Mathematics and precedence rules

[Saposhiente]

Note however that showing an event has probability zero is somewhat different from saying it can't happen. Applied strictly, [...]

Umm yeah, that's kind of exactly what I said.

Gambler's fallacy stuff

Here is the difference:
The probability of a fair coin coming up heads 50 times in a row is 1/2⁵⁰ (Very small). The probability of a fair coin coming up heads 50 times in a row given that the first 49 flips are guaranteed to come up heads (In this case, because you have already observed that they have. But you can also pretend that you're a time traveler and looked into the future before flipping. Etc...) is 50/50.

One thing to note though, is that while the number of times heads has come up in a row doesn't change the probability of the next flip being heads on a fair coin, it does increase the probability that the coin is not actually fair. So if given even odds on a real-life coin, after 49 headflips, you should bet on heads. Thus, the gambler's fallacy suggests the opposite of the correct move*

*Unless you're using a guaranteed-fair pseudorandom computer RNG. RNGs use an algorithm to effectively just cycle through a long list of random numbers where each number comes up the same number of times. Thus, if a bunch of the numbers in the list that cause one thing to happen come up at the same time, those instances of those numbers aren't going to come up again until the list is run through (which is almost never; instead it's generally when you restart your computer/program). Of course the list is so large that the effect is negligible, and other more complicated things can negate the effect; thus unless you are absolutely certain that the coin is indeed fair you should still take the advice opposite of the Gambler's Fallacy.
(If none of that made any sense, try replacing the coin with a deck of cards with face cards removed, with heads being even numbers and tails being odd numbers. The Gambler's fallacy will hold true with this for similar reasons.)

Casino stuff: House edge isn't the probability that the house wins; it's how much more likely it is for the house to win (Adjusted as necessary if the house gets less money when it whens than it has to give out if the player wins). Eg. even the best blackjack player can still only win ~47% of the time without cheating. The 47% where the player wins cancels out with 47% that the house wins, leaving 53%-47%=6% where the house gains the bet in money on average.

nedz: It should approach pi, however rounding methods can change this.