Gambler's Fallacy

Believing that dice/coins have memory, or that independent events will occur in "streaks". If a coin has just landed on heads four times in a row, surely it's much more likely to get tails this time, to even things out... or alternatively, heads is on a roll and will appear next time, too. See also Random Number God and Artistic License Statistics. In fact, if you toss a previously untested coin and (say) heads come up, there's a larger chance to get heads on a second roll, because the coin might be biased, although not very much larger, unless the coin is so warped that the imperfection is clearly visible.

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 same as any expected sequence of two numbers. 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 two 20's in a row, it's less likely than usual (< 1/20) that they'll get another 20. In reality, once they've rolled two 20's in a row, it's just as likely as ever (1/20) that they'll roll a 20 again. This also, most notably, works the other way around - if they've lost many bets in a row, they aren't any more likely to win the next bet.

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." This is not true.

Another factor is that many people confuse "a number of independent events" (where any of a number of permutations will do) with "a series of independent events" (where only one permutation will do). If we flip a coin twice, we have a 50% chance of getting heads and tails in some order (heads-tails and tails-heads; the other two possibilities are heads-heads and tails-tails). But if we specify that we want the series to be "heads-tails", the probability that that particular series will come up is only 25% — the outcome tails-heads no longer fits the criteria. (Of course, any series has the same chance of coming up. You have as much chance of flipping heads-tails as you do tails-heads, heads-heads, or tails-tails; namely, 25%.)

Also, stuff really does even out over time. Just not in the way some people might think. Say that you have flipped a coin and you have had 4 heads and 1 tails come up. Heads has come up 80% of the time. Now, you get the "normal" (more common) sequence, where 5 heads and 5 tails come up, bringing a total of 9 heads and 6 tails. You then have only 60% heads, so while this is a smaller number, it didn't exactly "even out."

To explain the above in another way, flip a coin 10 times, and the chances that heads was flipped 4 times or more is 82.81%. Flip it 1000 times, and the chances heads was flipped 400 times or more is 99.99999999%. But even if it was less than 400, the next flip will still be 50\50. This is also the reason why playing a high number of low-stakes games in Casinos increases the chances of the house making money; the house advantage only affects who wins a small percentage of the time, but this advantage "evens out" over the long haul. Unless you're a good card counter, taking advantage of free stuff, or just enjoy playing, you're more likely to be successful with a small number of high-stakes events.

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).

Live Action TV

 * In Only Fools and Horses, after beating Boycie at poker, Del Boy offers him double or nothing on the spin of a coin. Boycie's response is "I've beaten you on a spin twice, Del. By the law of averages, you've got to win this time." (The coin isn't fair, as it happens, but Boycie doesn't know that.)

MMORPGs

 * City of Heroes utilised a system called the Streak Breaker. This mechanic in the attack calculations sometimes forces attacks that would normally be misses to instead hit, based on the current number of misses in a row versus your chance to hit. In short, it breaks streaks of misses.
 * This one also shows up among players of World of Warcraft, in particular with the rare dragon whelp pets that drop out in the world, with many players assuming the more you kill the whelps, the higher your chance grows of finally getting a drop, while in reality the chance is independent of each past or future kill.
 * Because people refuse to accept that improbable does not mean impossible or certain (or simply because the large variation in time required was annoying), World of Warcraft developers actually modified this detail to conform to players' expectations. Your chances for a drop do gradually increase the more you kill.
 * This was also implemented because of some "kill this mob and loot this item off them" quests, where the drop rate was not 100%. Some of these quests had unusually low drop rates, and you could spend an hour or more trying to finish a quest. With this change, the chance goes up and up with each "loot", and then resets itself to the default rate with every success.
 * In fact, many video games play around with the RNG like this, both because players expect it, and because it's really, really annoying to be on the receiving end of a string of bad luck, unless it's the kind of game where being Crazy Prepared for bad luck is expected of the player.
 * A pokemon has a 1 in 8192 chance of appearing as shiny. Way back when, people thought that this meant 1 Pokemon out of every 8192 was shiny, when in fact, the odds never decreased. It's possible to go your entire career and never see a shiny.

Newspaper Comics

 * In a Peanuts comic, Lucy uses this fallacy to convince Charlie Brown to try to kick the football again.

Tabletop Games

 * Many Warhammer Fantasy Battle gamer or tabletop roleplayer will tell you that this is absolutely true. Others will perform astounding feats of Mathhammer in mid-game and tell you exactly how many units of each side should die in an assault, what is the expected variance, and whether or not the assault makes sense in terms of points of enemy units destroyed versus own loses.

Theatre

 * Discussed at length in 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. (Technically, Guildenstern is right in that given a fair coin, a series of 85 heads is exactly as probable as any other single series of that length [namely, 1 in 2^85]; however, if a coin should actually land the same 85 times, it's a good reason to believe that such a coin [or flip] is NOT fair.)

Web Comics
"Narrator: By now, the die is rolled. It's a 1; Qui-Gon dies. Pete (the one who prepared the die): "Awesome! That die will be even luckier next time!""
 * 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?

Real Life

 * The Martingale system works if three conditions are met: the player must have access to infinite reserves of capital, the player must be willing to endure a losing streak of any length, and the house must tolerate a table bet of infinite size. So it doesn't work. Casinos can hire mathematicians too; they'd never allow any system which can beat them, but what they will do is design it to look beatable.
 * Now, losing ten times in a row is pretty rare right? Well, it's a 1 in 1024 chance, which means it's likely to happen before you win 1024 times, and almost certain to happen before you win much more than that. And that's assuming 50\50 odds. Basically, it's like the house is buying under-priced lottery tickets from you and you're hoping they don't win. Also, since you're playing so many games to do this, the "evening out" effect of the house advantage makes your chances worse than some other strategies (more information on that near the end of the trope description).
 * The most famous example of this fallacy is the posting of roulette history in casinos. It's designed to trick people into falling right to this thought. For example, people might see the last few hits were red and so they bet on black. But in fact, there's still the same chance of landing on either color (18/38, assuming a double-zero layout).
 * This assumes that the wheel is fair. In reality, some gamblers were able to win big money by observing the bias of individual wheels (nowadays, analyzing the results with a computer, too), and betting accordingly.
 * There is a gambler's saying: If a coin is flipped 10 times in a row and comes up heads each time, the layman will assume tails is "due" and bet on tails. The mathematician will assume each flip is an independent event and not bet either side. The gambler will assume that something weird is going on and will not bet either side with the person flipping the coin—but will bet a third party that the next flip will come up heads.
 * This thread on Operation Sports.

Looks like this fallacy but is not

 * When the events are not independent: If you draw 10 red cards from a shuffled deck without replacing them, then the next one really is more likely to be black than red because of the 42 cards remaining, 26 are black but only 16 are red. This sort of situation in the real world is, in fact, hypothesized to be how humans developed the intuitions that lead to this fallacy in the first place.
 * It's also how counting cards in Blackjack works, but seriously, between the 6 to 8 deck sabots, constant reshufflings, Rainman-like mental math needed and pit bosses kicking out players for doing it? Not worth the hassle. Stick to mastering the basic game principles and play a perfect game, it's far easier.
 * If it has not been established that the trials are fair, then a significant deviation from the expected results could count as evidence that they are biased somehow. If a die rolls a 6 at least 10 times in a row, then it is possible that this is because the die is weighted.
 * Of course, one must also be careful not to take the possibility that it is weighed as the certainty that is weighed. It can never be 100% proven through statistical analysis alone, but it can be proven with 99.9999999% confidence.
 * It's also fully possible a die is weighted simply by poor quality control rather than deliberate action. Outside of casinos quantity of die purchased is preferred to quality and many die wind up with wildly varying sizes and edges. A "lucky die" or "cursed die" may very well have favor those numbers.
 * And that's science in a nutshell.