dtgreene: Also, many people don't understand probability, and some things are counter-intuitive.
For example, the birthday paradox: If 23 people are in a room, there is a 50% chance that at least two of them share the same birthday. Doesn't that number seem rather small? (I believe this calculation assumes that nobody is born on February 29, and that all birthdays are equally likely; two assumptions that aren't correct, but which serve to make the calculation simpler.) This chance increases to 99.9% at 75 people, and 100% at 366 (or 367 if you count 2/29) people (thanks to the pigenhole principle).
Stevedog13: I had a mind blowing realization the other day. There are two dozen people in my office and every single one has a birthday within six months of mine! I'm gonna need an abacus or something to figure out the odds of that.
Sometimes, in mathematics, obvious facts are, indeed, the most useful. Also, it's easy to forget obvious facts sometimes.
For example, let x, y, and z be integers, and that x < y < z. It is rather obvious that z - x must be at least 2.
Similarly, if x <= y and y <= x, then x = y. (It doesn't matter what x and y are, as long as they come from the same ordered set.)
Another example: You can't have an infinite descending sequence of positive integers. Hence, if you have shown that such a sequence exists, and you didn't make a mistake, then an assumption you made must be false. (This can, for example, be used to prove that the square root of 2 is irrational.)
