://doi.org/10.1137/S0036144504446436

That's actually not completelly accurate. The study protocol (https://osf.io/hkv8p) describes the procedure in greater detail.

People were pressing one button for heads and another button for heads (which we deemend less error prone and less likely to be subcontiously influenced). The trick was that the next coin flip started the same side-up as the previous landed. Therefore there was no need to record the start (and we randomized the starting position of every 100th flip)

We also did some auditing of the video recordings (trying to decode the outcomes from the videos) and they showed quite consistent degree of bias as the original responses.

I personally did 20,100 flips and I can assure you I have no clue how to control the flip. I centrally got much better at flipping and catching the coin in hand without dropping it---which takes some practice on its own.

(I know that there are techniques for adding the wobble to the toss, but I didn't study them and I have no clue how to do them. I think it is safe to say you don't discover them intuitevelly.)

There was indeed a lot of variation in the height of the tosses. I however disagree with the conclussion: two of my friends at the video had the most different height of tosses (one tossed thrice as hight as the other one), yet both of them had exactly the same bias (0.505). The amount of spin is unfortunatelly very misleading from the 30fps videos--the coins often seem like not spinning at all but that's just a result of the poor video quality.

You are absolutely right :)

We told people that the coin has to flip at least once (which would bias it for the opposite site). Whenever instructing people, I tried to explaining that the coin flip should look like you were trying to determine an outcome of a bet. You can find the complete experimental protocol here: https://osf.io/hkv8p

Also, I wish I had (any) budget to hire proffesional skilled tossers haha.

Not neccessarily because spinning and bouncing coins are often much more biased then flipped coins. (Unequal weight distribution on the side can bias a spinned coin while it doesn not bias a flipped coin. There are a couple of studies on it too.)

About a year ago, we embarked on a quest to answer one of the most intriguing questions:

If you flip a fair coin and catch it in hand, what's the probability it lands on the same side it started?

Today, we are finally ready to share the results. Thanks to my friends, collaborators, and even strangers from the internet, we collected flippin 350,757 coin flips. We ran several "Coin Tossing Marathons" (e.g., https://youtu.be/3xNg51mv-fk?si=o2E3hKa-ReXodOmc) and spent countless hours flipping coins.

In short, we found overwhelming evidence for a "same-side" bias predicted by Diaconis, Holmes, and Montgomery 2007: If you start heads-up, the coin is more likely to land heads-up and vice versa. How large is the bias? In our sample, the mean estimate is 50.8%, CI [50.6%, 50.9%].

We also found considerable variance in the same-side bias between our 48 tossers. The bias varied with a standard deviation of 1.6%, CI [1.2%, 2.0%], in our sample. The variation could be explained by a different degree of "wobbliness" between our tossers.

If you bet a dollar on the outcome of a coin toss 1000 times, knowing the starting position of the coin toss would earn you 19$ on average. This is more than the casino advantage for 6-deck blackjack against an optimal player (5$) but less than that for single-zero roulette (27$).

The manuscript is at arXiv: https://arxiv.org/abs/2310.04153 And the open data, code, and video recordings at OSF: https://osf.io/pxu6r/.

Diaconis, P., Holmes, S., & Montgomery, R. (2007). Dynamical bias in the coin toss. SIAM Review, 49(2), 211-235. https://doi.org/10.1137/S0036144504446436

Hi all,

I'm one of the authors of the reply and it was very interesting reading so many diverse thoughts and comments. I would love to respond to all of them, but it would take ages. Luckily, Stuart Ritchie (@StuartJRitchie) wrote an awesome post on his substack (https://stuartritchie.substack.com/p/nudge-meta) that goes much deeper and adresses many questions and the fair critique raised here.

Also, note that there is only a limited amount of information and nuance you can comprise into a strict 500 words reply limit in PNAS, which is the reason we focus only on one aspect of the original meta-analysis -- publication bias.

Cheers, Frantisek