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).
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.
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 :)
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.
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. :)
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...
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.
> 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.
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.
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.
They linked to the same video, but to a specific timestamp within it - by adding '?t=325' to the URL, which tells Youtube to play the video from 5m25s rather than from the beginning.
This can be really relevant in various fields, statistics, gambling, and decision-making. I like the fact that they imply the importance of considering potential biases in seemingly random events.
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.
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.
Randomly how? By a coin toss? Who will toss then? How many times? How skilled the participants should be? All these important questions must be decided by some authority. Sort of a Department of Equal Distribution. Or a Ministry of Fair Tosses. Wait a second...
There is a [video presentation of the paper](https://www.youtube.com/watch?v=-QjgvbvFoQA) which does a good job of explaining the inspiration for the study within the first few minutes.
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?
Yes and what immediately jumps out to me as a source of bias is that they asked this small group of 48 coin flippers to flip thousands of times each. I would’ve thought it would be obvious that when you ask people to do something thousands of times they might do it in a different (and biased) way than someone doing that thing only once.
Get a hundred thousand people to flip a coin once each and then see what happens!
Get a hundred thousand people to flip a coin once each and then see what happens!
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...
What's more, from the numbers cited it sounds like they had 48 people do nothing but flip coins for 8 hours (avg. 15 flips/min). Whether continuous or with breaks, there's no way you won't become seriously consistent. 7000 flips is many more flips most people will perform in their entire lives.
But is that the case? The only way I've ever seen people flip a coin is by flicking it in the air with their thumb and either catching it or letting it hit a surface. I've never seen someone flip a coin like it was a die.
The real lesson is probably that if you're skilled enough, and/or train for long enough, you can influence the odds significantly without anyone ever noticing anything.
The paper is an experimental validation of a previous paper that presented a statistical model. The experiment found the exact results predicted by the model. The reason for the non 50/50 result is precession of the coin.
Actually, I think it's more sound to approach this with clustered standard errors. Basic intuition is similar, but the sample size is what it is per person, and your observations aren't independent across draws but are across people.
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.
If you programmed a machine to flip a coin in the same exact way every time, would you not expect the coin to land the same way every single time? If you program some randomness into the machine to simulate human flipping, then you'd simply move scrutiny from the coin to the machine's programming.
I think the result could be better described as "humans tend to flip fair coins to land on the side they started".
But if you get someone to flip a coin thousands of times for a boring reason, I would lose confidence that they are flipping in the same way a normal human would.
If you are doing anything with human subjects, even something dumb like having them flip coins for an hour while recording the results, you need approval from your local ethics board.
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 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:
That's not tossing a coin, that's barely throwing it in the air.
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.
Is it unlikely? If I didn't read your comment I wouldn't see any problem there. I never saw anyone flipping a coin in a different way. It's just not done much around me.
I think it's still noteworthy that what many people consider a "fair toss" is not in fact a fair toss. In other words it's interesting from an applied psychology perspective even if the physics of the phenomenon isn't particularly interesting.
This was my first objection as well. However, if most people flip coins like that, then the measurements are valid -- the conclusions are about what average people will do, not a perfect mechanical coin flip. Otherwise you're falling in the no true coin flip fallacy.
Yeah, if I'm actually forced to use a coin instead of a computer system, I try to ping the thing off the ceiling and at least one wall (not in that order). Hitting various other things is a benefit, not a downside.
Your point about the coin hitting other things to be more unpredictable reminded me of an interesting blog post[1] about generating cryptographically secure random numbers. The memorable part for me is the suggestion of using five coins of different shapes and sizes so they get shaken a consistent number of times in a large cup.
The guy in the grandparent YouTube video suggests shaking the coin in a closed hand (or better, a box) to randomize the starting side and then transferring it unseen to someone else to flip it
Craps is also brought to mind where the dice have to bump the back wall
This makes me feel like, similar to everything else, even science is actually a spectrum. Based on how much insanity to put into the testing.
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.
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.
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.
They 'need' to fill slots not that the IN awards have become an annual media event (presumably yielding some profit) so they've taken to mocking perfectly legitimate research as long as it is in some way scatalogical or counterintuitive. I lost interest in the Ig Nobel prize as a result; they've gone from an intermittent amusement to a celebration of ignorance.
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.
The Ig Nobel has always been for serious science that sounds silly. Their website begins with
>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.
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.
This is anecdotal evidence but Dennis Rodman (the pro basketball player) was the greatest rebounder of all time. One of his teammates related to how he would watch guys shoot (usually during warmups) and count the rotation of the ball. Based on how many times the ball would rotate, he knew if it was going in or not and then would position himself to get the rebound.
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.
Sandy Miller is widely considered to be the best volleyball player of all time. He would famously wear the same unwashed shorts every game for good luck. Maybe this was his trick.
I knew someone else who could do this pretty reliably. He said it was a “feel the timing” thing. Best analogy he had was maybe like landing an ice skating triple jump, or a complex dive. It happens too fast to be consciously controlled. Instead the trick is to train the body to get a feel for success and then just let the body do it.
It's easy. All you need to do is rotate (yaw) your hand when flipping so that the coin spins but never actually flips, or a little slower so it flips only once. A watchful eye can detect it happening, though.
You can preview the effect by spinning a coin slowly on a table.
This is a common problem in intro Physics Mechanics class.
That is referenced in both the paper and the video in fact. Apparently Diaconis presented a model which predicted about 51% preference for "Same side" and also did 2500 flips and said that about 250k flips would be needed to get 3 sigma of significance. So this paper decided to test it empirically and got to about exactly that number after 350k flips from a team of researchers.
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?
Seems likely it would change. Here's another way to think about it:
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.
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.
Maybe you were like one with the coin and always pushed it the optimal way for like the same type of movement and direction and rotation for the same amount of rotations in air etc like perfected an initial condition and kept it stable like it rotated 6 times and landed the same way
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.
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.
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.
>If indeed it's happening, the only explanation can be something to do with very deep Quantum Mechanics including multiverse theory
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.
Now that so many physicists and legitimate experts (non-quacks) believe in Simulation Theory, we've sort of "merged" physics and Religion. The general agreed upon definition of God is "whatever thing is simulating the universe". Of course all the Religious dogma and mythology stories are things that most of them don't believe.
The fact that some people cause it and some people don't (the coin flip bias) can have an explanation something like having to do with their impact on the causality chain if our universe/timeline. It could be anything from which one of them is older, to which one of them has a future offspring that does something big that has a big impact on the universe (in terms of Butterfly Effect kind of knock-on effects).
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.
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.
Yes, that's because a coin toss is not intrinsically random, but just pseudo-random due to its chaotic behaviour which is especially notable at relatively "extreme" starting conditions.
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.
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.
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?
Probably not. A reasonable Kelly calculation would make the attempt negative utility. Too much overhead. Also, depending on who's betting against who, deviating from the very particular protocol in the study would be highly incentivized.
> 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.
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.
Or how when you look for something it always ends up in the last place you look, if it weren't there would have been some number of places you looked that were completely unnecessary.
Personally, I like to keep looking for the thing long after I've found it simply to prove the adage wrong. My keys weren't in the last place I looked because I checked three more places after I had them in my hand.
I don't think that's true, isn't this tested in a way to obviate that psychological effect? I've done coin-flipping in computer simulations and that doesn't happen. (And yes it was a bit more realistic vs a single element, multiple linked elements flip more realistically. No air resistance simulation though.)
How good are you at Bayesian statistics, conditionalization, and understanding various biases? The simulation here should be good (it's better than mine).
The very verse I was about to post! (Though I was going to quote it as more customarily and literally translated, “The lot is cast into the lap, but its every decision is from the LORD.”)
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.
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.
I don't think this is a real, non-psychological effect in general. For this coin flipping of this very particular method, yes the physics simulations look right, but in general it's not something I think exists, or would even reduce the compute needed for the universe.
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).
If anyone wants to look up why this might work, it's a Whitening transform [0]. I can't find the name of the algorithm itself being describe in the parent but there's more than just that for accomplishing the same thing.
It seems like he did everything! I first heard of Von Neumann in international relations & economics classes as the person who established game theory, then later in CS classes as the creator of mergesort, cellular automata, Von Neumann architecture, etc.
The odds are important to know because if someone gave you a trick coin that always lands on heads, you will be flipping coins until the end of the universe. And I'm sure you have more important things to do than that.
Then it's impossible to trust the coin in the general case.
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.
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.
That's the opposite of what the paper says. If the coin was biased you'd expect it to land on heads more often regardless of what side it starts on. The coins land on the side they start on more often.
No, first of all due to imperfections in the manufacture of real coins, there are actually no fair coins.
Also the bias in the probability affects the first throw as well as all the rest. If your dataset is composed of first throws/rest of the throws, you’re going to see they are correlated.
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".
That's not the problem. You can test that by using a highly secure random number generator, e.g. /dev/random in Linux, to select the initial side. Keep track of that initial side, record the side it lands on. This paper shows a same-side bias, not a heads bias.
How? I described how to randomize the initial side. Boolean true for heads, boolean false for tails, for example. Keep pulling those from the Kernel's secure RNG.
A coin with a heads bias is more likely to land on heads no matter how it's thrown.
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.
If you take a specific coin and find that when you prepare it to be flipped showing heads up, that it is more likely to land heads up, and that when you prepare it to be flipped tails up, it is more likely to land tails up, it seems confusing to call that coin 'heads or tails biased'
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.
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 :)
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.
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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. :)
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...
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
How do you control against the prospect of your coin flippers being biased in terms of the videos people choose to upload?
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.
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.
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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.
Some hi-res footage of an NFL toss: https://youtu.be/-sjAyQcv0oM
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.
link to the "wobble flip" trick https://youtu.be/-QjgvbvFoQA?t=325
I think you accidentally linked to the same video as the parent comment...
I bet this is the video you mean? https://www.youtube.com/watch?v=A-L7KOjyDrE
They linked to the same video, but to a specific timestamp within it - by adding '?t=325' to the URL, which tells Youtube to play the video from 5m25s rather than from the beginning.
Ah man, please use Bayesian statistics there... Well, the presenter says he doesn't know much about statistics.
The paper does use Bayesian statistics. Presenter is a pure maths PhD.
I don't think I was clear, but I was only talking about the presenter's attempted explanation of the statistics of this problem.
This can be really relevant in various fields, statistics, gambling, and decision-making. I like the fact that they imply the importance of considering potential biases in seemingly random events.
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.
testers that don't flip properly
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.
I was more thinking we'll need a Department of Randomness (or Ministry of Randomness for Westminster countries)
Perhaps whether it's a Department or a Ministry could be decided randomly.
Randomly how? By a coin toss? Who will toss then? How many times? How skilled the participants should be? All these important questions must be decided by some authority. Sort of a Department of Equal Distribution. Or a Ministry of Fair Tosses. Wait a second...
The obvious solution is to hand it off to the Department of Catch 22. Or the Ministry of infinite recursion. Wait..
> testers that don't flip properly
I think that's the point. It shows that people don't usually flip properly, leading to biased results.
There is a [video presentation of the paper](https://www.youtube.com/watch?v=-QjgvbvFoQA) which does a good job of explaining the inspiration for the study within the first few minutes.
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?
Yes and what immediately jumps out to me as a source of bias is that they asked this small group of 48 coin flippers to flip thousands of times each. I would’ve thought it would be obvious that when you ask people to do something thousands of times they might do it in a different (and biased) way than someone doing that thing only once.
Get a hundred thousand people to flip a coin once each and then see what happens!
Get a hundred thousand people to flip a coin once each and then see what happens!
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...
Having stats on the outcome of coin tosses in sports wouldn't help, because it's unlikely that the state of the coin before flipping was recorded.
What's more, from the numbers cited it sounds like they had 48 people do nothing but flip coins for 8 hours (avg. 15 flips/min). Whether continuous or with breaks, there's no way you won't become seriously consistent. 7000 flips is many more flips most people will perform in their entire lives.
In some circles, they'd make a post about how their "AI" flipped a coin 8000 times.
Waiting for the HNer that likes electronics hacking to Show HN: My coin flipping robot I built over a weekend for consistent flips.
Except that flipping a coin hundreds/thousands of time in a row is not a representive of how people will flip a coin a single time/few times.
But is that the case? The only way I've ever seen people flip a coin is by flicking it in the air with their thumb and either catching it or letting it hit a surface. I've never seen someone flip a coin like it was a die.
The real lesson is probably that if you're skilled enough, and/or train for long enough, you can influence the odds significantly without anyone ever noticing anything.
That has been known for decades. It's not the lesson of this paper.
The paper is an experimental validation of a previous paper that presented a statistical model. The experiment found the exact results predicted by the model. The reason for the non 50/50 result is precession of the coin.
Actually, I think it's more sound to approach this with clustered standard errors. Basic intuition is similar, but the sample size is what it is per person, and your observations aren't independent across draws but are across people.
https://en.m.wikipedia.org/wiki/Clustered_standard_errors
> only 48 testers/flippers
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.
If you programmed a machine to flip a coin in the same exact way every time, would you not expect the coin to land the same way every single time? If you program some randomness into the machine to simulate human flipping, then you'd simply move scrutiny from the coin to the machine's programming.
I think the result could be better described as "humans tend to flip fair coins to land on the side they started".
One would expect chaos effects to come into play.
One might, but that would be wrong.
But if you get someone to flip a coin thousands of times for a boring reason, I would lose confidence that they are flipping in the same way a normal human would.
1-2 flips should just invalidate the toss. Anyone in a real scenario upon seeing this would call shenanigans.
We need some minimum flippage for the toss to count.
> the role of some might have been more test subject than author
The reason is because it was used as incentive:
> Intrigued? Join us and earn a co-authorship
Per the linked youtube video.
If you are doing anything with human subjects, even something dumb like having them flip coins for an hour while recording the results, you need approval from your local ethics board.
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 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.
That's not tossing a coin, that's barely throwing it in the air.
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.
Is it unlikely? If I didn't read your comment I wouldn't see any problem there. I never saw anyone flipping a coin in a different way. It's just not done much around me.
If you claim to do a research on coin tossing, the minimum is to be aware on how people toss coin.
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".
No, the whole point of the paper (and the physics model it is verifying) is to see what happens in normal human coin tosses.
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 don't know what a normal human coin toss is. Does the paper contain evidence/argument to justify their way of flipping a coin as "normal"?
That still sound valuable if people generally tend to do it badly? If only to provide an argument for doing it properly.
I think it's still noteworthy that what many people consider a "fair toss" is not in fact a fair toss. In other words it's interesting from an applied psychology perspective even if the physics of the phenomenon isn't particularly interesting.
a coin is likely to land on the same side. it was flipped from if it was tossed by a machine at low RPM and height consistently*
there's your paper
I'm sure you will find similar behavior with dice if you just gently let them fall from your hands instead of throwing them across the table.
This is silly.
Somebody’s grant money getting thrown down a hole…
This was my first objection as well. However, if most people flip coins like that, then the measurements are valid -- the conclusions are about what average people will do, not a perfect mechanical coin flip. Otherwise you're falling in the no true coin flip fallacy.
Yeah, if I'm actually forced to use a coin instead of a computer system, I try to ping the thing off the ceiling and at least one wall (not in that order). Hitting various other things is a benefit, not a downside.
Your point about the coin hitting other things to be more unpredictable reminded me of an interesting blog post[1] about generating cryptographically secure random numbers. The memorable part for me is the suggestion of using five coins of different shapes and sizes so they get shaken a consistent number of times in a large cup.
[1]: https://blog.sia.tech/generating-cryptographically-secure-ra...
The guy in the grandparent YouTube video suggests shaking the coin in a closed hand (or better, a box) to randomize the starting side and then transferring it unseen to someone else to flip it
Craps is also brought to mind where the dice have to bump the back wall
Let's abandon coin flipping in favour of coin shaking then
It's a shake and then a flip. Put your hand on your hip and bend your knees in tight.
This makes me feel like, similar to everything else, even science is actually a spectrum. Based on how much insanity to put into the testing.
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.
but they did?
here's the video https://youtu.be/-QjgvbvFoQA?si=ZTT1LWWJC8T4LIQZ
That's the "the video", that's a video by a third party about the study, and it doesn't include all footage or all participants.
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.
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...
From this year's Iq:
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.
They 'need' to fill slots not that the IN awards have become an annual media event (presumably yielding some profit) so they've taken to mocking perfectly legitimate research as long as it is in some way scatalogical or counterintuitive. I lost interest in the Ig Nobel prize as a result; they've gone from an intermittent amusement to a celebration of ignorance.
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.
The Ig Nobel has always been for serious science that sounds silly. Their website begins with
>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.
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
Well, I suppose if you need fodder to fill your CV, this is one way to do it.
Especially on LinkedIn!
A single person would write 17000 posts about their "amazing journey" coin flip outcomes, and another 17001 "humbled by success" coin flip outcomes
Exactly :-)
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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.
If this is done with a quick toss and the coin is flipping rapidly in the air, that's pretty impressive.
This is anecdotal evidence but Dennis Rodman (the pro basketball player) was the greatest rebounder of all time. One of his teammates related to how he would watch guys shoot (usually during warmups) and count the rotation of the ball. Based on how many times the ball would rotate, he knew if it was going in or not and then would position himself to get the rebound.
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.
Sandy Miller is widely considered to be the best volleyball player of all time. He would famously wear the same unwashed shorts every game for good luck. Maybe this was his trick.
I knew someone else who could do this pretty reliably. He said it was a “feel the timing” thing. Best analogy he had was maybe like landing an ice skating triple jump, or a complex dive. It happens too fast to be consciously controlled. Instead the trick is to train the body to get a feel for success and then just let the body do it.
It's easy. All you need to do is rotate (yaw) your hand when flipping so that the coin spins but never actually flips, or a little slower so it flips only once. A watchful eye can detect it happening, though.
You can preview the effect by spinning a coin slowly on a table.
This is a common problem in intro Physics Mechanics class.
FWIW, there is also a 2007 paper [1] that offers a physical explanation.
[1] https://www.stat.berkeley.edu/~aldous/157/Papers/diaconis_co...
That is referenced in both the paper and the video in fact. Apparently Diaconis presented a model which predicted about 51% preference for "Same side" and also did 2500 flips and said that about 250k flips would be needed to get 3 sigma of significance. So this paper decided to test it empirically and got to about exactly that number after 350k flips from a team of researchers.
as long as a machine is used for consistency*
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?
Seems likely it would change. Here's another way to think about it:
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.
The paper requires that the coins flipped at least once to be counted.
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.
There's a "fair coin", and then there's a "fair flip". It's actually pretty difficult to do a truly "fair flip".
A fair coin is just a coin. There is no such thing as an unfair coin, unless its third side is so huge that it can't be reasonably called a coin.
You're going to have a lot of fans amongst con men. ;-)
Unfair coins very much do exist: https://izbicki.me/blog/how-to-create-an-unfair-coin-and-pro...
Sounds like maybe you were doing something like this? https://news.ycombinator.com/item?id=42184069
Maybe you were like one with the coin and always pushed it the optimal way for like the same type of movement and direction and rotation for the same amount of rotations in air etc like perfected an initial condition and kept it stable like it rotated 6 times and landed the same way
the Force is with them
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!
Winner of the 2024 Ig Nobel prize in probability [1]. A nice read as well!
[1] https://improbable.com/ig/winners/#ig2024
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.
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.
>If indeed it's happening, the only explanation can be something to do with very deep Quantum Mechanics including multiverse theory
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.
Any relatively new field of physics gets the same treatment as religion.
Now that so many physicists and legitimate experts (non-quacks) believe in Simulation Theory, we've sort of "merged" physics and Religion. The general agreed upon definition of God is "whatever thing is simulating the universe". Of course all the Religious dogma and mythology stories are things that most of them don't believe.
The fact that some people cause it and some people don't (the coin flip bias) can have an explanation something like having to do with their impact on the causality chain if our universe/timeline. It could be anything from which one of them is older, to which one of them has a future offspring that does something big that has a big impact on the universe (in terms of Butterfly Effect kind of knock-on effects).
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.
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.
Yes, that's because a coin toss is not intrinsically random, but just pseudo-random due to its chaotic behaviour which is especially notable at relatively "extreme" starting conditions.
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.
What in the Universe is an entropy generator?
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.
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.
If your cryptographer's code is based on coin flips, you probably want to find a new cryptographer.
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
Probably not. A reasonable Kelly calculation would make the attempt negative utility. Too much overhead. Also, depending on who's betting against who, deviating from the very particular protocol in the study would be highly incentivized.
I think I figured this out when I was about 6 years old. It pretty much is always true.
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.
That only works for a fixed bias, it's gameable if the person tossing the coin controls the bias.
That is outside the preconditions of the paper: „if the person tossing the coin controls the bias“
Let me explain.
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.
The paper is discussion regular people (not malicious people) tossing a coin, and under this precondition and assuming a fair (unbiased) coin.
It is not about intentional favoring on result.
This won't fix the bias. It would be biased towards the obverse of the starting side up of the first flip.
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.
Or how when you look for something it always ends up in the last place you look, if it weren't there would have been some number of places you looked that were completely unnecessary.
Personally, I like to keep looking for the thing long after I've found it simply to prove the adage wrong. My keys weren't in the last place I looked because I checked three more places after I had them in my hand.
That's a dangerous game to play. What if you find the thing a second time?
I don't think that's true, isn't this tested in a way to obviate that psychological effect? I've done coin-flipping in computer simulations and that doesn't happen. (And yes it was a bit more realistic vs a single element, multiple linked elements flip more realistically. No air resistance simulation though.)
Oh sure, let's doubt the evidence of our senses in favor of convention. That's good science.
How good are you at Bayesian statistics, conditionalization, and understanding various biases? The simulation here should be good (it's better than mine).
Next you'll cite Bible verse.
“We toss the coin, but it is the Lord who controls its decision.” - Proverbs 16:33 (TLB)
The very verse I was about to post! (Though I was going to quote it as more customarily and literally translated, “The lot is cast into the lap, but its every decision is from the LORD.”)
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.
Aye, always at the ready with His noodly appendage.
I don't think Bible verses are related.
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.
So if computation in the enclosing universe got more expensive, they'd enable more aggressive optimizations, and we'd see the effect get stronger?
I don't think this is a real, non-psychological effect in general. For this coin flipping of this very particular method, yes the physics simulations look right, but in general it's not something I think exists, or would even reduce the compute needed for the universe.
Does this explain the rarity of antimatter?
anyone else thinking about Pokemon TCGP...
Misty's flips are not fair, that's for sure
Yes… but the choice of which side they start is a random one!
And a toast covered in jam lands 100% of the time on the jam side.
And cats always land on their feet. In combination, these facts can be exploited to achieve perpetual motion: https://youtu.be/Z8yW5cyXXRc
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).
If anyone wants to look up why this might work, it's a Whitening transform [0]. I can't find the name of the algorithm itself being describe in the parent but there's more than just that for accomplishing the same thing.
0: https://en.wikipedia.org/wiki/Whitening_transformation
That's Von Neumann Whitening.
Thank you. This was useful to learn.
I’ve seen this attributed to John von Neumann, of all people
It seems like he did everything! I first heard of Von Neumann in international relations & economics classes as the person who established game theory, then later in CS classes as the creator of mergesort, cellular automata, Von Neumann architecture, etc.
Wait til you hear about what he did in Math and Physics...
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)
I consider LLMs to be the first successful non-von-neumann architecture in many decades
Importantly - you don't have to know the odds of the coin ahead of time, or which side is more likely. You only need to know that it is consistent.
The odds are important to know because if someone gave you a trick coin that always lands on heads, you will be flipping coins until the end of the universe. And I'm sure you have more important things to do than that.
> you will be flipping coins until the end of the universe
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
Nah, you can put in a rule to stop. It would be better to know ahead of time, but you don't need to.
Each flip would need to start with the same side up though, if this paper is correct.
What if consecutive unfair coin flips are not independent?
Then it's impossible to trust the coin in the general case.
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.
It's also impossible to prove that a given coin is not being controlled by an AI. (Or a deity.)
Yes, which is why you can only trust abstract coins that exist in a formal system which assumes independent tosses :)
If you require true randomness without any assumptions this is not the universe for you.
Just perform the same coin toss in two universes.
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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.
Also, there is fair _coin_ and fair coin _flip_, two different things.
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.
That's the opposite of what the paper says. If the coin was biased you'd expect it to land on heads more often regardless of what side it starts on. The coins land on the side they start on more often.
No, first of all due to imperfections in the manufacture of real coins, there are actually no fair coins. Also the bias in the probability affects the first throw as well as all the rest. If your dataset is composed of first throws/rest of the throws, you’re going to see they are correlated.
I think you're missing the fact that you don't have to chain coin flips literally right after another.
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).
You're missing the point.
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".
That's not the problem. You can test that by using a highly secure random number generator, e.g. /dev/random in Linux, to select the initial side. Keep track of that initial side, record the side it lands on. This paper shows a same-side bias, not a heads bias.
A same side bias is either a heads bias or a tails bias.
How? I described how to randomize the initial side. Boolean true for heads, boolean false for tails, for example. Keep pulling those from the Kernel's secure RNG.
Its not, its a bias towards which side the coin started on.
Which is either heads or tails.
A coin with a heads bias is more likely to land on heads no matter how it's thrown.
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.
If you take a specific coin and find that when you prepare it to be flipped showing heads up, that it is more likely to land heads up, and that when you prepare it to be flipped tails up, it is more likely to land tails up, it seems confusing to call that coin 'heads or tails biased'