What are some interesting un-intuitive problems in probability aside from Monty Hall? Does anyone know of some interesting and almost strange problems in probability? 

I know that probability is sometimes notorious for being mind-bending and un-intuitive! (Monty Hall is already an obvious one)
Shuffle a deck of cards (standard $52$-card bridge deck) and deal one card. What's the probability that it's an ace? The answer is $\frac1{13},$ nothing unintuitive about that.

Shuffle a deck of cards and deal one card after another until an ace is dealt, then deal one more card. What 's the probability that the last card is an ace? Intuitiveness is a matter of opinion; personally, I find it kind of unintuitive that the probability is still $\frac1{13}.$

(More generally, for $n\ge k\ge2,$ if balls are drawn one by one without replacement from an urn containing $n$ balls of which $k$ are black, then the probability that the first black ball is immediately followed by another black ball is equal to $k/n.$)
An interesting introductory probability problem is one of basic geometry.  Say you have a circle with an inscribed equilateral triangle.  The question is, if you were to randomly draw a chord on this circle, what is the probability that it will be greater than the side length of the triangle?

You will come to 2 (or possibly more) conclusions.  Thus, this problem emphasizes the importance of being precise when stating a question in probability.  Saying "randomly selecting a chord on the circle" is the ambiguous statement that allows one to arrive at several conclusions about the probability.