Even though you would expect that gamers would tend to be rational and good at math, I have noticed that gamers have a lot of superstitions and misunderstandings of probability.
For example, the sheer variety of dice-related superstitions is legendary. Frequently gamers will refer to some dice as being "lucky" or "unlucky." This in itself may sometimes make sense because it is possible that a die might be defective and so produce uneven results, but there are lots of superstitions that can't be explained that way. For example, one gamer told me that his group didn't want him to touch their dice because any dice he touched would suddenly "have their luck drained" and "roll horribly for months." And if you do a google search for "dice superstitions" you will find a whole lot more, like people who put dice that roll poorly in a freezer to "teach them a lesson."
Another common category of misunderstandings is not understanding independence of events. Frequently when someone gets a very good roll on something unimportant he will express regret at "wasting" a good roll. Of course getting this roll in no way impacts any future rolls. Last night I was playing Magic: The Gathering. It was a "draft" format which means you pass booster packs around, "drafting" cards out of the packs, and making a deck with the cards you draft. My opponent had a "Tome Scour" card in his deck that "mills"a target player five cards - i.e. makes him put five cards from the top of his library (draw pile) into his graveyard (discard pile). The main use of this type of card is in "mill decks" that try to win by emptying out the opponent's library (because if you have no cards to draw at the start of your turn you lose.) My opponent didn't have kind of deck but did say that he liked that card because "in the last game, I milled out lots of really good cards." Of course, milling out cards does not affect the average quality of the cards coming up - it's just as likely you will mill out poor cards and leave the opponent with the good ones.
Of course, sometimes the players aren't the ones that get the probability messed up - sometimes the players know more about probability than the game designers. My problem from a while ago, "
Mathematically Challenged," is based on the actual skill challenge system in D+D 4th edition. If you didn't allow the "aid another" trick described in the problem (and the rules actually said you weren't supposed to allow that trick) I think the calculation was something like the characters had about a 7% chance of succeeding at an average difficulty skill challenge. After this was discovered, they released errata* that changed all the difficulty levels and challenge rules so that instead of it being a 7% chance of success, it was more like a 99.7% chance.
*There are currently over 100 pages of errata to D+D 4th edition, most of them to fix things which proved to be too powerful or easy to abuse. One recent example was Wormhole Plunge, a power that creates a one-square zone where whenever an enemy is in that zone, you can teleport him three squares. The trick was to teleport him three squares
straight up, so he falls down and takes falling damage. After this he is then in the same square as before, so you can repeat the process. And the teleport is a free action, so you can do it as many times as you want in one turn, so you can keep going until the monster is dead, no matter how tough the monster is. And this is a power you can get at level 1. The fix was to make it so you can only do the teleport once per round.
