Talk:Gambler's fallacy

This Gambler's fallacy has a fallacy within itself
What if I imagine I bet 0 unit on the first several game? You cannot get it logically right — Preceding unsigned comment added by 220.232.202.2 (talk) 07:43, 15 January 2015 (UTC)

Good article
Before nominating again, fully source the article with references that comply with WP:MEDRS. --LauraHale (talk) 08:41, 27 April 2012 (UTC)

Slot machine jackpots can in most cases be "due"
In most jurisdictions, Slot Machines etc have Payout schedule / percentage required by law and predetermined by the manufacturer. http://en.wikipedia.org/wiki/Slot_machine#Payout_percentage. Perhaps modify the example in the second paragraph to Roulette or something? — Preceding unsigned comment added by Billyoffland (talk • contribs) 15:14, 6 October 2011 (UTC)

Pim says: Totally agree! This sentence suggests slotmachines make random choices, while all modern slotmachines are programmed to be predictable.. — Preceding unsigned comment added by Pimnl (talk • contribs) 19:32, 16 October 2011 (UTC)

Whoa, with respect, the above two comments are absolutely incorrect. Slots do in fact have a payback percentage set but this does not ever mean a machine can be "due". The payback percentage reflects what will statistically happen given the chance of each reel stop and the payback set to particular combinations of stops. This payback percentage is in no way guaranteed to occur in any set period of time. A slot machine might be set with a 95% payback schedule but may payback 50% one day and 150% the next. It could even payback well above or below its payback percentage for years, however statistically unlikely. The only guarantee is that in an infinite number of spins the machine will average its set payback percentage. The random number generators in slots are independent trials and do not have "memories" and the computer inside will not, and cannot, influence the RNG to try and match the payback. This is based on 15 years of experience as a casino employee. I've worked with game manufacturers and have read plenty of PAR sheets. For another source see: http://www.casinocenter.com/?p=846 AlmtyBob (talk) 06:24, 17 February 2012 (UTC)

To put all that another way: If a slot machine fails to pay out for a whole year, it is not the case that some agent of government will knock on the casino's door to talk about it, nor will the casino be expected to do anything in particular. They might generally be require to demonstrate the machine's fairness ante facto, but not post facto. (If it were the latter than it might make some sense to consider a result "due", especially if it's the last day within a year during which the machine is officially required to pay out 1% of earnings, and it hasn't paid out all year.) 141.158.199.150 (talk) 20:30, 18 May 2013 (UTC)

Here's some more information in support of AlmtyBob. The truth is that programmers calculate slot machines payout rates OVER THE LIFETIME OF THE MACHINE, not to an annual or any other arbitrary standard. Read any of the industry journals and you'll see this corroborated. Also, I called the gambling commission to check this out in my home state of WA, as I'm working on a white paper on this very subject. And my state commission agrees with what has been stated in every published book on the industry (I've looked this up in 4 of them), that payout chips on the motherboard of the machine chassis are not set to "pay out" at any time in particular, that they function almost (but not quite) as randomly as the random number generator over the life of the machine, and that this generally random behavior is required in the base game by state and federal law. The machines in my jurisdiction even on Indian land are regularly inspected by commission agents. Any changes on the gaming floor must be registered with the gambling commission first for departmental and public review, so none of this is secret. When I asked the department rep on the phone if slot machines are set to pay out by a particular date in the year, she just laughed. Here's both an online and published source, widely respected, that refutes the contentions in the original unsigned comment above: https://www.americancasinoguide.com/. Laguna greg 21:09, 31 August 2018 (UTC) — Preceding unsigned comment added by Laguna greg (talk • contribs)

Coins
Allow me to point out that euro coins don't have an equal distribution of weight hence the national side of the coin comes up more often. Try it for yourself and see. —Preceding unsigned comment added by 86.45.154.157 (talk) 19:56, 8 September 2009 (UTC)
 * I cannot find any verifiable evidence for this, but I can find verifiable evidence that coins, in general, are fair when flipped, regardless of the distribution of weight. You cannot "weight" a coin like you can "weight" dice, unless you are spinning, rolling, or otherwise allowing the coin to contact a surface before stopping it. Eebster the Great (talk) 02:06, 9 September 2009 (UTC)

I have always been unconvinced of the lack of bias in coin-tossing. Any coin is of a finite size. When tossed, finite energy is pumped into the flick that gives it lift and spin. The coin, on any trial, will then rotate a certain finite number of half-spins while falling upwards, and a possibly larger number when it pauses and falls downwards towards the landing surface. Humans have a tendency to flick coins with a roughly similar thrust on each toss and each human is, of course, the same height and has the same arm length (roughly) on every toss. That leads to the anecdotal conclusion that all tossings of the same coin will have roughly the same number of half spins. The only "random" factor in the process is then the starting condition; whether the coin is heads-up or not just prior to the toss. I have the feeling, completely unsubstantiated by any evidence save personal observations of my own behaviour on a very few occasions when I remembered to note it, that an human tossing a coin repeatedly will force a seemingly random selection of heads-up positions before tossing the coin "just for fairness' sake". In short, people fiddle the game to produce expected results. While this cooking of the books may not invalidate a single toss I suspect it biases experiments in repeated coin-tossing in favour of "randomness". I would really like to know if I'm right but I see absolutely no way of unequivocally proving the point either way. Qordil (talk) 17:11, 27 August 2013 (UTC)


 * The initial conditions that vary from one toss to the next, by the same person, are not just which coin face is initially pointing up. The precise direction of hand, thumb, and arm movement inevitably varies -- it's impossible for a person to make these identical on different tosses; also, the precise amount of energy imparted to the coin inevitably varies from toss to toss. And the nature of the coin toss is that the outcome is very sensitive to these slight variations in initial conditions. Duoduoduo (talk) 17:38, 27 August 2013 (UTC)


 * I agree that the initial conditions are unlikely to be identical apart from the obvious state of which side of the coin is facing up as the toss starts, but I have never been sure that coin tosses are sufficiently sensitive to small variations in thrust and angle to affect the outcome. I may, of course, be entirely wrong; intuition is an exceedingly poor guide to probability and randomness, as this very discussion and the subject matter under discussion show. Coins being tossed tend to be large, they tend to be fairly heavy and they tend not to be tossed with great force so I suspect the initial forcing conditions don't affect the result too much. Whether "too much" is "enough" to produce a chaotic system that is sufficiently sensitive to initial conditions to mimic a random "fair toss" or not is exactly what I would like to see tested. If it could be.

This intuitive feeling that coin tosses are sometimes forced by the tosser has bothered me sufficiently that I thought I would mention it. I would be exceptionally pleased if Science could put the notion to bed finally and forever someday. Qordil (talk) 17:56, 27 August 2013 (UTC)


 * You could test it yourself: flip a coin once, remembering how you flipped it and what the result was. (To make it fair, make sure it rotates at least three or four times.) Then do 100 more flips, each time trying to replicate your original flip. The null hypothesis is that the flips are independent, in which case the total number of heads is binomially distributed with probability parameter = 1/2. By Binomial distribution, the number of heads is close to normally distributed under the null, with mean = 50 and standard deviation = 5. So there's a 95% probability under the null that the number of heads will be in the range 50 plus or minus 1.96 × 5, or between 41 and 59 inclusive. If you're trying for heads and you get more than 59 heads, that's unlikely to have happened with fair independent coin tosses. Duoduoduo (talk) 20:01, 27 August 2013 (UTC)

circular logic
The offered proof in the article is fallacious itself, because it is Begging_the_question

consider "Explaining why the probability is $$\scriptstyle\frac{1}{2}$$ for a fair coin":

The fallacious gambler works under the assumption that probability is ever-changing, depending on the previous outcomes. Thus he would not assume

but rather something like
 * probability of 20 heads = 0.520
 * probability of 20 heads = [(1-x)*(0.5)]20

since for the fallacious gambler the probability of Heads decreases with every Heads.

The corrolary is that for the fallacious gambler a fair coin does not exist unless it has previously produced perfectly even results and even then it becomes biased again after the very next toss. The fallacious gambler cannot within his logic calculate 2 or more coin tosses using the same probability for each.

Hence the fallacy cannot be disproved using the toss of a fair coin, since the existence of such a coin is already contradicting the gambler's fallacy and it is rather unsurprising that any subsequent reasoning would do the same. BharatKulamarva (talk) 11:12, 29 November 2009 (UTC)


 * It seems like your argument is that the "fallacious gambler" uses "fallacious logic".....which is evidently true and is the basis of the Gambler's fallacy.


 * Certainly you must admit that fair coins exist. You can test it yourself by flipping coins and recording the number of heads/tails.  So indeed, as you said, the existence of such a coin contradicts the fallacious gambler's logic.  This does not mean such coins do not exist, rather it means that the gambler's fallacy is exactly just that, a fallacy.  Paul Laroque (talk) 01:21, 21 December 2009 (UTC)


 * Agreed. As I said: a fair coin does not exist for the fallacious gambler. Thus if a fair coin exists, the fallacious gambler is proven to be - well - fallacious.
 * My point though was: If I was a fallacious gambler, the paragraph in question would not be a proof to me, since it assumes the existence of a fair coin, which - being a fallacious gambler - I would deny.
 * Herein lies the heart of the problem of proving the fallacy. The existence of a fair coin is technically just an assumption, albeit an assumption that most reasonable people (including me) agree to be true and that the whole of classical probability theory builds upon.
 * Thus my reasoning that you cannot - strictly speaking - prove the fallacy, just illustrate its insanity. For example it would follow that - since all coins are biased - you could prepare dice for a competition by purchasing a lot of them and rolling them beforehand at home. You would then keep the dice that are "due" (e.g. had very little sixes) and take them to a competition. This absurd example should illustrate the fallacy especially since it exactly contradicts the concept of "lucky dice" which many gamblers believe in as well. --BharatKulamarva (talk) 11:58, 22 December 2009 (UTC)

Ok, so it is very obvious that if we have a set of fair coin flips of TTT that the next flip has a .5 chance of being tails. But among the next two flips we have a more complex set of possible outcomes, i.e. TT, TH, HT, HH. So, the odds of getting a single heads in the two flips is 3/4, while the odds of getting only tails is 1/4. Am I missing something about the gamblers fallacy or does it only really apply to expectations of the initial or next result? If I'm not horribly misunderstanding the argument here, it should be clarified by linking to other articles, etc. And, I'm perfectly willing to help with clean up. — Preceding unsigned comment added by Andwats (talk • contribs) 07:21, 24 March 2012 (UTC)


 * BharatKulamarva is correct, the logic is circular, the thing being proved "the next toss has 50/50 chance" and the assumption "a fair coin exists and has a 50/50 chance on each toss" are the same, therefore it is question begging. Paul Laroque's example of how you can test for yourself if there exists "a fair coin" actually favours the gambler's fallacy because if you tossed a coin n times, and the result was 50/50, it doesn't need to show that each toss was 50/50 it could show that the nth toss was "due" as the gambler's fallacy tells us, and brought things back into balance. The fallacy doesn't lie in the math, but in the creation of the observed set but this is OR. The fact remains that the "proof" offered is circular and therefore worthless.

From a gambler point of view.
The theory is true, the math its accurate but in the real world and from a gambler point of view it doesn't work exactly like that. No physical system is 100% random, A coin or dice will have infinitesimal variations in the distribution of mass, shape... A roulette table would have hundreds if not thousands of variables affecting the odds, a poker slot machine has a pseudo random number generator.... The list goes on

For instance, a very well know method to bit the odds in roulette is to expend days or even weeks on a given table writing down the numbers, after you have obtained a significant sample its only a matter of entering the data on a computer and run an statistical analysis. You will always find a deviation, the ball has a slightly bigger tendency to fall on certain area of the wheel, then you calculate your playing strategy according to those statistics, if you play smart and long enough the house looses.

Casinos of course hate this kind of thing, they will ban you if they find out what you are doing. Roulette makers spend a great deal of time fine tunning the tables in order to minimize the effect and make the system as random as possible, random generators on gambling machines use huge base lists, dices are manufactured as uniformly as possible, shapes with tolerances on the 100s of millimeters... No matter how hard they try, Physical tolerances will cause a deviation from the mathematical odds. The goal is to make those variations small enough to prevent anybody from taking advantage of them, but they will always be there. Its an intrinsic characteristics of any real physical system.

I've been banned from casinos in Europe for playing black jack in the way they like less (Never cheated) and for using this tactic playing roulette, takes time and self discipline, They've got so good at building those devices that the money earned is in the best possible scenario just enough to make a living, because all the precautions taken the deviations are really small, a mistake will set you a long way back. Roulette is not a good game for a professional gambler but the method does work if done properly.

--70.186.170.117 (talk) 13:23, 15 January 2010 (UTC) Carl - Louisiana


 * I don't believe you have won money on roulette using this strategy. The idea that you could find the extremely marginal tendencies of a decent roulette wheel even with years of daily study seems very unlikely, but even if you could, this tendency would not begin to approach the house advantage. The house has a 5.3% edge in roulette--this is quite significant. Obviously the wheel is not so skewed that it is six percent more likely that the ball fall into a given pocket than it should. That would be one seriously warped wheel. Besides, the amount of time you would need to invest into studying it would be enormous to get a statistically significant sample. No, if you really have won money playing roulette, you are just very lucky.


 * Blackjack is different, because the house edge is very small if the player makes all the right bets (as determined by computers), only about 1% for a normal pack. By counting cards, one can theoretically wait to play until the pack is rich with aces, tens, and face cards, making blackjack likely enough that the odds are actually, temporarily, in the player's favor if the player bets correctly. So in principle, one can make only extremely low bets when the deck is not in her favor, and then bet enormous sums of money when it is, therefore making money overall. However, this is not usually a viable strategy for a number of reasons. For one thing, all casinos have table maximums and minimums, so you usually cannot vary your betting enough to overcome the house edge. Furthermore, casinos that suspect players of counting cards usually kick them out, so it must be very surreptitious, meaning one cannot simply sit there betting the minimum and then suddenly start betting thousands of dollars when the deck is good. That said, a few MIT kids did manage to pull it off a few times, and they are not the only ones. So it is impossible, but rare and difficult, and not viable in many casinos.


 * But neither of these has anything to do with the gambler's fallacy. That fallacy has to do with the belief that, say, a roulette wheel that has spun red five times in a row is bound to spin black next, whereas what you are describing is actually the opposite (since the roulette wheel spins red so much, it must be skewed toward red), and not significant enough to affect practical betting anyway. Eebster the Great (talk) 06:27, 16 January 2010 (UTC)


 * I don't agree with the article. Yes, while each time you get red, it was 50% chance, but consider that for you to get 6 reds in a row, is much smaller chance, than getting 5 reds and 1 black. If you know statistics, it would be smart of you to pick black, even though the events are unrelated, the chance of getting 6 reds in a row is still much smaller. This doesn't mean that you cannot get 6 reds in a row, there is a 50% chance, so you could very well lose. It's not logical at all to assume 100% that the next one will be black, but it is more likely. Things like 20 heads in a row, are more rare than 10 heads and 10 tails interlaced. 96.231.249.80 (talk) 20:42, 15 June 2011 (UTC)


 * It is true that 5 reds and 1 black in some order is more likely than six reds, but that's because 5+1 can be RRRRRB, RRRRBR, RRRBRR, RRBRRR, RBRRRR, BRRRRR — each of these is exactly as likely as RRRRRR but collectively they are six times as likely. Given RRRRR_, the final sequence is either RRRRRR or RRRRRB, equally likely, and your belief to the contrary is the gambler's fallacy. —Tamfang (talk) 21:59, 15 June 2011 (UTC)


 * Why is it a fallacy? RRRRRB is simply more likely, than RRRRRR... There is still a 50% chance that R or B will be the 6th. But usually, objects and physics being random, and R and B having equal chance of getting picked, it seems likely that eventually a B should come up, or something is broken with the roulette table. 96.231.249.80 (talk) 04:34, 20 July 2011 (UTC)


 * It is not a fallacy to say that after five Rs, a B is likely to eventually come up. Actually, it's not just likely for B to eventually come up, it's 100% certain. Doesn't matter if the wheel is biased, doesn't matter what the previous spins were. The key however is the word "eventually". If you spin it ten thousand times you can bet your life that you'll get at least one B! The "gambler's fallacy" concerns whether or not B is more likely than R on a particular spin, not just "eventually". Because when a gambler bets, he is betting on a particular spin. I mean, you can't go into a casino and make a bet that "at least one of the next ten thousand spins will be B"! :-) --Steve (talk) 06:48, 20 July 2011 (UTC)


 * RRRRRB is not more likely than RRRRRR. To suggest otherwise is the very essence of the fallacy. It's easier to think in terms of flipping a coin: how would the coin know that heads came up five times in a row so that it should spin toward tails this time? It simply doesn't. If anything, RRRRRR is more likely than RRRRRB because a string of identical outcomes may indicate that the apparatus is not actually statistically unbiased. (However, a string of five even-chance events is not enough to suspect bias for a real coin or a professionally produced roulette wheel.) Probability is difficult to learn, and even those of us who have studied it for years have to actively suppress our cognitive biases at times. Evolution has wired the human brain to misestimate the probabilities of certain types of events. Understanding probability offers substantial benefits in our lives, so it is smart to learn from to the experts rather than saying they're wrong. The truth is sometimes unintuitive, but it's still the truth. (BTW, @Steve: I think it's fairly safe to say that nothing in the real world is actually 100% certain. However, I completely agree that it would be prudent to bet your life against the chance of 10 k sequential lands on red—provided you know that the wheel is fair and that it will remain fair. :) Gerweck (talk) 13:56, 10 August 2013 (UTC)


 * While RRRRRB is not more likely than RRRRRR, it is still a fact that eventually the colour will change. And if the string is already unusually long, we can assume that it will end soon.--77.12.23.146 (talk) 19:18, 28 December 2019 (UTC)


 * "It is still a fact that eventually the colour will change". No it isn't. The probability is 18/37 on each spin, and "unusually long" means very little. Believing things like this is the reason why the fallacy occurs.-- ♦Ian Ma c M♦  (talk to me) 06:58, 29 December 2019 (UTC)


 * I think this is actually a very good point that is being raised. Gamblers do indeed practice this strategy, and it is also documented thoroughly in the statistical/mathematical literature on the subject.  There are several tests for bias documented in the statistical literature.  This prediction method is dealt with in the article in the section entitled "Non-example: Unknown probability of event", though this section could certainly be expanded with some more work.  The paper referenced in that section explains why betting on the most common outcome is the optimal prediction method under the assumption that biases may exist in the process (and certain other plausible assumptions about this bias).  SCF71 (talk) 8:26, 7 August 2010 (UTC)


 * This is not a good point. Roulette wheels are very precisely machined to have the least bias possible.  I'll admit that it is impossible to make a wheel completely without bias.  The miniscule default bias would exploitable if not for the casino edge.  Beating the 2.7% edge of a single-zero wheel would require a very large bias and a very large number of trials.  Wheels do gain even more bias over time, which is why casinos balance them on a daily or weekly basis.  Even if the wheel is only balanced monthly and the game is dealt at a reasonable speed (40 spins/hour) for 24 hours a day AND every single spin was tracked, that's still only 28,800 spins to detect a bias.  It's incredibly unlikely, even in those ideal circumstance, to detect a bias large enough to overcome the house edge. AddBlue (talk) 06:44, 17 February 2012 (UTC)

Non-examples of the fallacy
The statement "This is how counting cards really works, when playing the game of blackjack." is erroneous. The (spurious) skill of card-counting for profit is not based on either remembering which individual card values have been previously dealt, or on calculating the ongoing probabilities of individual card values appearing. That this follows an example that uses a Jack (specifically, in lieu of a 10-value card generally), only serves to compound the error.
 * I would ask former members of the MIT Blackjack Team whether the statistical advantage conferred by card counting is spurious. I do agree that it is difficult to beat the house, both because few people possess the required faculties and because casinos take effective countermeasures. It's also not true that individual card values are ignored by all types of card counters. In particular, ace tracking has been used successfully by some professional gamblers. Gerweck (talk) 14:21, 10 August 2013 (UTC)

Very first sentence false?
The first sentence of the article is "The Gambler's fallacy, also known as the Monte Carlo fallacy (because its most famous example happened in a Monte Carlo casino in 1913)[1] or the fallacy of the maturity of chances, is the belief that if deviations from expected behaviour are observed in repeated independent trials of some random process then these deviations are likely to be evened out by opposite deviations in the future."

I have a major problem with the way this is stated. In a very specific and quantitative sense, it IS true that deviations from expected behavior are likely to be evened out by future results - not by opposite deviations exactly, but simply by virtue of the fact that future results will average to the mean, and there will eventually be many more than them than the original deviation. That's called the law of large numbers, and it lies at the base of all of statistics.

So I suppose the article's first sentence isn't exactly wrong, but I think it's potentially very misleading. It ought to be re-phrased to make it clear that the fallacy is believing that the future results are in any way influenced by those already obtained, or to highlight more clearly the fallacious part in the sentence as is (which is that the deviations will be evened out not simply by more data, but specifically by opposite deviations).

Unless someone else has any objection, I'll re-write the first sentence to something like this: "The Gambler's fallacy, also known as the Monte Carlo fallacy (because its most famous example happened in a Monte Carlo casino in 1913)[1] or the fallacy of the maturity of chances, is the belief that if deviations from expected behaviour are observed in repeated independent trials of some random process, future deviations in the opposite direction are then more likely." (Can anyone think of a better term than "direction"?)   Waleswatcher (talk) 03:01, 19 February 2011 (UTC)


 * "it IS true that deviations from expected behavior are likely to be evened out by future results" Nope. They don't even out. Things only even out if you average them, but in many cases you're interested in (for example) the amount of money in your pocket, and that is not an average, it's a sum. That's not the same thing. The average converges, the sum diverges, it follows a drunkard's walk in fact, and gets further and further from the mean with repeated gambles.Rememberway (talk) 03:23, 19 February 2011 (UTC)


 * "Nope. They don't even out. Things only even out if you average them" Sorry, but the latter is *exactly* what I said.  You took part of the middle of a sentence from my comment, dropped the beginning and end, and then repeated the rest in your own words - so I'm not sure what your point was supposed to be.  My point is that someone reading the first sentence of the article as is now might well conclude that it's a fallacy to believe that the average will even out, when in fact it isn't.  Do you object to my proposed re-wording?  Waleswatcher (talk) 13:58, 19 February 2011 (UTC)


 * Sorry in advance for being a bit of a pedant, but that first sentence is still slightly incorrect. Gambler's Fallacy is the belief that a past independent trial will affect a future independent trial.  The difference there being that many people falsely believe a streak will continue, not just that the opposite will happen.  Examples of this in a casino abound, from people who believe they are on a winning "streak", particularly in craps, to baccarat players that wait for "runs" of Player or Banker wins and then bet that side, assuming the streak is bound to continue.  It's not incorrect to the point that I'd edit it, but since you brought it up... AddBlue (talk) 06:52, 17 February 2012 (UTC)

The story of the events at Monte Carlo Casino in 1913 is itself questionable. Something of this nature would surely have been reported in the press at the time, yet I have searched several online newspaper archives without finding any references to the event. — Preceding unsigned comment added by StylusGuru (talk • contribs) 15:24, 10 December 2016 (UTC)

Inverse gambler's fallacy
I removed a link to the inverse gambler's fallacy. The article with that title describes it as drawing the conclusion that there must have been many trials from observing an unlikely outcome. The rather different concept this article was referring to was the belief that a long run of heads means that the next roll is outcome is likely to be heads. MathHisSci (talk) 16:38, 7 April 2011 (UTC)
 * Yet let's have a look at the article content. "The probability of getting 20 heads then 1 tail, and the probability of getting 20 heads then another head are both 1 in 2,097,152." So, the probability of me getting heads that many times is ~ 1 in 2 million? Without ANY evidence to support either position, would the more rational assumption be the coin is not fair and heads is weighted? Since we talk about "fair coin" (no such thing, varying degrees of debris, weighting of the print, etc) and we didn't even consider fair toss, while it doesn't invalidate the fallacy, would you not agree it brings into question some of it's more common applications? — Preceding unsigned comment added by 87.112.178.244 (talk) 15:39, 25 May 2011 (UTC)


 * Yes, the inverse gambler's fallacy, as this article defines it, can be seen as more rational than the usual version in practice, though not in the formal mathematical model. MathHisSci (talk) 21:39, 8 August 2011 (UTC)

Psychology of Gambler's Fallacy
Here are some sources that I'm considering for this page, and what they will contribute to the page:

Burns, B.D. and Corpus, B. (2004). Randomness and inductions from streaks: "Gambler's fallacy" versus "hot hand." Psychonomic Bulletin and Review. 11, 179-184. These researchers found that people are more likely to continue a streak when they are told that a non-random process is generating the results. The more likely it is that a process is non-random, the more likely people are to continue the streaks. Useful explanation of the types of processes that are more likely to induce gambler's fallacy.

Croson, R. and Sundali, J. (2005). The gambler's fallacy and the hot hand: Empirical data from casinos. The Journal of Risk and Uncertainty 30, 195-209. This is an observational study rather than an experiment, observing the behaviors of individuals in casinos. I found it interesting that they also observed the "hot hand" phenomenon in gamblers as well - and that it's not just restricted to basketball.

Oppenheimer, D.M. and Monin, B. (2009). The retrospective gambler's fallacy: Unlikely events, constructing the past, and multiple universes. Judgment and Decision Making, 4, 326-334. This article introduces the retrospective gambler's fallacy (seemingly rare event comes from a longer streak than a seemingly common event) and ties it to real-world implications. The researchers tie it to the "belief in a just world" and perhaps even hindsight bias (the article talks about how memory is reconstructive).

Rogers, P. (1998). The cognitive psychology of lottery gambling: A theoretical review. Journal of Gambling Studies, 14, 111-134. Ties the gambler's fallacy in with the representativeness and availability heuristic. Defines gambler's fallacy as the belief that chance is self-correcting and fair.

Roney, C.J. and Trick, L.M. (2003). Grouping and gambling: A gestalt approach to understanding the gambler's fallacy. Canadian Journal of Experimental Psychology, 57, 69-75. Explains that simply telling people about the nature of randomness will not eliminate the gambler's fallacy. Instead, the grouping of events determines whether or not gambler's fallacy occurs. Very interesting, and possibly a good source for a possible "solutions" section.

Sundali, J. and Croson, R. (2006). Biases in casino betting: The hot hand and the gambler's fallacy. Judgment and Decision Making, 1, 1-12. Correlates hot hand and gambler's fallacy - people who exhibit one will also exhibit the other. Introduces the possibility of a construct underlying both of these.

One idea I had for possibly altering the structure of this article: dividing the "psychology" section into subsections by each psychological concept - biases, grouping, etc. Songm (talk) 21:40, 7 March 2012 (UTC)

If any of you would like to see some of the edits I'm planning for this page, you can check out my sandbox here. —Preceding undated comment added 01:11, 28 March 2012 (UTC).

philosophical point of view
It is entirely possible that the universe does have a 'memory' of events and that probability theory and the idea of randomness are not actually correct.

There is no way to prove probability theory. You can't prove probability theory by, for example, tossing a coin and counting results and comparing to expected results because you would actually have to use the theory to do that comparison. The argument becomes circular. It is just one of the axioms we just accept in science.

I work with probabilities and stats so I'm not saying it is wrong. I'm pretty sure it is right and it's a great tool. But, I do find it fascinating that it may well be false and there is no way of knowing if it is or isn't. There is, for example, no way to demonstrate or prove 'randomness'. We must simply state that a coin toss is random and accept it. There are tests for randomness, but there are many sets of numbers that pass randomness tests that are in fact not random, the famous example being the Mandelbrot set. These sets exist in nature frequently.

So maybe the gambler's fallacy is not so false after all.

Still I think it *probably* is :-p  — Preceding unsigned comment added by 129.78.32.22 (talk) 05:47, 7 August 2012 (UTC)

Childbirth
"The probability of having a child of either gender is still regarded as 50/50." The idea that the probability is 50/50 is a fallacy. It is only roughly true, with respect to large populations. In human populations we see that there are slightly more male births than female ones, and it is believed that this difference is because more boy babies die before reaching reproductive age than girl babies. Thus the population has reached some sort of equilibrium, but it is an equilibrium of 50/50 at reproductive age, not birth. This is called Fisher's Principle. http://en.wikipedia.org/wiki/Fisher%27s_principle According to Fisher the roughly 50/50 outcome is entirely dependent on having some individuals in the population who are genetically predispositioned to produce boys as offspring, while others are genetically predispositioned to produce girls. The outcome for populations says nothing about the expected sex ratio of offspring of individuals. Thus people who after having a series of children of the same sex who keep trying for the other, because they think in some sense 'they are owed one' are making the gamblers fallacy. But people who keep trying because they think that 50/50 are good odds, may be making a different error. They may be strongly biased to produce the sex they already have produced, and the odds of them getting the other one may be very small. As of now we have no way to tell, because the mechanisms which determine the sex of offspring are largely unknown. — Preceding unsigned comment added by 85.224.196.196 (talk) 13:49, 17 September 2012 (UTC)

Merge law of averages into this article
As far as I understand, this article is talking about the same concept as law of averages. Since the other article is shorter, would anyone be opposed to merging that article into this one? 173.181.83.109 (talk) 21:59, 24 June 2013 (UTC)


 * As far as I understand it, the Gambler's Fallacy is a psychological phenomenon, it is a failure to understand and to correctly use the "laws" of probability. As a matter of human psychology, it is not identical to the law of averages so having one article for each would seem to be more appropriate. It is like the subject of risk and the inability of people to correctly assess risk which would fall under psychological failings.Qordil (talk) 18:10, 27 August 2013 (UTC)

Preaching to the choir
I'm just now writing a response to someone who's brought up a difficult and fallacious argument. It's related to this topic, so I wanted to see if I could refer them here. I cannot. The problem is that the article is not written for the casual reader, but for the accomplished statistician. There needs to be extended text at the beginning that explains in simple and compelling language why this is a fallacy.

From the backtalk in the comments, there are a fair amount of people who are convinced they are still right. (For example Carl from Louisiana, on Jan 15, 2010.) Eebster the Great responded, but was more or less taking the (correct) and somewhat sterile party line. I don't feel that's effective explaining to Carl and people like him why he's in error. For Carl, one of four things is happening. Either the gambling object is not quite "fair" because of manufacture or wear, or the house is intentionally cheating in his favor, or he is observing patterns where none have statistical significance, or he is mis-remembering what happened. Naturally, the last argument isn't going to sway any reader, but the others should alert casual readers that what they see as compelling evidence may be flawed. Leptus Froggi (talk) 20:31, 11 October 2013 (UTC)

I recently developped a simulation tool to discuss the gambler's fallacy with a friend. It may be used to oppose some "factual" arguments to people who still consider their statement as true. If you think this simulation can be used in any way as an argument in this article, feel free to do it. I would be glad it serves a wider audience.

For more informations, refer to the README.md file and the section Examples, especially the `least` strategy which refer to the strategy of the gambler's fallacy.

Project : https://framagit.org/carrieje/gamblers-fallacy (User carrieje (unregistered)) 18:08, 31 December 2021 (UTC+1) — Preceding unsigned comment added by 109.12.161.143 (talk)

What if...
Suppose we have 6 heads in a row. What if the gambler bets not on what the next toss will be (with a resulting 50/50 probability), but on whether or not there will be a series of 7 heads in the game he plays (with the next toss completing the series)? Since the probability of 7 heads in a row is low, wouldn't it still be prudent to bet against it? And if not, why not? Try not to repeat the formulation of the fallacy or restate that the probability is still 50%. Explain, why a bet on the probability of 7 heads in a row is meaningless. What if we're talking about 69 previous heads (still assuming a fair coin) and the next toss is no. 70?89.12.84.0 (talk) 15:44, 12 October 2013 (UTC)

OK, after a bit of thinking I got where I went wrong. The "unlikely" part has already happened. And while it could in principle get even unlikelier, it is divided from that further improbability by a small step of 1/2. Ignore the previous stuff :-)89.12.84.0 (talk) 16:20, 12 October 2013 (UTC)
 * I see you answered your own question, but the key issue is conditional probabilities. Even people who can do a decent job of estimating probilities can get tripped up when the issue is coonditional probablities.-- S Philbrick (Talk)  16:44, 23 January 2018 (UTC)

False example for retrospective gambler's fallacy.
Another variety, known as the retrospective gambler's fallacy, occurs when individuals judge that a seemingly rare event must come from a longer sequence than a more common event does. For example, people believe that an imaginary sequence of die rolls is more than three times as long when a set of three 6's is observed as opposed to when there are only two 6's. This effect can be observed in isolated instances, or even sequentially. A real world example is that when a teenager becomes pregnant after having unprotected sex, people assume that she has been engaging in unprotected sex for longer than someone who has been engaging in unprotected sex and is not pregnant.[17]

The example is completely false. It compares looking at a dice roll and concluding something about previous dice rolls to pregnancy, but the analogy doesn't work at all. Being pregnant is instead analogous to "successfully" rolling 3 6's anytime in the sequence, not just as the very last roll. This is because having sex and not being impregnated that particular time doesn't undo being impregnated previously. — Preceding unsigned comment added by 98.164.193.204 (talk) 23:57, 14 October 2013 (UTC)

I'm not sure how pregnancy is relevant to the fallacy either. From what the article is describing, the fallacy occurs when a previous event is perceived to have an impact on the outcome of the next event. Pregnancy doesn't work this way. Having unprotected sex multiple times increases chances of pregnancy because 'you're buying more tickets' but also increases the chances of each individual attempt. Recalling from old lecture slides in university, the chance of pregnancy from an isolated case of unprotected sex is around 10%, however if you have unprotected sex each day for 30 days for example, the chances for each individual case goes up to 40% at least — Preceding unsigned comment added by Etceteralol (talk • contribs) 23:44, 3 August 2014 (UTC)

It appears the wording has been changed since then to be unambiguous (i.e. "on a given night"). Recommend archiving this section. 153.142.16.144 (talk) 04:45, 25 October 2019 (UTC)

Recent Paper Supports the Gambler's Fallacy as Fact
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2627354

When you observe short sequences, the probability that a sequence will alternate is greater than 50%. Paraphrasing part of the paper, consider these coin flips (generated just now using a random generator at: http://www.mathgoodies.com/calculators/random_no_custom.html): 1001010101110 So we have: 6 zero's, and 6 one's. A perfect 50/50 as expected. But, the gambler's fallacy is not this probability, it is the probability that given a short sequence and a 1, what is the chance the the next value will also be a 1? In the above example, out of the 6 "ones", only 2 are followed by another "one". In the sequence above, two thirds of the time you will be better off switching, a predictive power of the next flip that is better than 50/50! If this sequence were extended out, then this probability would converge back to 50%. On the other hand, if you were to get more sequences of the same length, then these skewed probabilities would likely remain. — Preceding unsigned comment added by Dvanatta (talk • contribs) 08:50, 26 October 2015 (UTC)
 * Re this edit: the first problem is that the Social Science Research Network looks like one of the many places where people can publish research online with little if any peer review. Unless probability theory is badly wrong, the probability of a 50-50 bet being successful remains at 50-50 every time. People have often tried to convince themselves otherwise, but that's the way it goes.-- ♦Ian Ma c M♦  (talk to me) 15:30, 23 January 2018 (UTC)
 * I've only read through page three, but the short answer is that your math is wrong. You aren't counting things corrrectly.-- S Philbrick (Talk)  15:37, 23 January 2018 (UTC)

While an interesting way to summarize the experiment, you are correct to say the possible proportions are: {0, ½, 1}

But the probabilities are: {3/8, 2/8, 2/8}

not {3/6, 1/6, 2/6}

The expected value, of course, is:

0*3/8+ .5*2/8+1 *2/8= 0+.25+.25= .50

Exactly as one might expect.-- S Philbrick (Talk)  15:48, 23 January 2018 (UTC)


 * Very short sequences can be misleading. It takes a large amount of trials to approximate a normal distribution curve accurately. The authors of the paper don't seem to have taken this into account.-- ♦Ian Ma c M♦  (talk to me) 16:07, 23 January 2018 (UTC)

Monte Carlo Casino example
Jonah Lehrer’s How We Decide is no longer available, as it was pulled by the publisher in March 2013. It has been removed from the article as a source. As for the Monte Carlo example, where black came up 26 times in a row in August 1913, some sources regard this as an anecdote with doubtful authenticity. At 1 in 136.8 million, it is very unlikely, but of course not outright impossible. Here is a roulette wheel that came up red ten times in a row, but at 1 in 1347 this is much more plausible.-- ♦Ian Ma c M♦  (talk to me) 13:51, 6 February 2018 (UTC)

I've been looking around for a proper source on this event. The new citation is a 2015 BBC article. If it's such a popular event, surely it must have a primary source? IrateSpecialist (talk) 23:52, 14 December 2022 (UTC)

Monte Carlo Casino Odds
The odds were stated to be 1:136.8 million. However, that is getting specifically black 26 times in a row. It's not relevant that it was black; it'd make no difference if it had been red 26 times in a row. The "first" time it landed on black wasn't actually the first time, unless it was opening day within the first few rolls (and everyone happened to bet on that one). Rather, the "first" black was completely insignificant to the gamblers, and only became the "first" by being followed by 25 more. Therefore, I've corrected the odds to a black happening 25 MORE times after the "first" as 18/37 chance per roll, so 1 / 0.486486486^25 = 66,564,294.2. Roguetech (talk) 12:38, 21 September 2018 (UTC)
 * This is correct, but if a person was betting on black continuously, the probability would be 136.8 million as stated.-- ♦Ian Ma c M♦  (talk to me) 13:12, 21 September 2018 (UTC)
 * Re this edit: when betting on red or black on a single zero roulette wheel, the probability of success is 18 out of 37, or p = 0.486. Writing it as $37⁄18$ implies that the fraction is 2.055, which seems obviously wrong.-- ♦Ian Ma c M♦  (talk to me) 16:21, 22 September 2018 (UTC)
 * I think we're getting confused because of the wording of the sentence. The probability is p = ($18⁄37$)26-1 which is 1 in 66.6 million. I've adjusted the wording to be consistent with the other sentences in the article where the probability is given.-- ♦Ian Ma c M♦  (talk to me) 16:38, 22 September 2018 (UTC)
 * Thanks for our hint concerning my edit, ianmacm, but I cannot agree: 1. There are no other sentences in the article where the probability (of 26 blacks) is given. 2. If the calculcation was to be consistent, it would have to be the same as in the coin example: The probability of 21 heads in a coin experiment is ($1⁄2$)21. The probability of 26 blacks in a roulette experiment is ($18⁄37$)26. As there is no first head only becoming first because of other heads following (if you start a coin toss sequence and get heads, it is the first, no matter which outcomes follow), there is no first black only becoming black because there are more blacks following. Even if a sequence has a length of 1, the first element remains the first. 3. Roguetech's calculation of the probability of "a black happening 25 MORE times after the 'first'" is correct, but they should have adapted the sentence as well. As of now, it states that the probability of getting 26 blacks in a row is ($18⁄37$)251:, which is false. It denotes the probability of getting 25 blacks after the first trial (no matter if the outcome of the first trial was black or red, by the way). So you'd either have to change the sentence or the probability. Guy Bukzi Montag (talk) 13:05, 11 August 2019 (UTC)
 * It depends on the wording. The probability of specifically 26 blacks in a row is 1 in 136.8 million. However, the crowd at the Monte Carlo casino would have been just as happy with 26 reds in a row, and when this is taken into account, the probability is 1 in 66.6 million.-- ♦Ian Ma c M♦  (talk to me) 13:23, 11 August 2019 (UTC)
 * Of course it depends on the wording. But the wording, as it is in the article, is wrong. Or the probability. They don't fit together. Guy Bukzi Montag (talk) 10:18, 12 August 2019 (UTC)
 * One way round this problem is to go back to the previous wording, which makes clear that the probability of black (and black only) 26 times in a row is 1 in 136.8 million. It starts to get more complicated when red or black is specified. However, I do believe that the current wording is correct, which is "the probability of a sequence of red or black occurring 26 times in a row is ($18⁄37$)26-1 or around 1 in 66.6 million." After a red or black has occurred with p=1, the follow up is red or black 25 times in a row. Possibly it could be changed to "either red or black" to make this clearer.-- ♦Ian Ma c M♦  (talk to me) 16:54, 12 August 2019 (UTC)
 * To be clear, the argument of "it doesn't matter if black or red" is just an easy way to look at/explain it. It simply doesn't make sense to count the "first" spin. If a casino only did roulette once per day, and only continued spinning if it landed on black, then there would be a 1:137 million chance of spinning it 26 times on any specific day. (This is the same as saying "if you flip a coin X number of times".) But that doesn't reflect reality of the "Monte Carlo" example, where they continue rolling regardless. So which is the "first" spin...? You can pick any of them.! For the sake of figuring "odds", you can pick any one as the "first" spin, with the only caveat being that you ignore all prior spins. If the spin you picked as being the first were red (or green), then the odds of getting 25 more blacks in a row would be 0%. So if you're measuring consecutive black spins, why pick red as the first one? There is no one specific set of 26 spins, rather for every 26 spins, there are 26 possible starting points. Another way to look at/explain it, there's no requirement for when the Monte Carlo players placed their bets. If caught in the "gambler's fallacy", you wouldn't bet on black if the prior spin was anything other than black. Ergo, the first spin must be black for the "gambler's fallacy" to apply. So in context of either a continuously operating casino, or any "gambler's fallacy", you would not count the first spin. I don't oppose it saying "red or black" (or "either red or black"), but only as a means to avoid arguments. @@Guy Bukzi Montag Note that the same argument doesn't apply to the "Example" of flipping coins, since it addresses specifically "a single toss", which must be a specific single toss. However, for the "Non-example" example of flipping coins, I agree that the odds should be computed on the standard of "more flips" rather than "total flips". Roguetech (talk) 18:41, 19 August 2019 (UTC)

In the sentence: "This was an extremely uncommon occurrence: the probability of a sequence of either red or black occurring 26 times in a row is $(18⁄37)^{26-1}$ or around 1 in 66.6 million", I think the emphasis "extremely uncommon" here is not right, as all 26 color sequences are equally rare $(18⁄37)^{26}$; it was not extraordinary to have 26 blacks in a row, I think the reason it induces the gambler's fallacy is, it being easily discernible and expressible for people in comparison to other equally rare 26 color sequences. The paragraph gives the notion of an extremely rare event occurred which I think can be a kind of gambler's fallacy. With respecting the discussions, I am offering an edit for the sentence as:

"Although the probability of any red/black permutation of size 26 is $(18⁄37)^{26}$ or 1 in 136.8 million, assuming the mechanism is unbiased, the streak of the same color creates a sense of extra-ordinariness and induces the gambler's fallacy." S.POROY (talk) 13:50, 15 December 2022 (UTC)

Different formulation for the probability
This was added on my talk page by :

Hello, regarding the change I made suggesting that the probability of a sequence of either red or black occurring 26 times in a row is $2×(18⁄37)^{26}$, since the probability of the first occurrence of a red or a black is not 1, but 36/37, then the probability of 26 occurrences in a row is not 1 x $(18⁄37)^{26-1}$, it's (36/37) x  $(18⁄37)^{26-1}$, which is $2×(18⁄37)^{26}$.

As 2A02:2F04:A001:C400:450:9681:1692:82EE points out, the current wording assumes that the probability of red or black on the first spin is 1 (completely certain) when it is actually $36⁄37$ on a single zero roulette wheel. I think this is the more correct wording. This makes the probability of red or black appearing 26 times in a row in any given 26 spins 1 in 68.4 million rather than 1 in 66.6 million.-- ♦Ian Ma c M♦  (talk to me) 19:19, 17 May 2024 (UTC)

What About Stochastic But Not Truly Random Phenomena?
A of systems in the physical world are stochastic but not truly random; e.g. economic growth, the stock market, sex ratios (I noticed that did get a mention), etc. Processes that seem random in the short and even the medium term will still converge back to a mean eventually, usually in no more than 10-20 years. I think the article should acknowledge this more. Is there really such a thing as true randomness outside of contrived human games and particle physics? Heck, I've read papers on random number generators that make truly random sequences, and it's not easy. — Preceding unsigned comment added by Joeedh (talk • contribs) 20:17, 13 November 2018 (UTC)


 * John von Neumann famously said that "Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin." For the purposes of everyday probability theory, events like the toss of a coin, the throw of dice and the spin of a roulette wheel are considered to be random. It's never quite that simple, but events like the ups and downs of the stock market cannot be examined with basic probability theory.-- ♦Ian Ma c M♦  (talk to me) 05:21, 14 November 2018 (UTC)

January 2019
Re this edit: there is a large amount of philosophy and very little mathematics in this addition. As far as mathematicians are concerned, the fallacy is real and occurs when a person falsely believes that non-dependent random events will even out in some way, which they are under no obligation to do. This is similar to previous attempts to undermine the fallacy with philosophical reasoning that has little to do with the underlying mathematics. I'm also worried by the use of Social Science Research Network as a source, because this is a preprint service, where people can upload anything they like without any effective peer review. Is this really suitable for the article?-- ♦Ian Ma c M♦  (talk to me) 17:47, 14 January 2019 (UTC)
 * Don't even bother engaging, this is a continuation of the editor's attempts from a year ago to "to determine how long it takes for an anarchic experiment in democratic knowledge accumulation to become, as eventually they always do, a cult of a few restricting knowledge to acceptable forms". Toohool (talk) 18:10, 14 January 2019 (UTC)


 * Here is my final two cents on the issue: If two events are assumed to be independent and random, assuming that they will even out over a given period is a fallacy. In fact it is worse than a fallacy, it is just plain wrong. Likewise, if events A and B are assumed to occur at random, there is no advantage in choosing A or B and the choice is arbitrary. There have been repeated attempts to muddy the waters on these issues by introducing philosophical lines of reasoning that ignore these two principles. This is not conceptual dogmatism, it is just how the mathematics works.-- ♦Ian Ma c M♦  (talk to me) 05:50, 15 January 2019 (UTC)

Selective reporting section
Referring to this revert. are you saying you're going to get the book? Because as I've indicated, I have read the book. I don't remember the precise numbers - for example it could be 6 championships in 8 years instead of 5 championships in 7 - but I remember the essence of what was written. Banedon (talk) 06:46, 8 June 2019 (UTC)


 * Going from memory obviously isn't ideal. I had hoped it would be on Google Books; it is but you can't look inside it.  The book can be purchased on Amazon  but I don't have a copy.-- ♦Ian Ma c  M♦  (talk to me) 06:52, 8 June 2019 (UTC)
 * Like I said, I do. The puzzle in question was about how one guy was attempting to impress a woman by boasting about how his team had won X championships in Y years. The woman sarcastically congratulated the guy for winning in year Y and commented that the guy must've been so happy after his team failed to win in year Y+1. The guy apologized for not recognizing an "expert" at the game. The woman said she knew nothing about it. Puzzle asked for how it's possible. The solution gives the explanation in the text, and said the same thing about politicians & unemployment. Banedon (talk) 08:01, 8 June 2019 (UTC)
 * If you have a copy of the book, then why are you going by memory? It looks like you are the editor who introduced the example of unemployment statistics to the article, and now you're saying that the same Guinness Book of Mindbenders you originally cited gives a better example of something not being an example of the gambler's fallacy? Toohool (talk) 01:23, 11 June 2019 (UTC)
 * Because I don't have the book anymore. I inserted the original section, then figured perhaps the other example is better and so changed it. Now tell me: why are you questioning my motivations? Why aren't you looking at the text and commenting on which is better? Banedon (talk) 01:58, 11 June 2019 (UTC)
 * Because I'm skeptical about the whole paragraph. A half-remembered passage from a book that doesn't look like a very reliable source for any particular topic, about an example that it's not entirely clear is relevant to the article. Toohool (talk) 02:24, 11 June 2019 (UTC)
 * I've ordered a copy of the book. Then we'll have a look at what it says. As a general rule, you should not use a book as a source unless you can lay your hands on it and quote from it directly.-- ♦Ian Ma c M♦  (talk to me) 05:05, 11 June 2019 (UTC)

The book says...
OK, I have now got a copy of the book. There are 109 puzzles in it, and I can't find anything that refers to the Gambler's fallacy. So I have added Failed verification and will remove it unless a specific reference can be found.-- ♦Ian Ma c M♦  (talk to me) 17:16, 13 June 2019 (UTC)
 * The book does not call it "gambler's fallacy" (neither does the Martin Gardner book also cited in the section, by the way). But I am 100% certain the puzzle described above is there. As I remember it was in the first half of the book. If you can't find it, I'll find it myself in about two months. Banedon (talk) 21:32, 13 June 2019 (UTC)
 * Sorry, but this now needs a direct quote from the book. I've removed it from the article.-- ♦Ian Ma c M♦  (talk to me) 05:17, 14 June 2019 (UTC)
 * What kind of direct quote are you looking for? Like, the entire text of the puzzle? Banedon (talk) 05:18, 14 June 2019 (UTC)
 * There are 109 puzzles. Which one is it? If you cannot say, it is because you do not have a copy of the book to hand.-- ♦Ian Ma c M♦  (talk to me) 05:22, 14 June 2019 (UTC)
 * You've proven to be sufficiently annoying that I went and acquired the book myself. It is puzzle 2. The fact that the puzzle is literally on the first page of the main text makes me question if you made a good faith attempt to find it. Banedon (talk) 05:38, 14 June 2019 (UTC)
 * It's in the solution of Cricket for Americans. So apologies if I missed this. The Guinness Book of Mindbenders is a book of puzzles and their solutions, it isn't really a mathematics textbook. Looking at the exact quote, it is not strictly an example of the gambler's fallacy because it is not based on probability theory. It is more about selective citing of statistics or cherry picking to make them look good.-- ♦Ian Ma c M♦  (talk to me) 05:53, 14 June 2019 (UTC)
 * That's why the text was in a section on non-examples of the Gambler's fallacy. Banedon (talk) 05:54, 14 June 2019 (UTC)
 * There are WP:TOPIC problems because it isn't an example of gambler's fallacy. It doesn't give specific probabilities for the events. If these remain constant, it is an example of the fallacy if a person believes that they are influenced by past results. Cherry picking is something different.-- ♦Ian Ma c M♦  (talk to me) 06:02, 14 June 2019 (UTC)
 * Yes, that's why it is a non-example. Consider this example since it's more direct. Suppose there are two teams, Oxford and Cambridge, who are equally skilled. They play each other every year. I tell you that Cambridge has won the last eight years from 2012-2019. What are the odds that Cambridge will win in 2020? That would be 50% - it's the essence of the Gambler's fallacy. What are the odds that Cambridge won in 2011? As this source shows, that probability is not 50%. Banedon (talk) 06:05, 14 June 2019 (UTC)
 * Sports betting is not the same as probability theory. The toss of a fair coin is always p = 0.5. If Cambridge has a better team than Oxford, they are more likely to win. In the 2018 men's boat race, Cambridge were quoted at 4/11, while Oxford were quoted at 2-1. 4/11 is an implied probability of 73.3%, while 2-1 is an implied probability of 33.3%. This might seem counterintuitive when there are only two teams, but sports betting works like this. The figures do not add up to 100% because the bookmakers always make a profit regardless of who wins, this is because the odds are over-round.-- ♦Ian Ma c M♦  (talk to me) 06:34, 14 June 2019 (UTC)
 * I don't see how that's related. The example I gave isn't about sports betting, and it explicitly assumes that the two teams are equally skilled. Banedon (talk) 06:39, 14 June 2019 (UTC)
 * Gambler's fallacy is based on probability theory, such as red and black on a single zero roulette wheel always being p = 0.486. In real world situations like football matches and tennis matches, it is never quite that simple. Laying odds on a single event is not the same thing as gambling in a casino all day long, as the 2018 Boat Race can happen only once.-- ♦Ian Ma c M♦  (talk to me) 06:54, 14 June 2019 (UTC)
 * At that I think we're at an impasse which can only be settled by the participation of more editors. Banedon (talk) 10:38, 14 June 2019 (UTC)

The paragraph should just be removed unless someone can cite a reliable source that shows the need to distinguish this type of example from the gambler's fallacy. As it is, the example has no overt connection to the article topic and its inclusion is original research. Toohool (talk) 17:50, 14 June 2019 (UTC)

Coil-Toss analogy not accurately applied to Gambler's Fallacy; real-life Gambler's Fallacy is not accurately depicted, anyways, and not really a fallacy
In the context of "runs" it is not required to have a 20-item run of the same outcome in order to be considered a run for gambling purposes:

Yes, it's true that the 21st coin flip has a 50/50 chance of being heads even if it was heads the last 20 times. However, the chances of generating 21 heads in a row is NOT equally likely as having 20 heads and a tails in terms of gambling. There is only 1 way you can have 21 heads in a row, but there are 21 different ways you can have 20 heads and a tails in the run. The traditional notion of "the chance of getting a tails is going up" after a long run containing mostly heads is actually a correct notion, not a fallacy, because truly-random game outcomes do eventually approach a normal distribution of outcomes and you need to compare your current run to a theoretical normal run in order to visualize the likelihood of future outcomes.

IN reality, the majority of gamblers consider a lucky run to potentially include some losses as long as the wins highly-outweigh those losses. The real "gambler's fallacy" is not that "The next one will be tails" but rather "I should be seeing *more* tails soon" because runs in gambling are most often mixtures of results, not straight runs of perfectly uniform outcomes. A "run" in gambling can be a mixture of wins and loses that leans heavily toward wins, and in a chance-based game the longer you play the more likely your outcomes will approach a normal distribution.

The real-world "gambler's fallacy" is the assumption that a string of random events will eventually achieve a normal distribution of outcomes - that's not a fallacy at all. 50.239.107.122 (talk) 18:33, 3 October 2019 (UTC)


 * The key word here is "eventually". There is no fixed amount of time that would guarantee a 50-50 split of heads and tails. The probability on each individual toss is always p = 0.5.-- ♦Ian Ma c M♦  (talk to me) 05:40, 4 October 2019 (UTC)

"Is the gambler's fallacy really a fallacy?"
Re this edit: the simple answer to the question "Is the gambler's fallacy really a fallacy?" is "yes". As the coin toss example in the article says, "When flipping a fair coin 21 times, the outcome is equally likely to be 21 heads as 20 heads and then 1 tail. These two outcomes are equally as likely as any of the other combinations that can be obtained from 21 flips of a coin. All of the 21-flip combinations will have probabilities equal to 0.521, or 1 in 2,097,152. Assuming that a change in the probability will occur as a result of the outcome of prior flips is incorrect because every outcome of a 21-flip sequence is as likely as the other outcomes." Considering a series of events does not alter the probability of individual random events. I'm also concerned about a potential conflict of interest here. This section is giving way too much prominence to a single paper, which leads to problems with WP:DUE.-- ♦Ian Ma c M♦  (talk to me) 19:18, 1 September 2020 (UTC)

Roulette is not a good example
because good shooters can increase the probability of hitting a desired number significantly by training. This skill is a part of their profession, to defeat systematic big bettors.--2400:4050:95C3:2B00:207A:3830:B1F7:3D98 (talk) 01:02, 1 March 2021 (UTC)


 * Modern casinos go to considerable trouble to ensure that roulette wheels are random, while the croupiers are instructed to spin the wheel and throw the ball with a different amount of strength each time. This suggestion also implies that the croupiers are acting in bad faith, and would require a reliable source.-- ♦Ian Ma c M♦  (talk to me) 08:04, 1 March 2021 (UTC)

Childbirth section
This section refers to Laplace having published A Philosophical Essay on Probabilities in 1796. The article Pierre-Simon Laplace (under Inductive probability) gives a date of 1814 for this work. In Chapter 1 - Introduction - of this edition, Laplace refers to the essay as a development of a lecture he had given in 1795. It looks like this reference may have conflated the lecture and the essay. How should we handle this? Autarch (talk) 01:39, 9 April 2021 (UTC)

d'Alembert's system
Wikipedia "d'Alembert's system" redirects to this page. I think that that is a bad mistake. d'Alembert's system is not such a bad system. It does not really have anything to do with the gambler's fallacy, as far as I know. I mean: I'm not sure that d'Alembert himself believed in that fallacy. Richard Gill (talk) 12:00, 15 April 2021 (UTC)


 * This source says (correctly in my view) that d'Alembert's betting system is "a prime example of the Gambler’s Fallacy". It is based on the assumption that 50-50 bets should even out over a period of time, but there is no guarantee that they will. In practice, the system has no memory and the probability remains at 0.5 every time the event occurs.-- ♦Ian Ma c M♦  (talk to me) 16:17, 15 April 2021 (UTC)

The source does not say what d’Alembert’s motivation for his source was. I agree that you could motivate it that way, and you would usually be wrong. Whether it’s a good or a bad system depends on what you are after. Money? Fun? Do you need a particular amount of money with the biggest possible chance, or something else? I would like to see a separate article about the system and reference’s to d’Alembert’s own work. Richard Gill (talk) 16:32, 15 April 2021 (UTC)


 * I've now discovered that the d'Alembert system is *erroneously* attributed to d'Alembert. It should not be called "d'Alembert's system" at all. Richard Gill (talk) 17:51, 16 April 2021 (UTC)


 * Also, d'Alembert comes out very well in a comparison with other gambling systems. If you must gamble, it is not a bad choice. "Table 3 thus reveals the magic of the d’Alembert. Though it lags behind in terms of mean upside return for small goals, it matches Labouchere in terms of martingaling ratio and requires fewer rounds of play on average to achieve the goal. But the d’Alembert improves consistently as the goal becomes more ambitious. For a desired gain of at least 5% of the initial capital, d’Alembert requires more time to reach the goal, but does so with higher frequency and by requiring the gambler to take less money out of his pocket. d’Alembert also competes with le tiers et le tout on mean upside return, even though TT is designed to produce large gains." https://www.researchers.one/article/2020-08-32 "Risk is random: The magic of the d'Alembert", Harry Crane, Glenn Shafer Richard Gill (talk) 18:00, 16 April 2021 (UTC)


 * The d'Alembert betting system may be an example of Stigler's law of eponymy. In Croix ou Pile d'Alembert wrote that in two tosses of a fair coin, the probability that heads will appear at least once is $2⁄3$. This isn't correct, as there are four possible outcomes, making the answer $3⁄4$. The betting system is also known as Martingale, and people are advised not to try this in real life because they can lose money very quickly.-- ♦Ian Ma c  M♦  (talk to me) 20:52, 16 April 2021 (UTC)


 * ah, interesting. The d’Alembert is sometimes grouped with other systems called collectively martingale systems, but only one of them is “the” classic martingale system. With *the* martingale people rapidly do lose a lot of money. But not so much with d’Alembert. See Crane and Shafer. Richard Gill (talk) 03:20, 17 April 2021 (UTC)

Clarification on "probability" definition & calculations
Relevant to several discussions I'm seeing here, and in anticipation of future discussions, there are a few points that may need clarification:

First, "probability" is about *future* or *hypothetical* events. It is the mathematical likelihood that an event (or combination of events grouped together) will happen in the future or in a hypothetical situation. After an event has happened, by definition the probability then equals 100%, and equations for probability no longer apply. In non-mathematical terms, probability is the degree of certainty that a future/hypothetical event will happen. 100% equals complete certainty. For an event that has already happened, the degree of certainty is 100%. The probability of an observed past event/outcome is automatically set to 100%.

Also, please note that probability equations will often state conditions (e.g.; "assume a fair coin"). For a real-life situation, this doesn't mean you should always assume the condition is true. It means that if the condition has been met (e.g.; a "fair coin" has been confirmed to be true), then the probability equation applies. If the condition has not been met (or is not certain) then the equation provides an estimate at best.

Additionally: As with all things mathematical, adding a human variable makes things messy. People are not machines. Physical tasks (e.g., a coin flip) depend on human physical mechanics which is often inconsistent. This type of variability becomes less significant if the task is repeated numerous times and/or by numerous subjects. Also, the stated assumptions for a mathematical equation (such as "fair coin") can only be met if the human is able/willing to meet the condition. In this example with gambling, ethically questionable behavior is a variable that can render a mathematical calculation useless. Basically, humans ruin everything. You're welcome. :-)

Hope that helps clarify probability equations and how/when they're applied. Betsy Rogers (talk) 22:37, 21 November 2022 (UTC)

Loot boxes and pity timers.
If you know the exact rate a pity timer pays out, then you can calculate the minimum number of high value items and not Gambler's Fallacy. For example, in Hearthstone there are two pity timers for Legendary Cards. One at the beginning of expansion requires a Legendary card within the first 10 packs. Other than that it requires a Legendary within every 30 pack openings. So buying 100 packs at the beginning of the expansion, your guaranteed 4 Legendary cards at minimum. One within the first 10 and 3 within the next 90 at minimum. https://esports.gg/guides/hearthstone/hearthstone-pity-timer/ 2605:A601:A103:BB00:79:2283:79B5:749F (talk) 22:58, 18 June 2024 (UTC)