12 ms·
The Coin Flip: A Fundamentally Unfair Proposition? (2009)
- lordnacho 9y agoInteresting, but I'm not sure I buy the argument. Which seems to be that HTHTHTHTH... will always have either more Hs or equal H and T in it. If the thing is tossed but stays in its initial state, is that a coin toss? What if you didn't hold it either H or T up? It's not impossible to hold it sideways and throw it up.
- rahimnathwani 9y ago"At any given point in time, either the coin will have spent equal time in the Heads and Tails states, or it will have spent more time in the Heads state. In the aggregate, it's slightly more likely that the coin shows Heads at a given point in time—including whatever time the coin is caught." This doesn't make sense, because they started counting the number of heads from when the coin is launched. But the first state (actually the first few states) are irrelevant, because you always wait a second or two before catching the coin. If you follow this line of reasoning, then the state after the customary delay is the one that has the higher probability. But that state is fair (50/50) and, because the customary period varies, not determined by the initial state.
- Ended 9y agoAgreed, I think this is an overly-simplified explanation. Skimming the paper it looks like the crucial thing which introduces bias is precession of the coin.
- deleted 9y ago[deleted]
- wakamoleguy 9y agoHow do you reckon that the state after the waiting period ends is fair, if, as the article states, any point in time is more likely to have the initial side up? It would follow that at the end of the waiting period, the initial side is more likely to be up. And going forward from there, that new initial state (measured from the end of the waiting period) will continue affecting the rest of the flip. This bias will diminish over time, which backs up Premise 7 that more revolutions reduces the bias. On a tangential note, this behavior reminds me of Benford's law, which is probably based on other statistical reasoning. https://en.m.wikipedia.org/wiki/Benford%27s_law https://en.m.wikipedia.org/wiki/Benford%27s_law
- rahimnathwani 9y agoYou're right. My comment doesn't make sense.
- Adverblessly 9y agoI don't really follow the logic presented in the article. For convenience, let's assume that the coin spins at 2 seconds per revolution. The coin begins in a Heads up position. After 0.5 a second, the coin is on its edge. The coin then takes another second to pass through Tails and again be on its edge. The coin then takes another 0.5 a second to reach its initial state of Heads up. So it sounds to me like the coin spent 0.5 a second in "Heads", then 1 second in "Tails", then 0.5 a second in "Heads" before starting all over again. If I borrow the numbering analogy from the article, it sounds like if we examine the coin in 0.5 second increments we get H T T H H T T H H T T H H, etc. which invalidates the article's claim. Furthermore, the article claims a 51-49 split between sides. Even if we accept its reasoning (i.e. H T H T instead of H T T H H T T H) To get that specific result, (if my math is right) the coin would have to flip around 25 times on average (i.e H T H T H T H T H T H T H T H T H T H T H T H T 0.5*H), which is a specific number not mentioned in the linked article. All of that to say that I have no intention of refuting the paper behind the article, nor do I think they are wrong, but just that the logic presented by the article (and not the paper which I didn't read!) sounds too simplistic to be correct to me.
- gowld 9y agoMore simply: Imagine a counter ticking 1,2,3,4... If you call stop at a random time, uniform over the whole tick time, there's a bias toward odds. But if you choose to call stop at moment 20.5 (and stopping at a fractional time forces the system to round to the nearest whole number), plus random variation due to real-world physics, evens and odds are equally likely.
- vorathing 9y agoI think what the parent post meant is that this reasoning is flawed by assuming the coin spends the same time face-up in the very first iteration as in every other one, and that's a fallacy. To put it simply, assuming a 4 second rotation, the first second will be spent in state A, the next second will be spent equalising the time in both states, the next second will actually put state B second ahead in face-up totals, then reequalizing, ad vitam eternum - which invalidates the article's point about always having one with time at most equal, and one with time at least equal, since they alternate in total face-up time. Edit : Basically, unless you start the coin in the vertical state, the odds-and-even comparison doesn't apply.
- etskinner 9y agoSomething about the evens/odds heads/tails comparison the author makes doesn't sit right with me. If you're confident that the coin will make more than one flip, then how much cumulative time it has spent in 'heads' position doesn't matter (gambler's fallacy), since the 'catch' timing is effectively random after that first flip.
- DiffEq 9y agoAnd additionally...for the purpose used, it is "random enough".
- an_account 9y agoThink about it as randomly picking (catching) from a bag that is 49% heads and 51% tails (time spent in that position in air).
- blatherard 9y agoImagine that the differences were magnified, say heads 90% of the time. Would the randomness of the catch erase that difference?
- seanalltogether 9y agoI think I figured out whats bothering me about it. We need to figure out at what point in the rotation of the coin it can be determined to have changed to the next face. Is it when the coin is perfectly vertical? Maybe even a few degrees before because the rotational velocity will likely carry it over to the next face after making contact with your hand? It seems that in any case the coin does spend more time statistically on the face that it was initially flipped from, but it's not exactly (num_rotations/2)+1
- wakamoleguy 9y agoThe actual paper linked from the article does a great job explaining this. They mention that the side facing up at any moment is a reasonable approximation for how it would land. The time each side faces up is significantly affected by the angle at which you flip, as the coin precesses during the flip. Page 4 has an excellent pair of graphs illustrating this.
- JepZ 9y agoActually, my advice for a good strategy would be to train coin flipping until you are as reliable as the coin flipping device ;-) > [...] an automated "coin-flipper" device capable of flipping a coin and producing Heads 100% of the time.
- iambateman 9y agoAside from the inherent probability in the coin, it is also possible for the coin flipper to exude bias if the coin is caught. I can time a coin catch to get 75-25 heads. Don’t let someone catch the coin if they flip it.
- Kroniker 9y agoHow? Do you see it as it falls? or do you feel it when you catch and then choose to do the "Flip on to your arm" thing or not?
- mikekchar 9y agoNot the OP but I can do it too. It's all timing for me. When you flip the coin it spins at a determined rate. If you practice, you can time catching it. It's surprisingly not so difficult.
- deleted 9y ago[deleted]
- jrimbault 9y agoFor me the trick is to know which side it was on when I threw it, and doing the exact same throw each time. You have to practice the same move again and again. Throw and catch throw and catch, like practicing music scales. If you're doing the same throw and the same catch each time, it just becomes muscle memory (again, like playing an instrument) and you can get a good success rate.
- rekshaw 9y agoIf you: -start with the same face pointing up - apply a fairly consistent force with your thumb, and - wait a fairly consistent time before catching
- have_faith 9y agoMinimise speed of coin spinning and get good at consistent timing of catching.
- yosito 9y agoWhen I flip a coin, I reach up and catch it in mid air, then slam it on to the back of my other hand. I feel fairly certain that those two actions would negate any bias discussed in this article, and quite possibly introduce their own. If you know enough about a particular way of flipping a coin, you could know whether the chances are off from 50/50, but if you know that much, you could either devise a way for the result to be in your favor most of the time or you could ask for the method to be changed to make sure the other person isn't cheating. In either case, coin tosses are often done in casual circumstances where close enough is good enough.
- aidenn0 9y agoNever, ever, allow someone you don't trust to do this. In a day you can learn to get about a 70%+ swing in your favor by timing the throw and catch, and if they call it in the air, you can either flip it twice or just once when you slam on your hand. All fair coin tosses should land on a flat surface. Felt or carpet is better than tile because it reduces the odds of the coin spinning when it lands, which is weight-biased.
- bio_end_io_t 9y agoWhat type of coin are we talking about? A coin with rounded edges won't land on its edge 1 out of 6000 tosses. A coin with a flat edge that is thicker than either of its faces (or just thick in general; doesn't have to be thicker than the faces) will land on its edge a great deal more often. Also, a coin with a heavier "tails" side will more often land on heads in a spin. I guess I should read the paper. Maybe it clarifies.
- bio_end_io_t 9y agoThe 1/6000 figure comes from a paper "Probability of a tossed coin falling on its edge" from 1993. I looked for it in the hopes it clarified the type of coin used, but the paper, as far as I could find, is behind a paywall. Nevermind, found a link that talked about the paper. They used a US 5¢ coin, a nickel which has a flat edge and I think the thickest edge out of all (common?) US coins. At least it has the thickest edge to face ratio in terms of width.
- eric_h 9y agoI have actually seen a £1 coin land on its edge, I imagine given that it's a fair bit thicker than any US coin, it's much more likely with a £1 coin.
- flarg 9y agoAs an aside, I practiced false flipping a whole ago and got quite good at it, assuming I remember the initial state I can always know the end state
- EGreg 9y agoCan you explain false flipping physically? On a related note, it's much harder to fake randomness over time: https://faculty.math.illinois.edu/~hildebr/fakerandomness/ https://faculty.math.illinois.edu/~hildebr/fakerandomness/
- flarg 9y agoMake the coin wobble instead of rotate when you flip it, looks like rotation but it isn't. Shiny coins work best.
- fastaguy88 9y agoFor an alternative view, check out Andrew Gelman's: "You Can Load a Die, But You Can’t Bias a Coin" http://www.stat.columbia.edu/~gelman/research/published/diceRev2.pdf http://www.stat.columbia.edu/~gelman/research/published/dice...
- gowld 9y agoGelman doesn't contradict. Gelman says that a properly flipped coin is always approximately 50% heads, regardless of coin design, but NOT exact, and NOT regardless of flipping style. The OP is about empirically observed real-world tosses.
- GuB-42 9y agoUse rock-paper-scissors instead. It can be considered a game of skill but any one player can turn it into a game of pure chance if he wants to, as such, is can be considered fairer. The reason why RPS can become a came of chance is that there is no way to gain advantage over a player who plays randomly with equal probability : the outcome will be 50/50 no matter what. In game theory, it's called a Nash equilibrium. Of course, if the opponent is playing sub-optimally (i.e. not randomly), you can try to take advantage of it.
- creaghpatr 9y agoThe opponent almost always plays sub-optimially, lots of research on this as well for example: https://www.washingtonpost.com/news/arts-and-entertainment/wp/2014/05/01/how-to-win-rock-paper-scissors-almost-every-time/?utm_term=.4eb4cfb852c4 https://www.washingtonpost.com/news/arts-and-entertainment/w... Need a random number generator or other computerized solution to create 'fair' 50/50 odds
- mikeash 9y agoHow can a human participant play randomly with equal probability without using a three-sided die or other such source of randomness?
- tdb7893 9y agoYou probably could get close enough by mentally stimulating a really simple prng
- GuB-42 9y agoYou can memorize a few random numbers beforehand. It is perfectly fine as long as they are used only once. You can look for external sources in order to increase your entropy, like the seconds hand of a clock. You can get additional numbers or limit your bias by mentally running a PRNG. A lagged Fibonacci generator is a good candidate for this. Otherwise, you are right, you simply cannot mentally generate good enough random numbers.
- tjoff 9y ago
- skizm 9y agoI am not even kidding when I say Bill Belichick is probably reading this article and will now have his captains try and game coin tosses.
- harryh 9y agoI generally don't pay too much attention to the coin toss, but Slater definitely called heads when the Super Bowl went into overtime. Maybe he already knows. EDIT: https://www.patspulpit.com/2017/2/7/14522972/patriots-captain-matthew-slater-has-called-the-same-side-for-every-coin-toss-for-6-years https://www.patspulpit.com/2017/2/7/14522972/patriots-captai... It sounds like it's been Slater making the all on his own but you never know....
- joosters 9y ago1% may not sound like a lot, but it's more than the typical casino edge in a game of blackjack or slots. That's an exceedingly optimistic view of casinos. Slot machines have a far higher edge than 1%, generally in the range of 3-15%.
- eof 9y agoLogically correct; as a game of blackjack following basic strategy is less than 1% ;-)
- tryingagainbro 9y agosure but that pays for salaries, building, drinks...etc. So they're lucky if they end up with 1%
- travisjungroth 9y agoThe comparison was made to casino games, not actually running a profitable casino.
- 3pt14159 9y agoPer pull? Really? I always thought casinos were dumb, but how could they be that dumb?!
- aidenn0 9y agoNote that Slots have the biggest swing for the house. A slot machine is basically a money-making skinner box. Blackjack with basic strategy (no card counting, just some 0 memory rules that would fit on a 3x5 card) is a narrow edge for the house, as is craps.
- joosters 9y agoCasinos have plenty enough slot machines to average out any swings in profit/loss. I doubt the variance is that large overall. Some time ago, I remember reading a talk given by a Las Vegas casino operator during a conference on risk. They analyzed their top ten largest losses for the previous year, and found none of them were associated with their gambling results. The events that actually cost them were completely unrelated occurrences, like their resident entertainer getting savaged by a tiger, large fines from the gambling commission because of an incompetent employee, and so on. The gambling side of the business generated remarkably smooth results.
- SubiculumCode 9y agoWith normal people, any bias in the coin is cancelled out by lack of knowledge of which bias the coin has.
- Robotbeat 9y agoThe easiest way to cheat is to just feel which side of the coin is which right after you catch it.
- jjnoakes 9y agoAnd then go back in time and call it in the air?
- CNJ7654 9y agoNot necessarily. You can choose to either open your palm, if you feel your side is up, or invert the coin onto the other hand, if your side is down :)
- aidenn0 9y agoI said this in another comment, but if you catch it and flip it down onto the back of your other hand, you can often surreptitiously add an extra flip in without being noticed.
- Robotbeat 9y agoExactly. I've done this multiple times (always letting people in on it afterward, of course). I have almost no skill in slight of hand, but it fools people every time.
- strong_silent_t 9y agoIf I'm understanding the article correctly, as long as the guesser can't see the coin, and chooses randomly, it is a fair 50/50. I don't think you even need to flip the coin. The coin could just be sitting on a table in the next room. If the guesser is ignorant and chooses randomly I don't think you even need the coin. You could just write heads or tails on a piece of paper, cover it, and let them guess.
- unfamiliar 9y agoI thought this was going to go in another direction: if someone proposes to flip a coin, then they have probably decided that a 50/50 chance is more favourable for them than any other potential decision method (like, say, discussing what the fairest or most reasonable decision to make would be). Hence if you agree to a coin flip that someone proposes, you have already lowered your chances of a favourable outcome for yourself.
- jimrandomh 9y agoNo, because other decision methods have costs (eg time spent discussing), separate from the favorability of the outcome. Most coin flips are done in low-stakes contexts, where people would rather let the other person decide than waste five minutes.
- LoSboccacc 9y agoI like the idea of the first kick in the world cup being decided by debate tho.
- microtherion 9y agoAre you a supporter of the Greek team by any chane?
- dwaltrip 9y agoIt seems a coin spin (on a table top) would avoid these issues.
- SamBam 9y agoThe article addresses that: it's much worse.
- dwaltrip 9y agoInteresting... thanks for the correction. Apparently the head side being slightly heavier do to the extra decoration and whatnot severely biases the coin to falling tails up when spun on its edge, up to 80% of the time with some coins.
- fractallyte 9y agoJohn von Neumann figured out a solution for getting fair results from a biased coin: 1. Toss the coin twice. 2. If the results match, start over, forgetting both results. 3. If the results differ, use the first result, forgetting the second. This has appeared on HN before, but no one's pointed it out so far in the discussion. More info: https://en.wikipedia.org/wiki/Fair_coin#Fair_results_from_a_biased_coin https://en.wikipedia.org/wiki/Fair_coin#Fair_results_from_a_...
- darkmarmot 9y agoThat is amazing! super biased coin and it works! var h = 0; var t = 0; for(i = 0; i < 1000; i++) { var a = Math.random() < .1; var b = Math.random() < .1; if(a !== b) { if(a){ h++} else {t++}} }; console.log(h,t);
- contravariant 9y agoYou could also just note that the two possible results where both coins differ are equally likely.
- samfriedman 9y agoIs there an extension to biased objects of more sides, say a die?
- contravariant 9y agoYou could wait until you got a run of all possible sides, but that could take a while.
- Steko 9y agoI think if you do 6 rolls and take the sum modulo 6 it works. edit: no, got confused with something else (game I used to play was subject to widespread paranoia about the dice including the RNG source and whether other players had hacks. The dev ended up jumping thru all sorts of hoops to satisfy people -- taking random.org, letting each client add their own modifier and then using VN) I think the straightforward way to extend VN is rolling 6 dice and reroll all if there are any duplicates, then use the first one (or whatever). Probably more efficient to turn the die into a coin and then use a few coins to approximate the die.
- aidenn0 9y agoIt seems that for all biases except the weighting, all that is needed is for the caller and thrower to be opponents, and call it in the air. Then the thrower will want the result to be as unpredictable as possible since they cannot predict what the caller will call.
- nixpulvis 9y agoWhat the original paper really needs is an extension to the world of the spooky and quantum. Right!?