Two really cool factoids for you. (From http://gamebalanceconcepts.wordpress.com/)
1.) The Gambler's Fallacy is something we can exploit as gamers. People assume that long streaks do not appear random, so when trying to "play randomly" they will actually change values more often than not. Against a non-championship opponent, you can win more than half the time at Rock-Paper-Scissors by knowing this. Insist on playing to best 3 of 5, or 4 of 7, or something. Since you know your opponent is unlikely to repeat their last throw, on subsequent rounds you should throw whatever would have lost to your opponent's last throw, because your opponent probably won't do the same thing twice, so you probably won't lose (the worst you can do is draw).
2.) There's a variant of the Gambler's Fallacy that mostly applies to sports and other action games. The Hot-Hand fallacy is so called because in the sport of Basketball fans started getting this idea that if a player made two or three baskets in a row, they were "running hot" and more likely to score additional baskets and not miss. (We even see this in sports games like NBA Jam, where becoming "on fire" is actually a mechanic that gives the player a speed and accuracy advantage… and some cool effects like making the basket explode in a nuclear fireball.)
When probability theorists looked at this, their first reaction was that each shot is an independent event, like rolling dice, so there's no reason why previous baskets should influence future ones at all. They expected that a player would be exactly as likely to make a basket,regardless of what happened in the players' previous attempts.
Not so fast, said Basketball fans. Who says they're completely independent events? Psychology plays a role in sports performance. Maybe the player has more confidence after making a few successful shots, and that causes them to play better. Maybe the fans cheering them on gives them a little extra mental energy. Maybe the previous baskets are a sign that the player is hyper-focused on the game and in a really solid flow state, making it more likely they'll continue to perform well. Who knows?
Fair enough, said the probability theorists, so they looked at actual statistics from a bunch of games to see if previous baskets carried any predictive value for future performance.
As it turned out, both the theorists and sports fans were wrong. If a player made several baskets in a row, it slightly increased their chance of missing next time – the longer the streak, the greater the chance of a miss (relative to what would be expected by random chance). Why? I don't think we know for sure, but presumably there is some kind of negative psychological effect. Maybe the player got tired. Maybe the other team felt that player was more of a threat, and played a more aggressive defense when that player had the ball. Maybe the crowd's cheering broke the player's flow state, or maybe the player gets overconfident and starts taking more unnecessary risks.