Gambler's fallacy

The gambler's fallacy (also the Monte Carlo fallacy or the fallacy of statistics) is the logical fallacy that a random process becomes less random, and thus more predictable, as it is repeated. This is most commonly seen in gambling, hence the name of the fallacy. For example, a person playing craps may feel that the dice are "due" for a certain number, based on their failure to win after multiple rolls. This is a false belief, as the odds of rolling a certain number are the same for each roll, independent of previous or future rolls.

The fallacy is a fallacy of false cause and an informal fallacy.

Reverse
The Reverse Gambler's Fallacy is a misunderstanding of the laws of probability, most notably the law of large numbers, to imply that independent random variables can show trends that can be extrapolated to the future (or to the past).

For instance, given four successive results of "heads" for a series of fair coin tosses, instances of the fallacy make the gambler think that a result of "heads" is more probable for the next toss (the coin "favors" heads) or that a result of "tails" is more probable ("the odds need to even out"). Sometimes, the gambler may consider that he or she is on a 'winning streak', or is having good luck, that will continue.

You do not commit this fallacy if you predict the trend will continue because you suspect the underlying variables are not random and/or independent of one another. For example, if you suspect the coin toss is not a fair (i.e. unbiased) trial, you do not commit this fallacy if you predict the next toss will yield another head (though you may be making the Texas sharpshooter fallacy in suspecting bias in the first place).

Inverse
You commit the Inverse Gambler's Fallacy if you deduce, from an unlikely outcome of a random event (e.g. a dice roll), that many such events (dice rolls) probably occurred before.

For example, if you roll two dice and get double sixes, and then reason as follows:


 * 1) It is unlikely that double sixes would be thrown in a single roll.
 * 2) It is more likely that double sixes would be thrown in a long series of rolls.
 * 3) So someone has probably been throwing these dice before I came along.

The Inverse Gambler's Fallacy is frequently mentioned in discussions of multiple universes and the anthropic principle. For example, many people reason thus:


 * 1) It is unlikely that a single universe would happen to have physics capable of supporting life.
 * 2) It is more likely that such a universe would exist if there are multiple universes.
 * So, there are probably multiple universes.

Amongst philosophers studying anthropic reasoning, it has been debated whether this particular argument is or is not a fallacy. Although structurally similar to the fallacious example at the top, some, such as note the following: In the latter case we wouldn't be here to observe the "dice roll" (i.e. the laws of physics) if it was anything other than "double sixes" (i.e. a "fine-tuned" universe). Thus, this argument is not the same.

Instead, they suggest the following analogy:
 * Suppose you're told that a pair of dice will be rolled until a double six appears. You will then be allowed into the room to see the double six. You are summoned to the room and see the double six. Should you conclude that there were probably some previous rolls?

In this case, yes.

Further examples
The lack of predictability holds most clearly for coin-tossing &mdash; if one tosses 1,000 "heads" in a row, the odds of the next toss are still 50:50, even though people may think "tails" is more likely because of the lack of tails in the past. In one respect, this is because we're often unfamiliar with what a truly random sequence can look like. We expect a random sequence of heads and tails to look evenly distributed between the two options, such as HHTHTTHTTH, while the sequence HHHHHTTTTT looks less random. While the latter might have more order as there are more combinations that are evenly distributed than combinations where all the heads are clustered together, it is not any less likely a combination. We simply infer that this is not random because of the higher order, even though this is mathematically incorrect. By the fact that we perceive a sequence such as HHTHTTHTTH as more random than HHHHHTTTTT, we expect a more closely packed alternation between heads and tails. A string of one result is expected to switch to the other within three or four coin tosses, not continue on as a streak.

Looking at how the odds work, we can see why the expectation of a change is wrong and why the gambler's fallacy is fallacious. The odds for any particular combination of ten coin flips is as follows:

$$ 1/2 \times 1/2 \times 1/2 \times 1/2 \times 1/2 \times 1/2 \times 1/2 \times 1/2 \times 1/2 \times 1/2 = 1/1024 \,$$

This is true for any potential combination. Therefore the combination HHHHHHHHHT is precisely the same as HHHHHHHHHH. Illustrated another way, we can look at the probability from the perspective after the first 9 flips. This is:

$$ 1 \times 1 \times 1 \times 1 \times 1 \times 1 \times 1 \times 1 \times 1 \times 1/2 = 1/2 $$

All the 50:50 probabilities have changed into 100% certainties. This is because probability represents uncertainty. After a fact, we no longer have any uncertainty &mdash; if we've flipped nine coins, they can be on the table ahead of us with a clear result, we're no longer unsure which ones will land heads and which ones will land tails because they've already done so. While flipping 9 heads in a row is a significantly unlikely event, as it's just one potential combination out of 512 it is no more special (except for the perceived ordering) than any other combination. After the event occurs, however, the probability of it occurring is 1.

Similarly, the odds of winning the lottery don't increase or even decrease every time you play &mdash; even though people may think that they haven't won a small prize for a while so one should be due. In this case, the root of the fallacy is revealed; people believe random events become "overdue" if their occurrences lie outside the given odds. The odds of an asteroid impact causing an extinction level event might be worded as "one every 65 million years" but this doesn't magically increase the certainty once 65 million years have passed.

The name "gambler's fallacy" comes from the fact that this can often lead to people wasting money on gambling just because they think they're on a losing streak that they must break sooner or later. Slot machines suffer the same fate, with people insisting that some machines are "hot" and "due" for a big payout because that's how they're programmed. This simply isn't true, as probabilities can be read from manufacturing specifications and legal documents showing no such programming. This isn't helped by the selective reporting of people who do go on such losing streaks only to turn it around later, while the ones who continued their streaks stayed silent &mdash; or turned their luck around later anyway.

Contrary examples
The Gambler's Fallacy doesn't always apply as a fallacy &mdash; there are plenty of incidents where past actions do affect the next move. In card games, from poker to blackjack, cards already played are sometimes not available for the next hand, or draw &mdash; so odds have obviously been changed. Hence a "card counter" is a player who assigns a value to each played card to keep track of probability changes as cards are dealt and used, taking advantage of statistical opportunities when they appear. However, this is not a true exception to the rule as the odds really are different each time around due to the act of cards being used up &mdash; if you draw an ace from a complete deck of cards, a 4 in 52 (7.6%) chance, then the odds of drawing an ace again are changed to 3 in 51 (5.8%), as there are fewer cards available. These are examples of dependent events; the gambler's fallacy is properly understood as only applying to independent ones.

This is also the case with black swan type events. As these high impact, and often unfavourable, events are unpredictable it's not easy to know what one will strike, or when, or how they will be initiated. However, once it has struck then the problem and cause may be identified (in fact this application of hindsight is part of the definition of "black swan") and solved. The odds of that specific event ever happening again are reduced significantly. For example, while the 9/11 terrorist attacks happened precisely because no one (other than the perpetrators) thought they were possible, the reactionary moves to lock and secure cockpit doors as well as tighten airport security to comical levels means that an identical attack is far less likely, if not impossible. Of course, the odds of those incidents that are still genuinely outside the realm of our common sense experience haven't been affected.

To muddy the waters further, some Video games (and other digital products) use a pseudo-random generator that works like the gambler's fallacy. In these instances, a random event is in fact due after enough failures to achieve it.

Finally, while black swan events do still apply, actual casinos deal in high enough volume relative to individual gamblers (and with a high enough take) that a casino manager can realistically expect that the house will always (eventually) win.