History Main / GamblersFallacy

:: Psychologically, this fallacy tends to come from the fact that the odds to replicate a pattern ''do'' go up cumulatively. The odds of rolling 20 on a d20 twice is 1/400. The odds of rolling the first is 1/20, and the odds of rolling the second is also 1/20. The fallacy occurs when someone assumes that once they've rolled a 20, there's a 1/400 chance of getting a 20 again, when in reality there's still a 1/20 chance of getting 20 again. Similarly, the odds of rolling a 5 and a 17 on a d20 in that order are also 1/400, unless you've already rolled a 5 and are counting on a 17, which gives you a 1/20 chance as per usual.

:: Psychologically, this fallacy tends to come from the fact that the odds to replicate a pattern ''do'' go up cumulatively. The odds of rolling 20 on a d20 twice is 1/400. The odds of rolling the first is 1/20, and the odds of rolling the second is also 1/20. The fallacy occurs when someone attempts apply the full cumulative odds to the next roll. Similarly, the odds of rolling a 5 and a 17 on a d20 in that order are also 1/400, so that number's pretty irrelevant.

** It's actually slightly worse than that. Because the 0 and 00 spots are considered neither red nor black, the odds of a black number coming up on any spin is actually 16/38, or 42.1%. Playing black on every spin is statistically going to bleed the player dry in the long run.

* Any ''{{Warhammer}}'' gamer or tabletop roleplayer will tell you that [[RandomNumberGod this is absolutely true]].

[[AC:MMORPGs]]

** Apparently, even tropers aren't immune to this fallacy. The point of the statistical analysis is that "psychology" isn't necessary to explain the idea of a "hot streak." If, for example, a baseball player hits .375, it might be expected over the course of 168 games, he will have, say, a 10 game streak where he gets a hit every game. When this happens, people seem convinced that something else must be going on: a new hitting strategy, psychology, or just a player magically getting "hot." However, when you look at the statistics, random chance explains these streaks equally well.

::Note that Gambler's Fallacy applies only to systems that both ''have no memory'', and ''are explicitly known to be fair''. Drawing cards without replacement (read: deck now has "memory") does alter the probabilities of the next cards drawn, and if you do not explicitly know that the event being tested is fair, you can use things like n-heads-in-a-row to draw conclusions of bias in the system (see Non-examples and Theater sections below).

::Note that Gambler's Fallacy applies only to systems that both ''have no memory'', and ''are explicitly known to be fair''. Drawing cards without replacement does alter the probabilities of the next cards drawn (with replacement makes it apply again), and if you do not explicitly know that the event being tested is fair, you can use things like n-heads-in-a-row to draw conclusions of bias in the system (see Theater section below).

::Note that Gambler's Fallacy applies only to systems that both: have no memory, and are explicitly known to be fair. Drawing cards without replacement does alter the probabilities of the next cards drawn (with replacement makes it apply again), and if you do not explicitly know that the event being tested is fair, you can use things like n-heads-in-a-row to draw conclusions of bias in the system (see Theater section below).

*** A minor point, but the confusion may also stem from the type of questions you can ask about the probability of events. While it is true that 85 heads in a row is as equally likely as any other 85-string combination by ''odds of an unbiased coin'', there is significant evidence that the coin is ''biased''. Based on the 85 trials, there is no uncertainty as to what the outcome of the flips will be. Ff you saw these results, ''did not know explicitly that the coin is fair'' (this is important), and had to predict the next outcome of the next toss, the safe money would be on heads; no evidence for this coin yet exists supporting that it is not biased towards heads. So, even though the result may be equiprobable to all other outcomes in an unbiased coin, you can still fairly safely conclude that the coin is NOT unbiased from these 85 trials.

*** A minor point, but the confusion may also stem from the type of questions you can ask about the probability of events. While it is true that 85 heads in a row is as equally likely as any other 85-string combination by ''odds of an unbiased coin'', there is significance evidence that the coin is ''biased''. Based on the 85 trials, there is no uncertainty as to what the outcome of the flips will be. Ff you saw these results, ''did not know explicitly that the coin is fair'' (this is important), and had to predict the next outcome of the next toss, the safe money would be on heads; no evidence for this coin yet exists supporting that it is not biased towards heads. So, even though the result may be equiprobable to all other outcomes in an unbiased coin, you can still fairly safely conclude that the coin is NOT unbiased from these 85 trials.

:: A similar misinterpretation is that if an event has the odds of 1-in-n, then you are guaranteed a success if you make n attempts. As an exaggerated example, the probability of a "heads" on an unbiased coin is 1/2, therefore, flipping a coin twice is guaranteed to get at least one "heads." [[Understatement This is not true]].

:: A similar misinterpretation is that if an event has the odds of 1-in-n, then you are guaranteed a success if you make n attempts. As an exaggerated example, the probability of a "heads" on an unbiased coin is 1/2, therefore, flipping a coin twice is guaranteed to get at least one "heads." [[Understatement This is not true.]]

--> '''Narrator:''' By [[http://www.darthsanddroids.net/episodes/0195.html now]], the die is rolled. It's a 1; [[YouShouldKnowThisAlready Qui-Gon dies.]]

* [[http://www.darthsanddroids.net/episodes/0099.html A Darths & Droids strip]] covered this one, with one player having a carefully prepared 20-sided die that had previously rolled two 1's — the chances of rolling three 1's in a row is only 1 in 8000, so surely another 1 is almost impossible, right?
--> '''Narrator:''' By [[http://www.darthsanddroids.net/episodes/0195.html now]], the die is rolled. It's a 1; [[YouShouldKnowThisAlready Qui-Gon dies.]]
--> '''Pete''' (the one who prepared the die): "Awesome! That die will be even luckier next time!"
* Any ''{{Warhammer}}'' gamer or tabletop roleplayer will tell you that [[RandomNumberGod this is absolutely true]].
* Discussed at length in ''[[RosencrantzAndGuildensternAreDead Rosencrantz & Guildenstern Are Dead]]'', in which Rosencrantz flips a coin 85 times in a row and gets heads every time. Guildenstern suggests that it shouldn't be surprising since each coin has an equal chance of coming up heads or tails. Rosencrantz is not satisfied with this explanation, and neither is Guildenstern.
** The reason Rosencrantz and Guildenstren are confused? If a coin ''does'' do something like that, then it is probable that there's a bias towards the result it's constantly getting. (That die that rolls 1's more often than pure chance would indicate? Use it when you ''want'' to roll a 1!)
** The chance of this series happening (with unbiased coins) is one in 38 septillion (that's million million million million), in case you are wondering. But the odds of getting ''any'' particular ordered set of 85 results is the same 38 septillion; the odds of getting exactly eighty-four heads followed by one tails is ''also'' one in 38 septillion.

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