# Gamblers fallacy **URL:** \ **Category:** Factual Questions\ **Created:** [April 12, 2004, 7:03pm UTC](https://boards.straightdope.com/t/gamblers-fallacy/239642 "2004-04-12T19:03:32Z")\ **Posts on this page:** 20\ **Page:** 3
**Author:** ![MonkeyMensch](https://avatars.discourse-cdn.com/v4/letter/m/82dd89/32.png) [@MonkeyMensch](https://boards.straightdope.com/u/MonkeyMensch)\ **Post date:** [April 13, 2004, 2:29pm UTC](https://boards.straightdope.com/t/gamblers-fallacy/239642/41 "2004-04-13T14:29:56Z")
**ultrafilter** , I’m surprised at you. 🙂 > [@ultrafilter](#): > > If you flip a fair coin a lot, the number of heads will be very close to the number of tails. People think that there’s some mechanism at work that corrects imbalanced sequences (like 10 tails in a row), but there isn’t: as you flip the coin more and more often, the longest imbalanced sequence becomes insignificant. If you filp a fair coin a lot the expected value of the difference between heads and tails _increases_ and is equal to the square root of N, with N = number of tosses. > [@](#): > > In fact, you can show that the probability of getting exactly 50% heads goes to 0 as the number of flips decreases. What is increasing to 1 is the likelihood that you’ll have very nearly 50% heads. The probability of getting exactly 50% heads is zero whenever N is odd and increases to 1 as N approaches zero. The probability when N is even is C(N, N/2)/2[sup]N[/sup]. This value approaches zero with increasing N. ---
**Author:** ![bordelond](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/bordelond/32/150_2.png) [@bordelond](https://boards.straightdope.com/u/bordelond)\ **Post date:** [April 13, 2004, 2:36pm UTC](https://boards.straightdope.com/t/gamblers-fallacy/239642/42 "2004-04-13T14:36:26Z")
> [@Telemark](#): > > I’ve heard of folks landing coins on edge. FWIW, I’ve seen a flipped nickel end up on its edge. It’s uncommon, but with a relatively thick-edged coin like a nickel, a smart bounce off of a tabletop can cause the coin to end up on edge. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [April 13, 2004, 3:34pm UTC](https://boards.straightdope.com/t/gamblers-fallacy/239642/43 "2004-04-13T15:34:47Z")
> [@MonkeyMensch](#): > > If you filp a fair coin a lot the expected value of the difference between heads and tails _increases_ and is equal to the square root of N, with N = number of tosses. I wasn’t clear. I was speaking of the relative difference, which has an expected value of 1/sqrt(N). That decreases to 0. My bad. ---
**Author:** ![World\_Eater](https://avatars.discourse-cdn.com/v4/letter/w/f04885/32.png) [@World\_Eater](https://boards.straightdope.com/u/World_Eater)\ **Post date:** [April 13, 2004, 4:08pm UTC](https://boards.straightdope.com/t/gamblers-fallacy/239642/44 "2004-04-13T16:08:42Z")
> [@AndrewT](#): > > What I don’t understand is why this seems a strange concept. Neither do I. 😃 > [@](#): > > Let me ask the opposite question of you: after 1,000 coin flips coming up heads, what force would act on the coin to make it come up tails? How would it “know” which way up to land, and how would it arrange that it land like that? Well before all these great explanations, _I would have said_, The chances of flipping 1,001 heads must be astronomical, it’s “due” to flip tails. Of course I understand why this is wrong now. **Princhester** , everyone else too, great posts. ---
**Author:** ![Hampshire](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hampshire/32/4887_2.png) [@Hampshire](https://boards.straightdope.com/u/Hampshire)\ **Post date:** [April 13, 2004, 4:18pm UTC](https://boards.straightdope.com/t/gamblers-fallacy/239642/45 "2004-04-13T16:18:09Z")
(Someone check my math and logic here) If I had $127,000 to use and wanted to make a quick $1000 would I have a 93% chance of doing it by betting black on roulette? Chances that roulette wheel will come up red 7 of 7 times are 1 in 14 or only 7% (disregarding 0 and 00). If I win on any spin I immediately stop and walk away with $1k. 1st spin bet - $1K 2nd spin bet - $2k 3rd spin bet - $4k 4th spin bet - $8k 5th spin bet - $16k 6th spin bet - $32K 7th spin bet - $64K Even with your 93% edge the house still has that 7% chance to take your $127K. Does it sound worth while? What are the odds with 0 and 00? ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [April 13, 2004, 5:01pm UTC](https://boards.straightdope.com/t/gamblers-fallacy/239642/46 "2004-04-13T17:01:21Z")
**Hampshire** , the strategy which you describe is called a Martingale. Using that strategy, you do in fact have a large chance of winning a small amount, and a small chance of losing a large amount. Your expectation is still exactly the same; for every dollar you put on the table you’ll get back, on average, 90-some cents. Nonetheless, it’s still probably a bad betting scheme (worse than, say, just putting $1000 on each spin), because you’re guaranteeing that you won’t get a big, exciting payoff (which is probably why you’re at the roulette wheel in the first place), you’re cutting short the amount of time you spend playing (you’re having fun, right?), and you’re leaving a very real possibility of losing all of your gambling money in a hurry (and then what fun would your Vegas vacation be?). ---
**Author:** ![js\_africanus](https://avatars.discourse-cdn.com/v4/letter/j/87869e/32.png) [@js\_africanus](https://boards.straightdope.com/u/js_africanus)\ **Post date:** [April 13, 2004, 5:38pm UTC](https://boards.straightdope.com/t/gamblers-fallacy/239642/47 "2004-04-13T17:38:59Z")
> [@Chronos](#): > > **Hampshire** , the strategy which you describe is called a Martingale. Using that strategy, you do in fact have a large chance of winning a small amount, and a small chance of losing a large amount. Your expectation is still exactly the same; for every dollar you put on the table you’ll get back, on average, 90-some cents. Worth noting, I hope, is that no betting scheme can improve your expectation. So if you use the Martingale or the Anti-Martingale or some other way of responding with your bet size to wins & losses, you still expect to walk away with a certain amount of money. What these schemes can do is change the odds of some particular outcome such as: How do I maximize my odds of taking as long as possible to go broke? How do I maximize my odds of winning a certain % in a given time? How do I minimize my odds of losing a certain % in a given time? These may be worth knowing for your Vegas Vacation—i.e. going as long as possible w/out going broke; however, no scheme will help you make a career out of gambling unless by “career” you mean becoming a casino owner. The only _reliable_ book I know of that discusses this is Epstein’s The Theory of Gambling and Statistical Logic. It is a heavy book; but the fact that you find it in the math section and not right next to _How to Win at Slots_ and _Beat the One-Armed Bandit_ by “Diamond” Bugs Murphy and Leftie McFee, respectively, says a lot IMO. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [April 13, 2004, 5:52pm UTC](https://boards.straightdope.com/t/gamblers-fallacy/239642/48 "2004-04-13T17:52:36Z")
> [@Phnord Prephect](#): > > Now, with this information, don’t we have to completely rethink the whole concept of coin-tossin’? No. The context of the “gambler’s fallacy” is a “fair coin”, which is by definition evenly weighted and flipped with equal chances for heads and tails. Even stepping outside this to “pattern finding”, only moves us from probability (loosely, predicting chances for events from theoretical considerations) to statistics (determining chances for future events by analyzing past events). So: if you’ve got a “fair coin”, the math is always right. If you’ve got a real coin and are incorrectly assuming it behaves like a fair coin, of course the predictions from probability will be wrong. If you throw it a bunch of times and use _statistics_, you can find an idea of the real probability it lands heads or tails (and a feeling of how accurate your idea is). Even so: now you’ve got an event with some probability P, and separate tosses are independant, which brings us back to saying that the gamblers’ fallacy is wrong. _There is nothing “pushing” a sequence of independant Bernoulli trials to succeed in proportion to the probability of success of each individual one, it’s a convergence result that is more and more likely to hold as the sequence gets longer without end._ ---
**Author:** ![Opus1](https://avatars.discourse-cdn.com/v4/letter/o/858c86/32.png) [@Opus1](https://boards.straightdope.com/u/Opus1)\ **Post date:** [April 13, 2004, 6:10pm UTC](https://boards.straightdope.com/t/gamblers-fallacy/239642/49 "2004-04-13T18:10:09Z")
Two quick, unrelated observations: 1. If you are absolutely forced into making a bet, you are betting off picking a “hot” number than a “due” number. There’s always a slight chance that the game is rigged, either intentionally or accidentally, so you might as well capitalize on it. If you’re wrong, you’re in no worse a position than if you had chose your bet randomly. 2. The British roulette guy was monumentally stupid. Even discounting the 0 and 00 on the roulette wheel, his odds were far from 50/50. He has to pay at least 35% taxes on his winnings. Since he’s a British national who won his money in America, he might have some complicated international income tax issues that will cause a major headache and incur some legal bills. (But given that he has no other income or possessions, it probably won’t be _that_ complicated.) And, virtually everything he sold had decreased significantly in value. He may have gotten only a few hundred bucks for his entire wardrobe, which will cost in the thousands to buy back. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [April 13, 2004, 6:17pm UTC](https://boards.straightdope.com/t/gamblers-fallacy/239642/50 "2004-04-13T18:17:43Z")
In re: the roulette guy. Have we determined that this was a real anecdote and not a misremembering of Kipling’s “If–”? ---
**Author:** ![PigArcher](https://avatars.discourse-cdn.com/v4/letter/p/bcef8e/32.png) [@PigArcher](https://boards.straightdope.com/u/PigArcher)\ **Post date:** [April 13, 2004, 6:54pm UTC](https://boards.straightdope.com/t/gamblers-fallacy/239642/51 "2004-04-13T18:54:11Z")
> [@Mathochist](#): > > In re: the roulette guy. Have we determined that this was a real anecdote and not a misremembering of Kipling’s “If–”? You mean [this guy](http://www.cnn.com/2004/SHOWBIZ/TV/04/12/roulette.win/index.html)? As to the Gambler’s Fallacy, I think it was best expressed by Clive Owen’s character in [**Croupier**](http://www.imdb.com/title/tt0159382/): > [@](#): > > Gambling isn’t about money. It’s about not facing reality; Ignoring the odds. ---
**Author:** ![Bytegeist](https://avatars.discourse-cdn.com/v4/letter/b/9dc877/32.png) [@Bytegeist](https://boards.straightdope.com/u/Bytegeist)\ **Post date:** [April 13, 2004, 7:12pm UTC](https://boards.straightdope.com/t/gamblers-fallacy/239642/52 "2004-04-13T19:12:00Z")
> [@Hampshire](#): > > Chances that roulette wheel will come up red 7 of 7 times are 1 in 14 or only 7% (disregarding 0 and 00). Minor correction: actually the probability of 7 reds in a row is (18/38)[sup]7[/sup] — about 0.535%, or 1 in 187. More relevant to your strategy though, the chances of 7 _non-black_ numbers in a row is (20/38)[sup]7[/sup] — about 1.12% or 1 in 89. This doesn’t change your strategy’s expected rate of return however, which is still positive in the casino’s favor. ---
**Author:** ![js\_africanus](https://avatars.discourse-cdn.com/v4/letter/j/87869e/32.png) [@js\_africanus](https://boards.straightdope.com/u/js_africanus)\ **Post date:** [April 13, 2004, 7:20pm UTC](https://boards.straightdope.com/t/gamblers-fallacy/239642/53 "2004-04-13T19:20:09Z")
> [@Opus1](#): > > 1. If you are absolutely forced into making a bet, you are betting off picking a “hot” number than a “due” number. There’s always a slight chance that the game is rigged, either intentionally or accidentally, so you might as well capitalize on it. If you’re wrong, you’re in no worse a position than if you had chose your bet randomly. Unless the scammer is waiting for a hapless punter to take advantage of. Then all bets are off, so to speak. > [@Max\_Castle](#): > > As to the Gambler’s Fallacy, I think it was best expressed by Clive Owen’s character in Croupier: “Gambling isn’t about money. It’s about not facing reality; Ignoring the odds.” Gambling and insurance are two sides of the same coin. They both trade in risk—casinos sell it to you and insurance companies buy it from you. It is perfectly reasonable to say that one enjoys playing the slots, even though one is virtually guaranteed to suffer a net financial loss from it over one’s lifetime, if one enjoys the risk of plunking nickels into the machine and hoping for a payout. That is a fair market exchange that one is free to make. The Gambler’s Fallacy is about not understanding the underlying concepts. Which is not to say that many of those who gamble aren’t described by the quote you provided. I think I’m probably just nit-picking at this point and should shut the hell up. ---
**Author:** ![World\_Eater](https://avatars.discourse-cdn.com/v4/letter/w/f04885/32.png) [@World\_Eater](https://boards.straightdope.com/u/World_Eater)\ **Post date:** [April 13, 2004, 9:12pm UTC](https://boards.straightdope.com/t/gamblers-fallacy/239642/54 "2004-04-13T21:12:34Z")
Great, just when I start to understand, you guys take it to the next level and start talking about Wink Martindale or something. 😃 What’s the story with 1st spin 1k, 2nd spin 2k, etc? ---
**Author:** ![PigArcher](https://avatars.discourse-cdn.com/v4/letter/p/bcef8e/32.png) [@PigArcher](https://boards.straightdope.com/u/PigArcher)\ **Post date:** [April 13, 2004, 10:05pm UTC](https://boards.straightdope.com/t/gamblers-fallacy/239642/55 "2004-04-13T22:05:49Z")
> [@js\_africanus](#): > > Which is not to say that many of those who gamble aren’t described by the quote you provided. I think I’m probably just nit-picking at this point and should shut the hell up. I should have specified that the quote refered to _compulsive_ gamblers (Owen’s character is a recovering one). I think most casual gamblers (myself included) realize that they will lose in the long run and only play for fun. ---
**Author:** ![Alan\_Smithee](https://avatars.discourse-cdn.com/v4/letter/a/2acd7d/32.png) [@Alan\_Smithee](https://boards.straightdope.com/u/Alan_Smithee)\ **Post date:** [April 14, 2004, 4:43am UTC](https://boards.straightdope.com/t/gamblers-fallacy/239642/56 "2004-04-14T04:43:04Z")
[QUOTE=Phnord Prephect] Please, allow me to throw a nice big wrench into the works of this otherwise logical and correct discussion: > **[The Not So Random Coin Toss](https://www.npr.org/templates/story/story.php?storyId=1697475)** > > Flipping a coin may not be the fairest way to settle disputes. A team of mathematicians claims to have proven that if you start with a coin on your thumb, heads up, flip it and catch it in your hand, it's more likely to land heads up than tails.... [QUOTE] Heh. I figured this out a long time ago as a kid. As a result, I usually got to pick where we ate out when my sister and I disagreed. I’d always look at the coin in our Dad’s had right before he tossed it, and quickly call out whatever side wasn’t showing (since he’d turn it over when he caught it). Very often the coin wouldn’t actually flip at all in the air. I wouldn’t be at all surprised if, when it did flip, it tended to flip a consistent number of times, just as a piece of buttered toast will typically flip approxamately .5 times in the distance from the kitchen counter to the floor. (Yes it was a typical, modern, kid-centric household, and my sister and I got to decide where we ate. Neither of us is in jail yet, and I at least have very catholic tastes for food.) BTW, great stories, **Jurph**. An excellent way of refuting the Gambler’s Fallacy. Did you make those up, or did you steal them from someone? (John Allen Paulos, maybe?) Either way, I plan to steal them from you if I ever have a discussion of the Gambler’s Fallacy. ---
**Author:** ![shijinn](https://avatars.discourse-cdn.com/v4/letter/s/cc9497/32.png) [@shijinn](https://boards.straightdope.com/u/shijinn)\ **Post date:** [April 14, 2004, 4:54am UTC](https://boards.straightdope.com/t/gamblers-fallacy/239642/57 "2004-04-14T04:54:36Z")
someone please disabuse me of this notion: using **Hampshire** ’s method and **bytegeist** ’s calculation, a 10 non-black streak would be about 0.16% or 1 in 625. so if i would to say, wait for 3 reds to appear then start betting, does that mean i have about 1 in 625 tries to lose the 127k? does that mean i can win 625k during which i will lose only 127k?? i know that the expectation remains the same and varying the bets does not change the odds. this is especially clear when you are on that 64k bet - you still have a 10/19 chance of losing it. so what is wrong about the notion exactly? ---
**Author:** ![Garfield226](https://avatars.discourse-cdn.com/v4/letter/g/9e8a1a/32.png) [@Garfield226](https://boards.straightdope.com/u/Garfield226)\ **Post date:** [April 14, 2004, 5:26am UTC](https://boards.straightdope.com/t/gamblers-fallacy/239642/58 "2004-04-14T05:26:22Z")
> [@shijinn](#): > > someone please disabuse me of this notion: > > using **Hampshire** ’s method and **bytegeist** ’s calculation, a 10 non-black streak would be about 0.16% or 1 in 625. so if i would to say, wait for 3 reds to appear then start betting, does that mean i have about 1 in 625 tries to lose the 127k? does that mean i can win 625k during which i will lose only 127k?? > > i know that the expectation remains the same and varying the bets does not change the odds. this is especially clear when you are on that 64k bet - you still have a 10/19 chance of losing it. > > so what is wrong about the notion exactly? Well, I’m not a math major, but until one comes along, let me try. If, at the beginning of 10 trials, you say, “What are the odds that the next 10 will not come up black?” the answer is 1 in 625. That’s correct. If, at the beginning of seven trials, you say, “What are the odds that the next 7 will not come up black?” the answer is 1 in 89. That’s correct. In other words, it doesn’t matter what happened in the past. If you jump in after any possible combination of any number of trials, the odds that the next 7 numbers will be non-black is 1 in 89. Always (assuming a fair wheel etc…) Look at it **jk1245** ’s way: Chance of first three (that have already happened) being non-black=1 Chance of next seven being non-black=.0112 Chance of first three (that have already happened) being non-black AND next seven being non-black= 1\*.0112=.0112 ---
**Author:** ![shijinn](https://avatars.discourse-cdn.com/v4/letter/s/cc9497/32.png) [@shijinn](https://boards.straightdope.com/u/shijinn)\ **Post date:** [April 14, 2004, 6:03am UTC](https://boards.straightdope.com/t/gamblers-fallacy/239642/59 "2004-04-14T06:03:42Z")
i think i understand what you’re trying to say, but why can’t i walk in with 127k betting that a streak of 10 will happen only 1 in 625 tries, so i can slowly win 1k by 1k in the meantime? ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [April 14, 2004, 6:28am UTC](https://boards.straightdope.com/t/gamblers-fallacy/239642/60 "2004-04-14T06:28:34Z")
If you walk in with 127K and use a Martingale scheme, then you probably will win a grand. But you might lose 127 grand, without winning anything. The first possibility is more likely, but it is not 127 times more likely, so the Martingale scheme (like every other gambling scheme) is a loser. [Previous page](https://boards.straightdope.com/t/gamblers-fallacy/239642.md?page=2) [Next page](https://boards.straightdope.com/t/gamblers-fallacy/239642.md?page=4)