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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

There's a nice presentation of the paper here https://www.youtube.com/watch?v=-QjgvbvFoQA

In essence the effect comes from "precession" - the tendency of the flip to not be purely vertical but to have some wobble/angular momentum which causes it to flip in such a way as to spend longer on one side than the other. Depending on the technique this will have a greater or lesser effect on the fairness of the coin toss, ranging from about p_same = 0.508 for the best technique to one person in the study actually exhibiting 0.6 over a large sample which is staggeringly unlikely if the toss was purely fair. In the extreme, it shows in the video a magician doing a trick toss using precession that looks as if it's flipping but does not in fact change sides at all, purely rotating in the plane of the coin and wobbling a bit.

The video is quite a nice one for setting out how hypothesis testing works.

The paper looks like it has a large sample size, but it actually has a sample size of only 48 testers/flippers. Some of the videos of those testers show very low, low-rpm coin tosses, we're talking only 1-2 flips. Where they also flipped thousands of times, presumably in the same way. So there is actually a very small sample size in the study (N = 48), where testers that don't flip properly (low rpm, low height, few coin rotations) can affect the results disproportionately.

Doesn't look like the study author backgrounds are particularly focused on statistics. I would presume with 48 authors (all but 3 of which flipped coins for the study), the role of some might have been more test subject than author. And isn't being the subject in your own study going to introduce some bias? Surely if you're trying to prove to yourself that the coins land on one side or another given some factor, you will learn the technique to do it, especially if you are doing a low-rpm, low flip. Based on the study results, some of the flippers appear to have learned this quite well.

If the flippers (authors) had been convinced of the opposite (fair coins tend to land on the opposite side from which they started) and done the same study, I bet they could have collected data and written a paper with the results proving that outcome.

I wouldn't be surprised if there is something to it, but I suspected they didn't use legitimate coin flips (because it seems like a large amount of people can't really flip a coin), and looking at the videos confirms it, at least for the flips done by Bartos:

https://osf.io/6a5hy/

They're very low RPM and very low time in the air. Nothing I would accept for any decision worth flipping a coin for.

This paper is also this year's Ig Nobel Prize winner:

> Probability: A team of 50 researchers, for performing 350,757 experiments to show that when a coin is flipped, it is slightly more likely to land on the same side as it started.

source: https://en.wikipedia.org/wiki/List_of_Ig_Nobel_Prize_winners...

This has been commonly known by magicians for decades. I doubt that any single magician had conducted 350k flips, but I know I personally did ~2,500 to test the effect when I was a kid.

And I'm sure if you got 30 magicians together to pool data we'd have a meta-analysis of about this size but with experiments a century ago

Not totally relevant, but I once discovered it's pretty easy to cheat a coin toss, at least with an Australian 20c coin. Flip the coin, catch in your hand, and in the process of transferring it to the back of your other hand, feel which way up it is, and optionally flip it.

With our coins, the head (the Queen's face at the time) is pretty distinct with a large smooth area, compared to the rough feel of the platypus and water.

So if ever you're flipping for anything that matters, make sure the coin lands directly on the ground.

What I’ve learnt from this thread is that the problem with fair coin flips is not if they’re fair it’s whether we count them as a proper coin flips. And so who gets to decide?

And if most people aren’t flipping like that then should we design a machine that flips the coins? And we try to control other factors as well? Or is a human—their imperfections included—flipping the coins inherently important to the idea of coin flipping, statistics and randomness?

I learned a trick with flipping coins from a barber at my grandpas shop when I was probably 6 or 7. Since then I've always been able to flip a coin and determine what the outcome is. It's really just being consistent with the flip and the catch.

FWIW, there is also a 2007 paper [1] that offers a physical explanation.

[1] https://www.stat.berkeley.edu/~aldous/157/Papers/diaconis_co...

I am curious how this changes if we condition on it flipping in the air at least once. Can we think of this result as a mixture distribution of a fair 50/50 chance of it flips at least once, and a delta function that is 100% at the side it started on, if not flipped at all?

Here's a little through experiment I use to come to this conclusion:

Let's say you start a counter from the number 0, and keep on incrementing it. The moment you stop it to look at the counter, is it going to be odd or even?

At any given moment in time, either the number of observed odd numbers is the same as the number of even numbers, or the number of even numbers is larger by 1 (such as going from 0 to 1 to 2). So in the end there's always a slightly larger chance at stopping on an even number.

I know it's more complicated, I use it just as an intuitive explanation.

Haven't read the paper yet but this is so weird because when I was a kid I noticed this phenomenon myself. I noticed I could reliably flip a coin such that when it landed it would land on the same side as it was flipped from. I was getting like 80% accuracy and I didn't even know what I was doing to achieve it. I could just usually feel when I flipped it that I "did it right". I used it a couple times to win coin toss decisions but then sorta forgot about it and relegated it to a statistical fluke. It would be amazing of there was some merit to it.

This is clearly the law of conservation of reality at work.

Likewise, when you hear a word for the first time suddenly you hear it five times in a row. Or if you see somebody once you suddenly start running into them all over the place.

It's because it's cheaper to repeat past realities than to create new ones.

Winner of the 2024 Ig Nobel prize in probability [1]. A nice read as well!

[1] https://improbable.com/ig/winners/#ig2024

Pr(same side)=0.508, 95% credible interval (CI) [0.506,0.509]

Toss it 100 times, overstating the effect you'd expect it to land on the same side it started 51 times, opposite side 49.

This seems to have been lost in much of the discussion. Employing this in professional NBA basketball you /might/ get one extra toss win per season out of your 100 games compared to any other way of selecting without taking into account the starting side.

Good luck using this!

Useful. This demonstrates that coin flipping merely amplifies noise in human manipulation.

A classic example in the old PSSC high school physics curriculum was a little catapult-like device which tossed a coin, spinning it a few times in mid-air, and repeatably landing it on the same side. It's a demonstration that Newtonian physics is repeatable.

I'm not sure I believe this coin flip bias, but I would if lots of other researchers can reproduce it.

If indeed it's happening, the only explanation can be something to do with very deep Quantum Mechanics including multiverse theory, where we're simply "more likely" to be in a universe where the coin ends where it starts. (But honestly it seems like it would take trillions of flips to detect, just as a hunch) So that would make this experiment, believe it or not, akin to the infamous Slit-Experiment in Particle Physics, where multiverses are one way that's theorized as an explanation. That is, we're sort of in "all universes" as s superposition until something interacts in a way forcing us into ONE universe. (i.e. wave collapse)

Along the same multiverse theme, I also have this other wild conjecture (feel free to ridicule it!) which is that AI LLM (Large Language Models) are "tending towards intelligence" during training because at each quantum collapse (of which Model Training has astronomically high numbers, with powerful computer data centers running for months) we're nudged just slightly more probabilistically into a universe where LLMs are "smart" as compared to "dumb", and so when you multiply it all up over months of churning, that puts us into a universe with dramatically smarter AI, because of the sheer number of computations, adding all the probabilities. I realize the training of AI is "deterministic" but nonetheless only quantum probabilities "determine" which universe we collapse into at each QM decoherence. So you can ask WHY is there this 'nudge' towards universes with smart LLMs? Probably because in all future universes we only exist because LLMs save us, or help us in some way, so other timelines/universes are "less" likely.

I think I can explain it:

Other side: 1/2 flip, 1 1/2 flips, 2 1/2 flips...

Same side: 1 flip, 2 flips, 3 flips...

It seems like there's equal chances, but my theory is that the 1/2 flip is the least likely thing to occur. When you take that into account, there's a slightly increased chance that it's going to land on the same side.

Curious if this is true for dice, whenever me and my family play monopoly, my dad likes to look at the dice (as he's shaking it) and he usually gets a high number if he can see a low one and vice versa.

My heart goes out the cryptographers. All that code, written over decades, that assumes coin flips are 50:50. So much updating and rewriting to do. Quite a few algorithms that will need a rethink to remain fair.

thought experiment: if we design a mechanical arm to enable coin flipping utilizing advanced tech to establish fine-grained adjustments and calibrations to effectively reproduce results with any given coin to and work out formulas to arrive at these results; are we currently or will we ever be able to say with absolute certainty what any given coin toss's result will be?

When I was a kid we played quarters (dating myself) a lot. I felt this was the case, but nice to see it studied.

you shouldn't bet on it though

I think I figured this out when I was about 6 years old. It pretty much is always true.

And a toast covered in jam lands 100% of the time on the jam side.

Yes… but the choice of which side they start is a random one!

Flip it twice. Once to determine which side is up at second throw. Reverse to counter bias at start of second throw. Then flip again for final result.

anyone else thinking about Pokemon TCGP...

statistics be dammed,I'll flip you for it.....heads I win tails you loose

what if they got evidence from 350.758 flips, would this impact anything

not enough flips

I guess our world has been run with unfair flips, LOL.

Easy way to get a fair result from an unfair coin toss: Flip the coin twice in a row, in this case starting with the same side facing up both times, so it's equally unfair for both tosses. If you get heads-heads or tails-tails, discard and start over until you get either heads-tails or tails-heads, which have equal probabilities (so you can say something like HT = "heads" and TH = "tails").

This works even if the coin lands heads 99% of the time, as long as it's consistent (but you'll probably have to flip a bunch of times in that case).

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In other news, probabilities again used to prove whatever conclusions we were planning to present anyway.

It is time to stop the show, probabilities cannot prove specifics. Aka they cannot prove that the coin I hold is fair or not. We can only get trends for big populations.

There is only one way to prove if a coin is fair. Measure the actual thing that matters. In this case mass distribution. And if the measurement is inaccurate, then count atoms. One by one.

I noticed phenomenon in poker as well. Someone who runs well ahead of the crowd continues to do so seemingly even playing randomly with no thought into traditional poker theory.

For example, if a strong pair starts off with a bad beat then it tends to continue that trend. The word trend doesn't mean its going to happen but that its likely to continue the past.

When someone continues exploiting this trend they have seemingly "broken" the game, it no longer functions like a calculated game of odds and when somebody plays like a maniac (like in the first scenario i mentioned) there is seemingly no other defense than to wait until the trend breaks but no matter how seasoned a player is they cannot shake the past and its perceived likelihood of continuing.

This effect is rampant in stock market as well when there is seemingly less "random" reinforcements and belief in the crowd which without fail has given rise to black swans/massive collective drawdowns of the world war causing variety.

This is probably just because the coins aren’t actually fair. If the coin is slightly biased towards heads, the first throw is more likely to heads, and so are all subsequent throws. Same for tails.