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jbandela1
ChocMontePy
If anyone wants to test it, someone wrote a short code that simulates doing that 100,000 times:
https://www.techiedelight.com/generate-fair-results-biased-c...
The coin is biased to come up TAILS 80% of the time, but using Von Neumann's method in the program I got HEADS 50.035%, TAILS 49.965%.
098799
Why would you test it?
Probability of two heads: p*p
Probability of two tails: (1-p)*(1-p)
Probability of head followed by tails: p*(1-p)
Probability of tails followed by heads: (1-p)*p
It's not difficult to notice that if you remove the first two, the last two form a 50/50 distribution
cantrevealname
> Why would you test it?
I recall conversations on Usenet decades ago about the Monty Hall problem[1] in which people gave elementary proofs that probabilities don't change by opening a door. Even from mathematicians and statisticians. People were very insistent that the analytical solution was simple and obvious and that switching doors didn't change anything.
The only thing that changed some people's minds was a program that simulated the Monty Hall problem. This was needed to get people to reconsider their proof when the claim was highly counterintuitive.
bob88jg
Why would you not - analytical solutions are the rare occurrences might as well approach everything with simulation...
nofinator
Ironically, this reminds me of a story (folk tale?) about Von Neumann himself.
A colleague told him about the Two Trains Problem (https://mathworld.wolfram.com/TwoTrainsPuzzle.html), and Von Neumann replied with the correct answer. When his colleague said, "Ah! You figured out he trick!", Von Neumann replied, "What trick? I just summed up the distances in my head!"
voidfunc
Some of us suck at math.
sidkshatriya
The thing about math is that you can do things in multiple ways.
Theory is useful but so is experiment.
toxik
I think about it this way
p(th) = p(t) p(h)
p(ht) = p(h) p(t)
Hence p(th) = p(ht) regardless of coin imbalance as long as both events actually will happen. QED.
skrebbel
> It's not difficult to notice that
Look I found the mathematician
jszymborski
Some folks have more faith in their ability to derive proofs than write simulations and vice-versa.
chaorace
Wow. As usual, Von Neumann makes it look easy
dfxm12
You should change your name to ChocMonteCarloPy :)
mewpmewp2
That's amazing, but I guess it won't help when the person can choose the bias?
Because according to the study the person can choose the bias by choosing which side start up.
So if the person wants tails based on what you've said, they should always
1. Do the first throw starting tails up.
2. If the first one is tails, then they now want to start second one heads up.
3. If the first one is heads, they will want to try and get heads again to dismiss the results. So they will do heads up.
So assuming for example that they have an ability to control bias 75% vs 25%.
Then there would be 75% chance of getting first as tails. After that 75% chance of getting heads.
So they will have 56.25% chance of getting it right the first 2 rounds.
The worst case for them would be if they get heads first (25% chance), and then are unable to get heads again. Which would be another 25% chance so 6.25% odds to lose with the first round.
So 56.25% chance of winning the first round of 2, or 6.25% losing and 37.5% of having to try again.
And I think the odds would converge at somewhere around 90% to 10%. I didn't do full calculations here, but overall it seems this strategy would increase the bias even more.
kqr
> That's amazing, but I guess it won't help when the person can choose the bias?
Alice writes on a piece of paper whether to use the result from the first or the second coin, Bob flips the coins however he likes, then once there are two different sides of the coins up, Alice turns over the paper and reveals to Bob which coin contains the result.
Though I guess that unnecessarily complicates the procedure – maybe Alice can just write "heads" or "tails" on a note and then Bob flips without having seen the note. It essentially replaces the second coin with Alice's mind which hopefully doesn't suffer from the same known bias.
sebzim4500
If you're going to go that way you can skip the coin flip entirely. Just get both of them to write heads or tails on a note and then compare. This technique is used in some crypto projects, except instead of writing on a note you share cryptographic commitments.
kibwen
At this point you can just play odds and evens: one person picks odd, the other picks even, they both hold up either one or two fingers behind their back, reveal them at the same time, then sum the result. This prevents the randomness from being in any one actor's control. If you're worried that your brain's RNG can be gamed, then put an odd-denominated coin in one hand and an even-denominated coin in another, and mix them up so that even you don't know which hand has which.
mewpmewp2
Suppose Alice needs to take the coin first to herself, to use the aforementioned strategy without intentionally introducing bias, and then using result of that, which would determine whether the first or the second result from Bob would be used. Because otherwise Bob may be able to make psychological "guesses".
amluto
There are nice protocols like this that don’t require anyone to visibly flip a coin. See, for example:
c22
° Put the coins in a cup
° Shake the cup vigorously and dump coins on the table
° If coins match, go back to step one
° If coins are opposed take the result of the southernmost coin
wesleychen
You can solve this easily by always flipping with the same side (doesn’t matter which) facing up for all flips.
havnagiggle
There is skill to coin flipping. You'd need to blind the flipper, either physically blindfold or make it so they don't know which result is the positive outcome ahead of time.
mewpmewp2
Unless they can also introduce bias using strength/technique of the throw.
Retric
The final calculation is easy 56.25/(56.25 + 6.25) = 90%, unless the persons skills change between rounds or something.
mewpmewp2
Yeah, thought so as well, interesting how easily those numbers worked out, but then again it's because 75/25=3 and 3x3 = 9 so the final difference must be 9x between the probabilities or 100 / (9 + 1).
I was still lucky with the numbers as for example with 80% vs 20% it would've been 4x4=16 and so 1 to 16 comes to 100 / 17.
asimpletune
But wasn't the bias in the paper something like 50.5% vs 49.5%?
mewpmewp2
Yes, but I used more extreme numbers for ease of calculation and to clearly indicate the direction of a probability.
ctenb
That's really cool. The key insight is that "The reason this process produces a fair result is that the probability of getting heads and then tails must be the same as the probability of getting tails and then heads, as the coin is not changing its bias between flips and the two flips are independent." So you have to make sure you always start with the same side up.
1980phipsi
You can load a dice, but you can't bias a coin: http://www.stat.columbia.edu/~gelman/research/published/dice...
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dentalperson
I love the math/stats history around gambling-related things, thanks for mentioning this. This method assumes the flipper can't introduce bias. ET Jaynes in his book Probability Theory also mentions that it is easy to learn to flip a fair coin in such a way that the result can be predetermined. I searched a tiny bit for this but couldn't find what he was referring to though.
aspenmayer
I’ve heard that, with many hours of practice, dedicated amateurs and many famous magicians are able to do this kind of thing. I wouldn’t call it sleight of hand, but it is similar, although it may fall under that category broadly. I’m not a domain expert but I was taught some simple coin tricks as a child by my artist mom’s artist friend who ran the local frame shop. I never tried or thought to try to favor the coin flip or introduce bias, but it’s definitely a skill that can be acquired.
kqr
I remember at one point in my childhood learning a trick where it looks like you're flipping the coin, but you're really only causing it to rotate and wobble, meaning it's guaranteed to land on whatever side faced up as you tossed it. I don't remember how I did it though, and a few minutes of trying to recreate the effect has failed.
ryanar
Lots of practice, use your thumb to hit the edge of the coin to give you more control, aim for a specific spot so you use the same amount of force and control the amount of times it flips. You can also use a surface that absorbs more so the coin is less likely to flip after hitting it.
nullc
The VN debiaser is very simple but it's not very efficient-- it loses a lot of your randomness.
Under the same IID assumption you can take N flips that returned M heads and map them to the N choose M possible ways that could have happened. The result will (under IID assumption, even in the presence of bias) be a uniform number on the range [0..N choose M). The ctz(N choose M) trailing bits can be used directly (as they will be uniform) but the rest would have to be converted to binary via something like an arithmetic coder or rejection sampling.
The result is muuch more efficient.
Less directly, VN debiasers can also be stacked. Each debiaser outputs three streams: the normal one, one that says if the normal one output anything, and one that says if it got HH or TT. Then run VN debiasers on those. Though it takes a fairly large tree to extract most of the entropy.
vikingerik
Also interestingly, this extends beyond a two-sided coin, to any number of possible results, like a die with N sides.
To get a fair result from a biased dN: Roll it N times. If you don't get all N distinct results, restart. If you do, then the first of those is your final result.
Terr_
Also-related are all the problems like "simulate an 11-sided die with a 6-sided die", where the solution involves identifying certain rolls that should trigger a retry. [0]
In both cases it has to do with shaping the combination of multiple rolls using some knowledge of outcome-symmetries and overlaps.
____
[0] In this particular case, that's something like: Roll the die twice to get rolls A and B; map those a number X in the range range [1,36] using x=(((A-1)*6)+B); if X<=33 then return (x%11)+1 which will be equally likely in the range [1,11]; if X>33 then start over.
mucle6
This is very interesting, but it assumes a coin is biased the same way every flip.
If a coin is more likely to land on the side it starts on, then the bias can change between flips. To fix this, we just need to make sure the coin starts the same side up before every flip.
fbartos
About a year ago, we embarked on a quest to answer one of the most intriguing questions:
If you flip a fair coin and catch it in hand, what's the probability it lands on the same side it started?
Today, we are finally ready to share the results. Thanks to my friends, collaborators, and even strangers from the internet, we collected flippin 350,757 coin flips. We ran several "Coin Tossing Marathons" (e.g., https://youtu.be/3xNg51mv-fk?si=o2E3hKa-ReXodOmc) and spent countless hours flipping coins.
In short, we found overwhelming evidence for a "same-side" bias predicted by Diaconis, Holmes, and Montgomery 2007: If you start heads-up, the coin is more likely to land heads-up and vice versa. How large is the bias? In our sample, the mean estimate is 50.8%, CI [50.6%, 50.9%].
We also found considerable variance in the same-side bias between our 48 tossers. The bias varied with a standard deviation of 1.6%, CI [1.2%, 2.0%], in our sample. The variation could be explained by a different degree of "wobbliness" between our tossers.
If you bet a dollar on the outcome of a coin toss 1000 times, knowing the starting position of the coin toss would earn you 19$ on average. This is more than the casino advantage for 6-deck blackjack against an optimal player (5$) but less than that for single-zero roulette (27$).
The manuscript is at arXiv: https://arxiv.org/abs/2310.04153 And the open data, code, and video recordings at OSF: https://osf.io/pxu6r/.
Diaconis, P., Holmes, S., & Montgomery, R. (2007). Dynamical bias in the coin toss. SIAM Review, 49(2), 211-235. https://doi.org/10.1137/S0036144504446436
noobermin
>If you bet a dollar on the outcome of a coin toss 1000 times, knowing the starting position of the coin toss would earn you 19$ on average. This is more than the casino advantage for 6-deck blackjack against an optimal player (5$) but less than that for single-zero roulette (27$).
This sounds like the plot of a western where a man travels from town to town and gleans a little cash from the local waterhole a little every time. I did the math though, in order to get just the $19, assuming you played a modest 20 times a day, it'd take 10 weeks (not including weekends), and by that point people would definitely figure out your trick. In order to make any profit quickly, you'd have to distribute the strategy, after which your secret would explicitly be out there. Even assuming perfectly honest colleagues, having that many parallel people using the same strategy in the open means that before you turn any real profit, people will find out. It's a fun idea to fantasize about though.
Anyway, cheers on the paper! Pretty cool result that you guys put the effort in in implementing.
kqr
The upshot is that as long as you only stake $1 at a time, you're unlikely to lose more than $50.
On the other hand, /if/ you do, you'll have to play for 6000 more flips until you can be fairly certain that you're even again.
What's worse is if, after having lost $50, you're down to your last $50, there's almost a 1/5 chance you'll blow all of it trying to recover if you wager $1 each time.
If you grow wise and start Kelly betting you'll get back to your starting $100 on average in 5000 flips, though. If you can take out a loan of $500 first, you can Kelly bet your way to even much faster, in an expected 700 flips. Whether this is worth the interest on the loan depends on how quickly you can find challengers to bet with.
mewpmewp2
Make a deal with all the banks or systems that can print money that would allow you to take an infinite loan from them. Then just double the bet every time you lose.
If they can print money, why not infinitely as you will always pay it back anyway, so you don't have to worry about introducing inflation. There will always be a point when you can just burn the money that you temporarily introduced.
cortesoft
You also have to factor in your time cost. Your hourly rate is going to be really low, better off just getting a job.
thih9
> in order to get just the $19, assuming you played a modest 20 times a day, it'd take 10 weeks (not including weekends)
What if one play session consisted of 10 coin tosses (each an independent $1 bet)? I guess 20 games like this per day would still be doable. Would that mean $19 per week?
Next we up the bet to $10 per throw.
nomilk
It becomes clear why there's a same-side bias when watching the video: https://www.youtube.com/watch?v=3xNg51mv-fk
These are fairly gentle coin tosses; barely going a foot into the air!
When I think of a coin toss, I think high and spinning fast (like the ones before sports games, where the coin goes into the air and lands on the ground, usually rolls a short way, and is collected on whatever side it landed). I would guess the 50.8% same-side bias would be much closer to 50% if the coins were tossed this way in the experiment.
fbartos
There was indeed a lot of variation in the height of the tosses. I however disagree with the conclussion: two of my friends at the video had the most different height of tosses (one tossed thrice as hight as the other one), yet both of them had exactly the same bias (0.505). The amount of spin is unfortunatelly very misleading from the 30fps videos--the coins often seem like not spinning at all but that's just a result of the poor video quality.
sixstringtheory
How do you control for bias coming from the same coin flipper? Do they usually flip their coin from the same starting height (whatever comfortable arm positioning they have, which I assume would also introduce bias by how they catch it as well) and to the same arc peak height? Or were they encouraged to try a different body position, strength and angle of launch for each flip?
glandium
With a 1 foot to 2 feet toss, landing in the hand, I can get the same side as the starting side more than 90% of the time, without even trying. I wouldn't trust such a toss to be fair. Landing on the floor would change the game.
gowld
Bouncing erases the bias due to flipping.
Semaphor
> but less than that for single-zero roulette (27$).
This reminds me of the casino I was at with a fair roulette. Betting Black/Red and getting a zero meant all bets stayed for the next spin ;)
My winning strategy was to always bet the opposite color of a friend of mine.
jonahx
From the Method section:
"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"
Presumably if you instead allow the coin to land and bounce on a hard surface, the bias would disappear?
fouronnes3
What is the physical explanation for this bias?
bradrn
It’s in the paper:
> The standard model of coin flipping was extended by Persi Diaconis [12] who proposed that when people flip a ordinary coin, they introduce a small degree of ‘precession’ or wobble—a change in the direction of the axis of rotation throughout the coin’s trajectory. According to the Diaconis model, precession causes the coin to spend more time in the air with the initial side facing up. Consequently, the coin has a higher chance of landing on the same side as it started (i.e., ‘same-side bias’).
[12] Diaconis P, Holmes S, Montgomery R. Dynamical bias in the coin toss. SIAM Review 2007; 49(2): 211–235.
Philip-J-Fry
After reading this the first thought I had was how do you stop people flipping the same way? Like, give me a baton and I could throw it at varying heights and control which side I caught it on. In theory the same applies to coin flipping. You can get quite consistent with your positioning and power.
You could probably control for it by making people alternate which side was face up before the flip.
That's my intuition anyway.
aspenmayer
Your comment reminded me of two-up.
> Two-up is a traditional Australian gambling game, involving a designated "spinner" throwing two coins, usually Australian pennies, into the air. Players bet on whether the coins will both fall with heads (obverse) up, both with tails (reverse) up, or with a head and one a tail (known as "Ewan"). The game is traditionally played in pubs and clubs throughout Australia on Anzac Day, in part to mark a shared experience with diggers (soldiers).
rstarast
I would guess it's rather mathematical. Each coinflip has some number of half-flips. Now analyze the distribution of that number. If this distribution were to start at its maximum with 0 half-flips and decay as it increases, summing over the even values (same side up) clearly gives more than summing over the odd values. Now the distribution isn't going to be like that, but I expect that it's generally "front-loaded" in a way that causes a similar effect.
mewpmewp2
Yeah, and seeing that the bias occurs only in some people, perhaps it occurs in people who do as little rotation as possible. Not sure if this study has a graph including amount of rotations occurred in general. E.g. you could take all coin flips where 0-5 rotations occurred and compare them to 6-11 rotations.
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alkonaut
> This is more than the casino advantage for 6-deck blackjack against an optimal player (5$)
I have seen that figure (roughly 0.5% edge) but that has to depend on how deep the shoe is dealt? I remember playing only the last hands with dealers playing down to between 1.0 and 0.5 decks left. That meant you could play hands where you knew almost all remaining cards were suited. I guess the average edge assumes constant bet and doesn't include betting strategies based on counting at all? (And those strategies obviously wouldn't work in any real casino because it's "frowned upon").
gala8y
This was my intuition in childhood. If you choose tails to be yours and start with tails then catch it, it is most likely to be tails. I came up with this observation myself. Weird.
kqr
Yeah, I think you were just lucky that your superstition happened to be true.
Trying not to be disrespectful but I don't believe you intuited a 50.8 % bias. So nothing weird going on at all.
gala8y
I know it's not on par with any real stats, but still... I just had this strong conviction that worked this way. The really interesting stuff is why it is so. I would think along lines of brain timing tossing and catching, eye-brain-timing rather than gravity and coin itself. Added: Yeah, now I remember actually manipulating timing of catch to achieve this.
> Trying not to be disrespectful
No worries, it is just me using high context communication style, where I assume that you know that I know this and I just share what was my experience in childhood (it was not like a single thought).
bryanbuckley
Noticed as a kid I could flip a quarter with a certain consistency, so I experimented a bit and quickly got to be >90% accurate with an ordinary (controlled) flip.
Pretty simple. In fact I just picked up a quarter and practiced (20+ years out of practice) and have some observations: 1) harder than when I was a kid, my fingers are lot bigger + stronger so it's not as precise from the start. A bigger and heavier coin would help. 2) the timing factor is bigger than I recalled.. essentially you can watch the coin flipping and get a subconscious/automatic/predictable sort of count/feedback to it. You can bring your hand up to the coin in the air at a precise moment pretty easily and "tell" (>90% accuracy today of the flips I just did that I considered successful before looking at the result) if the flip was predictable. Hand eye coordination, spatial awareness is very correlated to this skill, I suppose. 3) it really is the same side that comes up.. again I think because of the automatic watching/count/completion of full rotations, i.e. catching the coin at the end of a full rotation instead of a partial.
Came in handy occasionally.. if I knew I was going to be wrong (other person usually waits to call mid-flip) I could catch the coin a little lower to give myself a chance, or punk them by not putting it on the back of my hand as is more standard (they might demand a re-flip.. kind of like if you are playing rock paper scissors and one person goes on 3 and the other on 4).
gala8y
This. When I tried to comment briefly yesterday that the coin falls on the same side, only then did I remember that I actually did that and that it has nothing to do with physics but rather neuroscience (and innocently bent morality - which was also the object of my internal observations). then I remembered that I had actually considered different coin sizes, but I was never as thorough in my attempts to bend the results as you were. oh... the playgrounds of childhoods...
JKCalhoun
Fair coin, unfair flip.
helsinkiandrew
I think you can argue that the experiment wasn't representative of 'normal' coin flips.
On average, each flipper in that experiment flipped a coin over 7000 times, after that amount many people will have learned to flip in a comfortable way with less variance between physical action and force they use. I'd imagine that in that case the coin would more likely land with the same orientation.
I don't think this would be true if someone flipped without practice.
fbartos
I personally did 20,100 flips and I can assure you I have no clue how to control the flip. I centrally got much better at flipping and catching the coin in hand without dropping it---which takes some practice on its own.
(I know that there are techniques for adding the wobble to the toss, but I didn't study them and I have no clue how to do them. I think it is safe to say you don't discover them intuitevelly.)
SamBam
I think this is a great point. One flipper 7000 times is quite different than 7000 flippers one time, if the aim is to see whether there is an underlying bias.
misja111
I bet that with enough practice, you can learn to toss the coin so that you get a much bigger than 51% chance of landing it on the same side.
rootusrootus
I think you'd need more than practice. Most people would need a teacher. Just doing something over and over isn't automatically going to make you better. E.g. The old "10000 hours of practice makes you an expert" rule assumes deliberate practice. And even then, it's incorrect.
cantrevealname
I'm still looking for an intuitive or ELI5 explanation of the mechanism for this bias.
The original paper says:
The standard model of coin flipping was extended by Persi Diaconis who proposed that when people flip a ordinary coin, they introduce a small degree of precession’ or wobble—a change in the direction of the axis of rotation throughout the coin’s trajectory. According to the Diaconis model, precession causes the coin to spend more time in the air with the initial side facing up. Consequently, the coin has a higher chance of landing on the same side as it started.
Another coin toss experiment[1] site says this:
The basic reason is that, instead of rotating around a horizontal axis as one might imagine, a typical tossed coin is rotating around a tilted axis which is precessing in 3-space, and this entails a certain degree of "memory" of the initial parameters.
The Diaconis paper[2] has the definitive explanation but it's hardly intuitive. I got a feel for why it is, but I can't do an ELI5. The best I'm able to write is this: A human being is likely to introduce some precession in the coin toss. If there is precession, then the angular momentum vector is going to spend more time in the heads direction if starting from heads, and that accounts for the bias.
What I think would work well for an ELI5 is an animation of a coin toss showing the angular momentum vector sweeping out a region during its flight, and showing it spends slightly more time pointing toward heads.
[1] https://www.stat.berkeley.edu/~aldous/Real-World/coin_tosses...
Q_is_4_Quantum
Perhaps it helps to imagine someone had a "screwy thumb" and the coin only precesses when they "flip" it (in fact people can train themselves to do this, and its very difficult for you, the sucker, to see in the air that the coin is not rotating but just precessing!). Hopefully its obvious that whatever side is initially facing up will be the same one facing up when its caught?
The next step is not at all intuitive to me, namely that even someone trying to do a fair flip causes some precession, and that this isn't decoupled from the rotation.
robocat
Video of how to train yourself to precess the coin and cheat:
a_c
I always tell people that result of coin flip is highly start state dependent. Imagine a sequence of H(ead), T(ail), H, T, H, T, ... if the sequence starts with H first, in no way can the number of T exceed that of H, but the number of H might be 1 greater that that of T. I never tested my self, but I hypothesize that the propability will be more skewed if the number of revolutions is less, i.e. having a shorter Head-Tail sequence.
Edit: The sequence was meant to represent the sequence of head and tail facing up during the rotation. A sequence of [H, T] denotes one full rotation, starting with head. [T, H, T, H] denotes two full rotation, starting with a tail, ends with a head. I didn't mean the result of a flip. So the result of a flip is the final element of the sequence.
roflmaostc
Of course that is true, if you wait not significantly long enough.
Imagine the situation that you flip the coin (starting at H) and you grab it in air immediately. Of course, you will get H as result.
But let's say, the time to stop the coin can be a relatively long time T. Then, I think the probability is some kind of sum. Let's choose \Delta T= 10ms as time discretization:
P(H) = 1 / T * (10ms-0ms) + (30ms-20ms) + (50ms-40ms) + ... = 1/T \sum_{i=0}{floor(T / (2 * \Delta T))} \Delta T
P(T) = 1 / T ((20ms-10ms) + (40ms-30ms) + (60ms-50ms) + ... = 1/T \sum_{i=0}{floor(T / (2 * \Delta T)) - 1} \Delta T
For T -> \infty P(H) and P(T) getting more similar.
But, in practice you wouldn't wait equally distributed in time but more like a Gaussian distributed time period. Hence, each term of the sum would get weighted differently. And the variance and the offset of the Gaussian distribution can shift the probability in favor of H or T. It's really dependent of the concrete parameters. If you grab always after 35ms, then you'll always get T for example.
Adverblessly
If I were to dig through my comment history I would find I have already responded to this exact sort of comment before, so let me regurgitate :)
If the coin is resting on your hand waiting to be flipped, it is currently mid-way through being on side up. This is because the switch between being e.g. heads up to tails up is done when the coin is vertical. If it isn't a clear explanation, try imagining catching the coin and "flattening" it at different angles, while 50% of angles will match either side, at the moment the coin is flipped, it is already half-way through the angles representing the current side.
This means that the correct sequence you describe it not THTHTH but rather THHTTHHTTHH. Taken at even intervals, both sides will appear the same number of times. Taken at odd intervals, at half of the intervals there are more Ts and the other half have more Hs.
cabirum
Can you toss a coin without it flipping at least once (changing state)? I find it quite unlikely to happen, so if the starting state was H, your sequence will be THTH...
a_c
I can't quite define what counts as start of sequence. Maybe a sequence should always have at least one element, and toss straight up is allowed. But if some flipping is mandatory, then the start of sequence would mean the other side of the coin. All I could deduce before this paper is the probabilty of coin flip is skewed.
mucle6
I would have never considered this. Such an interesting way to think about the problem
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ss1996
Thank you for this. I'll make sure to request a best of 350,757 next time I'm deciding anything by coin toss.
LadyCailin
It’s 50.8% bias, so you only need to do best of 1015!
havnagiggle
I've always caught the coin and flipped it onto the back of my hand to display the result. So I guess I have an opposite-side bias.
quietbritishjim
That's the usual way of doing it.
rootusrootus
Huh. I thought everyone did it my way. Flip it onto the floor, hunt it down, and see which side came up.
I'm mostly serious. I know that's not ideal. But I've never been able to master the art of flipping a coin onto the back of my hand, and even catching it mid-air is hit-or-miss for me. The vast majority of time I or anyone I'm with has flipped a coin it's ended up on the floor.
quietbritishjim
> I've never been able to master the art of flipping a coin onto the back of my hand
That's not the usual way of doing it. Usually, you catch it in the palm of your throwing hand, and only then put it on the back of your other hand, flipping it once more in the process.
sjducb
How do I bias a coin flip?
Based on the paper it looks like 55% chance that it will land on the same side it started is possible. This was the most extreme subject.
The bias is caused by procession so I want my flip to process as much as possible. Maybe I offset my finger as far away from the center of the coin as possible. Also putting as much force into it as possible is probably a good idea.
Finally I have to catch it in a way that the top side is facing up when I reveal it.
Any thoughts?
curriculum
Give it some spin with your index finger. Imagine putting a coin between your index finger and thumb, heads side up, resting on your middle finger below — not too different than how most people start a coin flip. With your index finger, rotate the coin in its plane, so that it stays “heads up” but the head is rotating. Now try doing this and simultaneously flipping the coin by flicking it with your thumb at a point on the bottom, close to the edge. If you do it right, you’ll impart a spin. If you impart a modest spin, the coin will never actually flip over, but will just wobble, and will therefore land heads up. An observer will likely not know what you did, because it is hard for the eye to tell the difference between a flip and a wobble at high speeds.
hddqsb
@robocat linked a great video showing how to do this: https://m.youtube.com/watch?v=A-L7KOjyDrE
rootusrootus
> procession [...] process
Nitpick: Precession. This is one case where a minor mispelling really does throw off the meaning of the sentence. At least to me.
mihaic
Interestint, my intuition was inverse to the result: landing on the opposite side takes an odd number of flips and on the same side it would be an even number of flips.
Since for every number of flips either the number of odds is the same as the number of evens, or higher by 1, the chances of getting an odd number of flips is higher. There seems to be more at work here, and only testing validates a model!
alexmolas
I would like to see how TianqiP flips the coin. This user have a ratio of 0.601 [0.582, 0.619] heads, which is a lot. This is the type of skill that can get you a couple of bucks if played strategically.
jcoder
Or perhaps their coin was biased
avipars
They actually flipped a coin >350,000 times and the paper has 49 authors. Based on the paper https://arxiv.org/pdf/2310.04153.pdf, they recorded video footage of each coin flip
I thought they would have run a computer simulation instead.
fbartos
Yes, I mentioned that in the original paper but it got eddited for some reason.
hurtuvac78
This is incredibly puzzling.
Is there a minimum number of rotations per coin flip to consider it valid?
If looks like the bias was not evenly distributed across people. How did you protect your experiment from skilled bad actors who could influence the data with a few bad/skilled flips? Did strangers on the internet fare any differently than in-person attempts from trusted people?
fbartos
We told people that the coin has to flip at least once (which would bias it for the opposite site). Whenever instructing people, I tried to explaining that the coin flip should look like you were trying to determine an outcome of a bet. You can find the complete experimental protocol here: https://osf.io/hkv8p
Also, I wish I had (any) budget to hire proffesional skilled tossers haha.
cantrevealname
I assumed that the 1% bias was entirely due to coins that did not undergo any rotation at all. However, reading that you told people that the coin has to flip at least once, I think I assumed wrongly. It sounds like the bias is due to coins that have undergone an integral number of 360-degree rotations (not zero rotations). But what exactly is the physical mechanism causing this bias? It's easy to understand why zero rotations would introduce a bias, but I can't easily picture a reason for a bias toward an integral number of 360-degree rotations. Is there a simple and intuitive way you can explain the physical reason?
fbartos
It's not about the number of rotations at all. I doubt that you can control it at all even after dozens of hours coin flipping (I did more than 20h and I can't eveb guess how many rotations the coin made) Diaconis, Holmes, and Montgomery (2007) proposed a physical model of coin flipping that introduces the bias as a result of wobblines (i.e., off-axis rotation in the flips).
Diaconis, P., Holmes, S., & Montgomery, R. (2007). Dynamical bias in the coin toss. SIAM Review, 49(2), 211-235. https://doi.org/10.1137/S0036144504446436
morelisp
Why do you think this is puzzling? This bias has been analytically and dynamically predicted for years.
hurtuvac78
Because I find it counter-intuitive. And because I am not aware of scientific development in this field.
@bradrn on this thread kindly extracted the description of the proposed physical model from inside the paper, it is helpful: https://news.ycombinator.com/item?id=37830265
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Von Neumann described a very elegant way to get fair results from a biased coin.
1. Flip the coin twice
2. If you get the same result both times, goto 1
3. Now that you have different results for your pair of flips, use the first element of the pair of flips as your result.
https://en.wikipedia.org/wiki/Fair_coin#Fair_results_from_a_...