For example, witness the Monty Hall problem and the concept that 0.9999... equals 1.
In case you've never encountered the Monty Hall problem, it basically involves a word puzzle using the old Let's Make a Deal TV show as a setting. Here's the puzzle as it appeared in Marilyn vos Savant's "Ask Marilyn" column back in 1990:
Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1 [but the door is not opened], and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?
Pretty much everyone would answer "no" since door 1 and 2 each have an equal chance to have the car behind them. However Marilyn's answer, which has been backed up by professional mathematicians and logicians since then, is that switching to door 2 doubles your chance of winning the car.
Say what?! Bizarre, huh? This line of reasoning says that each door at the beginning has an equal (1/3) chance of having the car behind it. So when you choose door 1, you have a third chance of being right. That never changes. However door 2, which starts out with a third chance of having the car, increases to 2/3 chance once that door 3 is revealed to have a goat behind it.
As far as I'm concerted this reasoning is nuts. But it's statistically sound. Personally I'd think good ole Monty Hall was trying to screw with me and I'd never switch doors but that's just evidence that I'm cynical. :-)
Even if you don't subscribe to my cynicism though, why would the chances of door number 2 increase? Well, since there's no reason for the chances of door 1 to increase past 1/3 (a postulate I don't agree with) then once door 3 is revealed, the additional 1/3 chance that was originally ascribed to door 3 must go to door 2. (I don't buy that) And so you end up with door 1 having a one third chance of having the car and door 2 having two thirds.
There's a detailed run down of this on Wiki but I'm not convinced.
The idea that 0.9999 with infinitely continuing 9s equals 1 is another couterintuitive argument. I don't care how thin you slice it, a fraction of one does NOT equal one. However mathematicians disagree. Personally I think that they're just painted in the corner due to the argument:
One third times 3 = 1
one third = 0.3333
3 * 0.3333 = 0.9999
therefore 0.9999 = 1
To me that just seems silly. It might be by an near infinitely small amount, but 0.999999999 etc is still less than 1. Too bad almost all mathematicians disagree.