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Magician-turned-mathematician uncovers bias in coin flipping (2004)
- keepshoutingsir 6y agoTl;dr: There’s no particular bias to heads or tails, but a coin will land the same way it was when it was tossed 51% of the time.
- keithwhor 6y agoThe article is more interesting than just the finding, to those who want to delve deeper. It’s a short biography of Persi Diaconis, an accomplished mathematician with no high school diploma who didn’t start studying mathematics until he was 24. Really, really fascinating life and experience.
- sp332 6y agoI remembered the result that showed a coin toss was biased that way, but I didn't know that the degree of bias hadn't been nailed down. I guess it makes sense that it varies from person to person. Now I'm interested in an anthropological study of how different people toss coins...
- hnracer 6y agoI'm probably being dense but what do you mean by "the same way it was when it was tossed 51% of the time"?
- kilbuz 6y agoWhen you flip a coin, it is either heads side up or tails side up as it rests on your thumb. This initial position may slightly affect how it ultimately lands.
- deleted 6y ago[deleted]
- callamdelaney 6y agoPretty sure it’s because the heads is heavier.
- 0-_-0 6y agoI thought it was common knowledge that heads are heavier than tails. My head certainly is.
- prionassembly 6y agoHN is way too harsh on jokes, even when they’re really good - like this one.
- rootusrootus 6y agoI think good jokes can get appreciation here, but we try pretty hard to avoid the reddit-style puns and joke chains that distract from the discussion. They're entertaining sometimes... on reddit.
- umanwizard 6y agoThere are plenty of other sites you can go to if you want to read chains of joke comments.
- Spooky23 6y agoSome folks have big heads, others big tails.
- Closi 6y agoIt’s not, it’s because when flipped the number of flips the coin does in the air is not entirely random.
- morelisp 6y agoNot only is this (somewhat obviously given the discovered bias) wrong, it’s physically impossible to bias a coin flip by weighting one side. Weighting a coin will change its center of gravity but to bias a flip you would need to impart angular momentum while it is in the air; at any constant rate of spin (again somewhat obviously once pointed out) each side will spend 50% of its time "up" and 50% "down". https://www.tandfonline.com/doi/abs/10.1198/000313002605 https://www.tandfonline.com/doi/abs/10.1198/000313002605
- dheera 6y agoI think it's more random for me because I flip coins onto the floor, not onto my hand, or spin them on the table and see how they land when they stop spinning.
- eh78ssxv2f 6y agoDo we know the reasoning behind the bias?
- FpUser 6y ago>"Diaconis didn't even own a computer. (He still doesn't. He says he got sick of adjusting to new operating systems and noted, after not using computers for several years, that "nothing bad happened to me. And I said, 'Well, I could either learn the current system or learn differential geometry. I think I'll learn differential geometry this year.'")" Good for him. He made the right choice I think.
- crazypython 6y agoSounds like he would enjoy GNU Emacs, which hasn't changed its interface in decades. Even the copy-paste shortcuts are the same.
- _jal 6y agoHaving spent the majority of my days typing at various applications for the last 30 years, I have developed a great appreciation for UI longevity. I'm a vi person by muscle-memory, and the stability there means I've learned a thing or two about it. But I've used emacs enough that I'm comfortable there, too, at least to a point. Don't get me wrong, I love playing with new toys. But I also love not having to think about my tools - I'd rather think about what I'm building.
- simonh 6y agoThe same is true of Notepad. It’s not really the point.
- crazypython 6y agoEmacs has calendar, weather, email, games, todo, search, and almost everything else you need in a modern operating system.
- aj7 6y agoI’d rather give myself an enema. Downvote me.
- throwaway98797 6y agoAny of his books worth reading? Group Representations In Probability And Statistics Magical Mathematics: The Mathematical Ideas That Animate Great Magic Tricks — winner of the 2013 Euler Book Prize
- zests 6y agoPersi is always worth reading although I have only read his papers.
- iainctduncan 6y agoMagical Mathematics is great! Dense actually, walks the "layperson" through proving how some complex card magic works with combinatorics. Really fun read, some mind bending stuff in there! The math is harder than one would expect the cover...
- sdenton4 6y ago"Group Reps" is one of my favorite math books of all time. It is a fantastic read, and completely changed the way I think about representation theory. (In short, you can generalized the Fourier transform to functions in groups and take advantage of all the usual Fourier tricks and intuitions.)
- jaytaylor 6y agoLink to pdf: http://personalpages.to.infn.it/~billo/didatt/gruppi/reps.pdf http://personalpages.to.infn.it/~billo/didatt/gruppi/reps.pd... Caveat emptor: Not sure if it's the right book, though. No author listed.
- thoughtstheseus 6y agoWith practice most people can dramatically influence the outcome of a coin flip. I’ve gotten to >80% accurate when I was practicing.
- spiffytech 6y agoWhat do you do to bias the outcome?
- deleted 6y ago[deleted]
- nojokes 6y agoYou catch it at the right moment. You learn to feel the right timing by practice.
- deleted 6y ago[deleted]
- david927 6y agoFlip it with the right amount of force, like a juggler with bowling pins.
- alisonkisk 6y agoNo. You need to put a lot of horizontal torque on it so it never flips over at all.
- thoughtstheseus 6y agoThat’s not a coin flip! I agree that’s a method to achieve the outcome though. If you get the wobble right it’s almost imperceptible.
- iainctduncan 6y agono audience is going to fall for that...
- hardmath123 6y agoSee also: this short piece about detecting bias in coin tosses by listening to the “ping” they make when tossed. https://cs.stanford.edu/~kach/can-one-hear-the-fate-of-a-coin.pdf https://cs.stanford.edu/~kach/can-one-hear-the-fate-of-a-coi...
- iainctduncan 6y agoPerci is a fascinating and very accomplished guy. His book "Magical Mathematics" is really interesting. There's actually a long history of math and comp sci folks who were very good magicians. Alex Elmsley, Bob Hummer, Martin Gardner are a few other to check out if you're into that stuff. (says the guy who has spent the last few days alternating between hacking and vainly trying to master the perfect Faro shuffle... ).
- emcq 6y agoDon't forget Ron Graham who cowrote magical mathematics!
- iainctduncan 6y agoYeah! And Ron was a pro juggler too. (as have been I!) Sadly Ron died this year.
- jpcooper 6y agoFor those interested in games of chance, I've been playing around with Betfair's Exchange Hi Lo card game [1]. I came up with an algorithm to, given a game state, quickly compute the odds of all subsequent outcomes in polynomial time. Based on this algorithm, I've been building a bot [2] to integrate with Betfair's games API to play automatically for me. After finishing the first iteration, running it for a bit and losing a few Euros, I quickly realised that some of my assumptions as laid out in the README were wrong (I will be updating this soon). Namely, I wasn't accounting for Betfair's commission on winnings (silly me). I currently have an EU-based account, which has a 6.5 per cent commission on winnings. UK-based accounts have the minimum at 5 per cent. The commission has an effect on the cheapest profitable odds you can provide, which has an effect on whether people want to match bets at those odds. With this in mind, it quickly becomes clear that any unmatched bets you see in the market are far from the maximum profitable odds. This makes sense, and competing based on odds becomes impossible at this point. I am led to believe that the only way to make money is by having a UK-based account plus getting the bets to the market first. I have a few ideas on how to do this. If I can get the bets to match at odds that guarantee profit in the long-run assuming UK-based commission of 5 per cent, then I'm going to call it a day. Regardless, I'm not expecting to make too much money from this. I have a lot of free time right now (looking for a job), so if anyone wants to get into anything else available on Betfair together, then please get in touch at the email on my Github through [2]. My discrete probability and programming are okay, but it would be great to have someone who could share their statistics knowledge. [1] https://games.betfair.com/exchange-hi-lo/standard https://games.betfair.com/exchange-hi-lo/standard [2] https://github.com/jpcooper/betfair-exchange-hi-lo-odds https://github.com/jpcooper/betfair-exchange-hi-lo-odds
- lemonspat 6y ago> Diaconis himself has trained his thumb to flip a coin and make it come up heads 10 out of 10 times. I had better dexterity as a child, and after hours I was able to flip a coin to 'heads' about 8/10 of the time. Same position, same muscle reflex, same results. I would proudly show my family my skill Over time, I lost interest and the skill
- deleted 6y ago[deleted]
- thotsBgone 6y agoI did a high school math project that was like this but with dice. After thousands of die rolls, I found out that dice with pips are not very fair, but dice with engraved numbers are better.
- PragmaticPulp 6y agoThe key to breaking this potential advantage is to let someone other than the coin flipper call heads or tails while the coin is in the air. That’s how I was taught to do it as a kid when we were playing for fun. I never really thought about the reasoning until now. If you talk to people deep into the performance magic community, there are myths and legends of people who practice dice rolling to the point where they can influence the outcome of a dice roll. I suppose training for hours a day for years on end could have some such influence.
- pschw 6y agoIt is definitely doable. _Scarne on Dice_ (1945) details a number of methods for controlling dice throws. And Steve Forte's "Gambling Protection Series" videos show what the moves look like in action. Mind you, none of these moves will fly in a modern casino.
- wool_gather 6y agoAlso the book "Gambling Scams" by Darwin Ortiz. As I recall, the techniques generally involve not making normal "fair" free-tumbling rolls, but constraining the die movements somehow. Things like tossing them into a corner or making one of the pair spin flat, around only a vertical axis.
- daveFNbuck 6y agoThat doesn't break the advantage, it just mitigates it. The advantage is just small enough that it doesn't matter much even before the mitigation. The caller could still look at the coin before it's flipped and have a slight edge in calling it. It's not terribly hard to influence the outcome of a die roll. You can roll it end over end like a wheel so two of the faces remain on the sides and can't be the result.
- tzs 6y agoThere was an episode of the old "Breaking Vegas" series [1] on the History Channel about such "dice dominators". This site seems to have the episode online [2]. [1] https://en.wikipedia.org/wiki/Breaking_Vegas https://en.wikipedia.org/wiki/Breaking_Vegas [2] https://www.goldentouchcraps.com/HistoryChannel.shtml https://www.goldentouchcraps.com/HistoryChannel.shtml
- murkle 6y ago> Preliminary analysis of the video-taped tosses suggests that a coin will land the same way it started about 51 percent of the time. From 25 trials?
- morelisp 6y agoThey're measuring rotational velocity, not outcomes. As described in the article, actually flipping to determine the bias would have been a) arduous, b) probably prone to a margin of counting error larger than the possible bias (as they discovered for dice).
- kazinator 6y agoHalf-assed tosses of a flat object can obviously be biased. We can predictably flip an object to the other side with a light toss, not to mention flip a pancake. If you flip a coin through the air so that it rapidly rotates (say more than 15-20 times) before being caught, that seems like it would erase the bias. If there isn't decent spin, then "all bets are off" (quite literally). As a matter of procedure, there should be a rule: the coin tosser must place the coin onto index finger, which is coiled around the thumb, such that just the edge of the coin is struck from below by the thumbnail, when the thumb is flicked, sending the coin upward in rapid rotation. There should be additional rules that the coin should attain a height of at least one head height above the top of the tosser's head, or something of the sort.
- 10000truths 6y agoBetter yet, just bounce the coin off of two surfaces before it lands. Makes it almost completely unpredictable.
- forgotmypw17 6y agoI'm very surprised that a scientist who previously worked with shaved dice did not take into account or even mentioned what kind of coin they were working with. Many coins are not even close to equal weight or shape on each side.
- donpdonp 6y agoThe method to get fair flips from an unfair coin is interesting and was recently on hackernews. say a coin is heads(H) 70% of the time. that means it must be tails(T) 30% of the time. If you flip it twice, and calculate the odds of each pair of flips (multiply the probabilities) - HH 49%, HT 21% TH 21% TT 9%. The middle two outcomes are equal probability, aka a fair coin. So ignore HH and TT flips and use HT and TH as Head and Tail of the 'fair' coin.
- dragontamer 6y agoJust note: you're throwing away 58% of your results! The original coin flip runs 230% the speed of your new RNG (that throws away so many results). The Von Neuman method is a great way at demonstrating that removal of bias is possible, but not necessarily practical (especially at high bandwidth). In practice, you want to just cryptographic_hash(coin flips), which should maximize your entropy up to the entropy limit of 1/2 the hashsize (assuming you have a perfect cryptographic hash function). A 512-bit perfect cryptographic hash can only support 256-bits of entropy, due to the birthday attack.
- typicalset 6y agoIn the general case, you can compress a sequence of N biased coin flips with arithmetic coding. For a coin with known bias, this compression is (essentially) optimal and will therefore produce ~N*(Shannon information) unbiased bits.
- dragontamer 6y agoIn the "even more general case", the Shannon Theorem suggests that random_code(message) -> reaches the Shannon Limit. :-) The problem is that perfectly-random codes can't be decoded very easily... so this little factoid is kind of worthless in most cases. Fortunately, we don't actually care what the information looks like when flipping coins. We usually just want "some random message", so no need to decode in this application! ------ Cryptographers assume hash-functions are perfect random codes (they aren't in theory, but they are in practice... at least until they're cracked. Kinda funny how that works out). As such, the cryptohash(message) methodology should also reach the Shannon-limit, as long as your cryptohash remains secure... and you stay within the blocksize restrictions of the hash.
- nobrains 6y agoSlightly related, and of interest to minecraft players and speed runners: https://www.youtube.com/watch?v=1iqpSrNVjYQ https://www.youtube.com/watch?v=1iqpSrNVjYQ
- KwisatzHaderack 6y agoYears ago, I went to a talk Persi Diaconis gave on this exact topic. He was the most charismatic mathematician I ever met in person.
- smcameron 6y agoI remember when I was a kid, I had a book called "The Amateur Magician's Handbook", by Henry Hay, which described how lots of magic tricks and sleight of hand with cards, etc. were done, with detailed explanations and pictures. There was a section called "Heartbreakers", which contained tricks and techniques that were especially difficult to master. One of them involved predicting the outcome of a coin toss. The secret to it was that you just practice flipping and catching a coin to an extreme degree, aiming for consistency until you could replicate the action so precisely that the coin flipped the same number of times whenever you performed it. I practiced and practiced, but evidently not enough, because I could never do it. I found a bit of it online: "18. Heads-or-tails. This I take on the good and sufficient word of Eddie Joseph. He explains how it is possible to flip a borrowed coin, and call heads or tails correctly every time. "No trick to it -- just mechanical uniformity of action. It is obvious (when pointed out) that which face of the coin comes up depends simply on the number of revolutions the coin makes; and this depends on the weight of the coin, the height of the flip, and the force of the spin. "By always using one denomination of coin, you fix the weight. The uniform flip of your right thumb comes with practice. To get a uniform height, pick some marked point on the wall, and practice until you can toss the coin just that high every time. "You must also catch the coin on your left palm at exactly the same height each time -- your waist, naturally, is the easiest. "Once you attain true uniformity, you will discover that the coin always comes down showing the same face that went up, or else always the opposite face. Experiment with different heights of toss, keeping track of the way the coin comes down. Never vary the force of your thumb flip because that is the hardest to judge. "In performance, pick a mark such as a molding or picture frame on the wall of the room where you happen to be, make one trial toss to that height, and see whether the coin comes down the same as it went up, or opposite. Every time you toss to this mark afterward, the result should be identical.
- godmode2019 6y agoI could do this by playing a lot of chess, the trick is the coin makes a certain sound if you do it right, if you do it wrong its not the same sound
- skybrian 6y agoPeople normally think of probability as a way of understanding games of chance. But historically, games of chance evolved to make probability work, once people cared about that. Roman dice were lopsided. [1] When probability works, it's because people made it work by removing influences that would throw off the calculations. [1] https://www.atlasobscura.com/articles/dice-evolution-fate-fairness-archaeology https://www.atlasobscura.com/articles/dice-evolution-fate-fa...