Editing Gambler's fallacy (section)

==Non-examples==
===Non-independent events===
The gambler's fallacy does not apply when the probability of different events is not [[statistical independence|independent]]. In such cases, the probability of future events can change based on the outcome of past events, such as the statistical [[permutation]] of events. An example is when cards are drawn from a deck without replacement. If an [[ace]] is drawn from a deck and not reinserted, the next card drawn is less likely to be an ace and more likely to be of another rank. The probability of drawing another ace, assuming that it was the first card drawn and that there are no [[Joker (playing card)|jokers]], has decreased from {{sfrac|4|52}} (7.69%) to {{sfrac|3|51}} (5.88%), while the probability for each other rank has increased from {{sfrac|4|52}} (7.69%) to {{sfrac|4|51}} (7.84%). This effect allows [[card counting]] systems to work in games such as [[blackjack]].

===Bias===
In most illustrations of the gambler's fallacy and the reverse gambler's fallacy, the trial (e.g. flipping a coin) is assumed to be fair. In practice, this assumption may not hold. For example, if a coin is flipped 21 times, the probability of 21 heads with a fair coin is 1 in 2,097,152. Since this probability is so small, if it happens, it may well be that the coin is somehow [[Bias (statistics)|biased]] towards landing on heads, or that it is being controlled by hidden magnets, or similar.<ref name="Gardner1986">{{cite book|last=Gardner|first=Martin|title=Entertaining Mathematical Puzzles|url=https://archive.org/details/entertainingmath00gard|url-access=registration|year=1986|publisher=Courier Dover Publications|isbn=978-0-486-25211-7|pages=[https://archive.org/details/entertainingmath00gard/page/69 69]–70|access-date=2016-03-13}}</ref> In this case, the smart bet is "heads" because [[Bayesian inference]] from the [[empirical evidence]] — 21 heads in a row — suggests that the coin is likely to be biased toward heads. Bayesian inference can be used to show that when the long-run proportion of different outcomes is unknown but [[exchangeable random variables|exchangeable]] (meaning that the random process from which the outcomes are generated may be biased but is equally likely to be biased in any direction) and that previous observations demonstrate the likely direction of the bias, the outcome which has occurred the most in the observed data is the most likely to occur again.<ref>O'Neill, B.; Puza, B.D. (2004). [http://hdl.handle.net/1885/43089 "Dice have no memories but I do: A defence of the reverse gambler's belief".] Reprinted in abridged form as: {{cite journal|last1=O'Neill|first1= B. |last2=Puza |first2=B.D.|year=2005|title= In defence of the reverse gambler's belief|journal=The Mathematical Scientist|volume=30|issue=1|pages= 13–16|issn=0312-3685}}</ref>

For example, if the ''[[A priori and a posteriori|a priori]]'' probability of a biased coin is say 1%, and assuming that such a biased coin would come down heads say 60% of the time, then after 21 heads the probability of a biased coin has increased to about 32%.

The opening scene of the play ''[[Rosencrantz and Guildenstern Are Dead]]'' by [[Tom Stoppard]] discusses these issues as one man continually flips heads and the other considers various possible explanations.

===Changing probabilities===
If external factors are allowed to change the probability of the events, the gambler's fallacy may not hold. For example, a change in the game rules might favour one player over the other, improving his or her win percentage. Similarly, an inexperienced player's success may decrease after opposing teams learn about and play against their weaknesses. This is another example of bias.