Gambler's fallacy

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The gambler's fallacy, also known as the Monte Carlo fallacy^[1], the fallacy of the maturity of chances^[2][3] or, more scientifically, the negative recency effect^[4], 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).^[5] Or, in short:

“If you have been losing, you are more likely to win in future.” ^[6]

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.^[7]

Examples

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, the probability of getting $ n $ heads in $ n $ tosses is

$ \displaystyle \underbrace{\frac12 \cdot \frac12 \cdot \ldots \cdot \frac12}_{n\, \mathrm{times}} = \left(\frac12\right)^n = \frac1{2^n} $.

Let's assume that a player has flipped a coin five times and got five heads in a row. The probability of five heads is $ (\tfrac12)^5 = \tfrac1{32} = 3.125 % $. There are two main reasons why it is believed that the next toss will be more likely to show tails than heads:

“Ex-ante” probabilities

The probability of getting even six heads in a row is $ \tfrac1{64} = 1.5625 % $, so the probability of getting heads in the next toss could be assumed to be only 1.5625 %. While the first part of the sentence is correct, the conclusion is false.

The misunderstanding lies in not realizing that the probability is only correct before the first coin is tossed. After the first five tosses, the results are no longer unknown. The tosses are independent, and a coin has no memory, so a run of luck in the past cannot influence the odds in the future. The probability of heads in the next toss is $ \tfrac12 $.

If ex-ante probabilities are calculated, they have to be considered for both possible outcomes:

• The probability of 5 heads, then 1 tail is $ \displaystyle \underbrace{\frac12 \cdot \frac12 \cdot \frac12 \cdot \frac12 \cdot \frac12}_{\mathrm{heads}} \cdot \underbrace{\frac12}_{\mathrm{tail}} = \frac1{64} = 1.5625 % $.
• The probability of 5 heads, then 1 head is $ \displaystyle \underbrace{\frac12 \cdot \frac12 \cdot \frac12 \cdot \frac12 \cdot \frac12}_{\mathrm{heads}} \cdot \underbrace{\frac12}_{\mathrm{head}} = \frac1{64} = 1.5625 % $.

Actually, any possible sequence of heads and tails in six tosses has the same probability of 1.5625 %. There is no difference, because both heads and tails have the same probability of $ \frac12 $.

Law of large numbers

Another reason lies in the erroneous belief that the law of large numbers applies to small numbers as well, thus creating a “law of small numbers”.^[8] In other words, if five tosses of a fair coin have produced a sequence of five heads, people expect that the coin “ought to” have a 50:50 ratio of heads and tails in the long run and, as a result, more tails are “needed” to correct the deviation from that ratio produced by the first five tosses.^[6]

While it is true that the law of large numbers guarantees a 50:50 ratio in the long run, the conclusion is false. The misunderstanding lies in the term “in the long run”: The law of large numbers only states that the the expected ratio will be reached as the number of trials approaches infinity. It makes no predictions about a small number of trials.

Roulette

The alternative term “Monte Carlo fallacy” originates from a famous anecdote about the phenomenon, which occurred in a game of roulette at the Monte Carlo Casino on August 18, 1913,^[9] 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.” ^[10]

The explanation of the phenomenon works like in the coin example, only with different probabilities. A roulette wheel has 37 numbers: 18 black, 18 red, and 1 green (the zero).

• The probability for black is the same as for red: $ \tfrac{18}{37} $.
• The “ex-ante” probability that the ball will land on black 26 times in a row, is extremely small: $ (\tfrac{18}{37})^{26} $ or around 1 in 136.8 million.
• However, like in the coin example, the probability of getting 25 blacks and then one red is the the same as of getting 26 blacks: $ (\tfrac{18}{37})^{25} \cdot \tfrac{18}{37} = (\tfrac{18}{37})^{26} $.

The gamblers' chance of getting red or black was the same every time, no matter when they joined the table and what the previous outcomes were.

The fact that today's casinos install an LED marquee at roulette tables showing which numbers have occurred recently is no contradiction to this. On the contrary, it is to the casino's advantage if people believe in the gambler's fallacy and keep playing even after a streak of losses or wins: The law of large numbers guarantees that any streak by a player will eventually be overcome by the parameters of the game—which are always in favour of the casino.

Daily Kit Box

The Tech card of the Daily Kit Box from Asphalt 8 can serve as another example. Every Daily Kit Box grants exactly one Tech card, and this card can only be Advanced Tech (AT) or Mid-Tech (MT). As there are only two possible outcomes, the experiment can be compared to a coin toss, but with two differences:

• While the probabilites for a coin toss are known, there are no official drop rates for AT and MT in Daily Kit Boxes. Therefore, they have been inferred statistically from currently 254 boxes, showing an average ratio of 30 % AT and 70 % MT.
• As the probabilities for AT and MT are not the same, the analogy would be an unfair coin with a probability of $ \tfrac3{10} $ for heads and $ \tfrac7{10} $ for tails.

As AT cards are legendary and MT cards are rare (see rarity), players often regard the occurrence of an AT as success and that of an MT as failure. There has been a report of a 12 MT streak by a player looking for AT.^[11]

• The probability of getting MT from a Daily Kit Box is $ \tfrac7{10} $, while the probability of AT is $ \tfrac3{10} $.
• The “ex-ante” probability of a streak of 12 MTs is $ (\tfrac7{10})^{12} = 1.38 % $.
• In this case, contrary to the coin example, the probability of getting 11 MTs and then one AT is not the same as of getting 12 MTs, but less: $ (\tfrac7{10})^{11} \cdot \tfrac{3}{10} = 0.59 % $.

Players disappointed of not getting AT from Daily Kit Boxes should be aware that AT is always less likely to occur than MT, and that a streak of MT is actually the most probable sequence given the underlying probabilities:

$ \frac7{10} \cdot \frac7{10} \cdot \ldots \cdot \frac7{10} \cdot \color{limegreen}\frac7{10} $ is always greater than $ \frac7{10} \cdot \frac7{10} \cdot \ldots \cdot \frac7{10} \cdot \color{limegreen}\frac3{10} $.

Another comparison: Players would not be astonished to get a V6 Engine from an Extra Box. However, as of the Showdown Update, the probability of getting it is actually only 0.45 % which is even less than the 1.38 % probability of a 12 MT streak from Daily Kit Boxes.

Psychological aspects

The tendency of players to regard a streak of equal values as “atypical” or “non-random” has been the subject of various psychological studies.

First, a random process is not actively self-correcting, but the absence of such a law seems to conflict with people's everyday experience:

“Some familiar processes in nature obey such laws: a deviation from a stable equilibrium produces a force that restores the equilibrium. The laws of chance, in contrast, do not work that way: deviations are not canceled as sampling proceeds, they are merely diluted.” ^[8]

Second, it has been found that the notion of randomness itself mostly does not correspond with reality:

“People's notion of randomness is biased in that they see clumps or streaks in truly random series and expect more alternation, or shorter runs, than are there. Similarly, [when asked to produce a ‘random’ series] they produce series with higher than expected alternation rates.” ^[12]

This even leads to casinos in Las Vegas changing roulette weels more frequently than warranted. As soon as a wheel exhibits an usual run of reds, the wheel is changed, even if it is still operating perfectly fine (“randomly”),^[13] just to please the gamblers' expectation that streaks are unsual—and thus reinforcing and increasing their belief in the fallacy.

Regarding the Asphalt games, the incorrect notion of randomness also leads to a variety of “myths” among players, such as the game allegedly granting more or less of a desired card if players have performed certain actions or behave in a certain way. In reality, this perception is caused by mere coincidence—and by the fact that players usually do not keep track of all results of a random process, but only consider a small period of interest, mostly during special events when they need certain cards. These cards are then perceived as occurring less frequently, although their frequency does not differ from their normal drop rates.

See also

• Law of large numbers
• Wiikiproject Statistics: Myths

References