There are a few well-known examples where the probabilities of occurrences are contrary to intuition. One of them is the birthday paradox, with an example question such as "what is the probability that any two students in a class of 50 would have the same birthday?" Since there's 365 different birthdays (or 366 in leap years, though we ignore this for the moment), many people would think the chance of collision is not high. However, it can be shown that the probability of a collision is 1-365*364*...*316/(365^50) = 97% which is actually very likely.
Monty Hall paradox. There are three doors, A, B, and C, to choose from in this puzzle, and there's a prize behind one of the doors. The probability of choosing the door with the prize can easily be figured out to be 1/3. However, if after choosing, say, door A, and before checking if it contains the prize, door B is revealed to not contain the prize. At this time, you are free to stay with door A or choose door C instead. Most people would think that it doesn't matter, with both door A and door C having a 1/2 chance of containing the prize. This is however incorrect, and the real probabilities of door A and door C containing the prize is 1/3 and 2/3, respectively.
One other example is determining the probability of contracting a virus given the result of an imperfect virus test. Assume the probability of contracting the virus is one in ten thousand (i.e., 0.01%) and the accuracy of the virus test is 99% (that is, 99 of 100 that really contracted the virus would show positive results and one would show negative; also, 99 of 100 of those that did not contract the virus would show negative results, but one would show positive), what would be the probability of a person really contracting the virus if his/her test is positive? Intuition would suggest the probability to be ~99%, but in reality it is 1%.