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

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.

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.

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

> 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...

Botany: Jacob White and Felipe Yamashita, for finding that certain plants imitate the leaf shape of nearby plastic plants and concluding that "plant vision" is plausible.

This somehow doesn't fit the Iq award in my mind.

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).

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.

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

To me this kills the credibility of the entire study and of the authors.

Sure, there may be something to it, but people will have a very different thing on their mind unless they check the video, which I wouldn't have done without your prompting.

It's unlikely they don't understand how misleading it is.

And somehow I have the intuition a proper coin toss will not exhibit the same properties.

0: https://en.wikipedia.org/wiki/Whitening_transformation

There are multiple ways to ground Bayesian statistics without resorting to grounding in coin flips. The simplest one isn't that robust, there's a mathematical one but it's abstract and uses calculus, there's a quantum one but I'm not even going there, and there's a highly robust one that's too complex for me to understand.

Proof: Imagine the extreme case of the coin containing AI that knows exactly how you use it and how to manipulate each toss result. The coin itself can decide the outcome of your procedure, so it's impossible to trust it to generate randomness.

Very easy to claim he was the most intelligent human to ever live. Or perhaps he was never human...

https://en.wikipedia.org/wiki/The_Martians_(scientists)

Reminds me of one of my favorite movies, Rosencrantz & Guildenstern Are Dead, which opens with just such a scenario[1].

[1]: https://www.youtube.com/watch?v=RjOqaD5tWB0

If you require true randomness without any assumptions this is not the universe for you.

As the other commenter said, in between coin flips, use a highly secure PRNG to orient the coin randomly. This would correct for your bias (if true).

To add interest: there are plenty of people who firmly believe this, and make decisions by the drawing of lots, in various possible forms. I’m one. It’s taken me in interesting and unexpected directions this year.

A coin with a same side bias is more likely to land on heads if it's thrown with heads facing up, and more likely to land on tails if thrown with with tails facing up.

You said:

> 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.

Suppose I'm throwing the coin using your technique and I want to favor heads.

I hold tails up for the first throw, making tails more likely.

Then as per your rule, I put heads up for the second throw. Now, heads is more likely.

Choose the opposite starting face to make tails more likely. So, your technique does no prevent the coin tosser from being able to favor their desired outcome.

A single person would write 17000 posts about their "amazing journey" coin flip outcomes, and another 17001 "humbled by success" coin flip outcomes

A coin that is biased towards heads is one that would more often land on heads regardless of how you hold it when you start the flip.

The study finding is that every coin is more likely to land on heads if you start it with heads facing up, and will also be more likely to land on tails, if you start it that way instead. This bias, while small, is greater than the typical observed bias due to imperfections in manufacturing.

It's not about the "first throw" vs the "rest of the throws". It's about how you hold the coin when you go to flip it. That's what they mean by "started".

0 rotations is more likely than 1 rotation, since there is a wider range of rotation speeds that lead to exactly 0 rotations than to exactly 1. Similarly, 2 flips is more likely than 3, 4 is more than 5, and so on. So you're always biased towards an even number of flips and the starting side.

Take out the 0 case by your conditional, and you're left with 1 > 2, 3 > 4, 5 > 6, and so on, now biased towards an odd number and the non-starting side.

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

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.

Craps is also brought to mind where the dice have to bump the back wall

It is not about intentional favoring on result.

there's your paper

here's the video https://youtu.be/-QjgvbvFoQA?si=ZTT1LWWJC8T4LIQZ

This is silly.

The whole purpose of tossing a coin is randomness, so of course you want high and fast.

If the result was that no matter how high and fast you throw is you get this bias, it would have been interesting.

But now you just say "if you do things badly, things don't work".

I would imagine OP did something similar. Watch the coin as its rotating and then grabbing it and then flipping to the side he predicted.

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.

If you want to measure what happens specifically with high and fast coin tosses, then that’s an entirely different study to be done.

I think that's the point. It shows that people don't usually flip properly, leading to biased results.

If you are doing self-experimentation, you do not.

48 "authors" is a bit extreme, but it's the norm to do some light human research with a half dozen authors as the subjects.

I assumed they did these coin flips were done using a machine. But I guess they wanted to test if human flippers because they wanted to make claims about the human coin flip phenomenon.

I bet this is the video you mean? https://www.youtube.com/watch?v=A-L7KOjyDrE

We need some minimum flippage for the toss to count.

The reason is because it was used as incentive:

> Intrigued? Join us and earn a co-authorship

Per the linked youtube video.

It sounds like what they were intending to study is the actual variance that is introduced, on average, by imperfections in throws conducted by humans. Unless that's mistaken, it's a fair point to consider the n=48 here. Did they discover an average that can be generalized to humans or just to those 48?

https://en.m.wikipedia.org/wiki/Clustered_standard_errors

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.

Get a hundred thousand people to flip a coin once each and then see what happens!

The comment you replied to links to footage of one of the participants. You can see in that footage that the coin hardly leaves his hand.

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?

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.

Of all the stats we collect in sports, I wonder if someone has info on coin tosses in sports like American Football, Tennis, etc. I wonder if there are even rules regulating how a coin should be tossed in different sports...

I think the result could be better described as "humans tend to flip fair coins to land on the side they started".

Waiting for the HNer that likes electronics hacking to Show HN: My coin flipping robot I built over a weekend for consistent flips.

Clearly the coin flips at the beginning of sports fixtures need to be assessed by a panel of highly skilled judges who can pronounce on their validity. We'll also need local, regional, national, and international organizations to train, select, and maintain the quality of coin flipping judges and to maintain the integrity of the discipline while moving forward as new coins are minted and different sorts of flipping styles are proposed by. Membership of such organizations should be limited to those afilliated with the Ancient Order of Coin Flippers.

Incidentally the plant mimicry thing seems to be a defense against herbivorous mammals. It was previously theorized that the shape information was transmitted by symbiotic bacteria; the ability to imitate fake plants is a genuinely perplexing result imo.

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

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.

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.

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

>The Ig Nobel Prizes honor achievements so surprising that they make people LAUGH, then THINK. The prizes are intended to celebrate the unusual, honor the imaginative — and spur people’s interest in science, medicine, and technology.

There goal has never been to mock the award winners.

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

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 :)

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.

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.

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.

But if the tosser were to control, manipulate or just don't care enough about adding entropy to the toss, those random generation qualities of the object would start to fall apart.

PS: As I read before, dice/coins are not entropy generators but rather, entropy sinks+processors.

[1]: https://blog.sia.tech/generating-cryptographically-secure-ra...

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!

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.

Why would that be the only explanation? that seems like very low down on a long list of possible explanations.

I didn’t read the paper but the author was discussing how some people impart precession onto the coin which is a likely explanation for causing a bias.

But I just don't see a person being able to flip accurately enough cause this. No way. But I'm just playing along here. I don't truly believe this experiment is anything but either a hoax, or mistake.

Even if the testing was as many flips as possible over years and years of automated means, with a flipping machine that varies flipping power and angle, and detecting sub-millimeter wearing on the surface of a coin, and every single coin style/size in existence, of every single wear level possible from all positions and angles, through every different combination of typical earth-based air percentages... What does the result really mean? It doesn't actually come up with a "conclusion", its just an accounting of an exact series of events. You will still never use that into the future, you will still describe the act as having a probability of outcome.

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. :)

You can preview the effect by spinning a coin slowly on a table.

This is a common problem in intro Physics Mechanics class.

Unfair coins very much do exist: https://izbicki.me/blog/how-to-create-an-unfair-coin-and-pro...

1.0 flip, lands on side it started

1.5 flips, lands on opposite side

2.0 flips, lands on side it started

etc