User:Guy Bukzi Montag/sandbox/Gambler's fallacy

The gambler's fallacy, also known as the Monte Carlo fallacy or the fallacy of the maturity of chances, is the mistaken belief that, for random events, runs of a particular outcome (e. g., heads on the toss of a coin) will be balanced by a tendency for the opposite outcome (e. g., tails). Or, in short:


 * “If you have been losing, you are more likely to win in future.”

In situations where the outcome being observed is truly random and consists of independent trials of a random process, this belief is false. The fallacy can arise in many situations, but is most strongly associated with gambling, where it is common among players.

Coin toss
The gambler's fallacy can be illustrated by considering the repeated toss of a fair coin. The outcomes in different tosses are statistically independent and the probability of getting heads on a single toss is $$\tfrac12$$ (one in two). The probability of getting two heads in two tosses is $$\tfrac12 \cdot \tfrac12 = \tfrac14$$ (one in four) and the probability of getting three heads in three tosses is $$\tfrac12 \cdot \tfrac12 \cdot \tfrac12 = \tfrac18$$ (one in eight). In general, if Ai is the event where toss i of a fair coin comes up heads, then:


 * $$\displaystyle \underbrace{\frac12 \cdot \frac12 \cdot \ldots \cdot \frac12}_{n\, \mathrm{times}} = \frac1{2^n}$$.

One reason for this fallacy is an incorrect application of the law of large numbers. People argue that if heads and tails will occur the same number of times in the long run, which asserts that in the long run,

"ought to": 1.2. The gamblers’ fallacy: ‘‘If you have been losing, you aremore likely to win in future.’’People gambling on sports outcomes may continue todo so after losing because they believe in the gamblers’ fal-lacy. This is the erroneous belief that deviations from ini-tial expectations are corrected even when outcomes areproduced by independent random processes. Thus, peo-ple’s initial expectations that, in the long run, tosses of afair coin will result in a 50:50 chance of heads and tailsare associated with a belief that deviations from that ratio will be corrected. Hence, if five tosses of a fair coin haveproduced a sequence of five heads, the chance of tails on the next toss will be judged to be larger than 50%. This isbecause the coin ‘‘ought to’’ have a 50:50 chance of headsand tails in the long run and, as a result, more tails are‘‘needed’’ to correct the deviation from that ratio producedby the first five tosses

Roulette
The term “Monte Carlo fallacy” originates from the most famous example of the phenomenon, which occurred in a game of roulette at the Monte Carlo Casino on August 18, 1913, when the ball fell in black 26 times in a row.


 * “During that [...] run, most gamblers bet against black, since they felt that the red must be ‘due’. In other words, they assumed that the randomness of the roulette wheel would somehow correct the imbalance and cause the wheel to land on red. The casino ended up making millions  of francs.”

A priori: Roulette 26 black: This was an extremely uncommon occurrence: the probability of a sequence of red or black occurring 26 times in a row is $$(\tfrac{18}{37})^{26}$$ or around 1 in 136,823,184, assuming the mechanism is unbiased.

Daily Kit Box
Inverse fallacy (knowing the past): If a process is believed to be nonrandom, streaks are more likely to be followed. ###correct page when cited!

"clumped" data vs. "well distributed": Where was that? ###correct page when cited!