For example, how many people are required in order to have a better than evens chance that two or more of them have the same birthday. That is to say, we cannot give a mathematical answer until we know all the assumptions. If we assume that each individual's birthday is equally likely to be any of the 365 days of the year, and that one person's birthday has no effect on anybody else's, then the answer is that just 23 people suffice. This is rather a small number, and tends to come by surprise. This is a good example of getting the question wrong.
Surprises emerge from calculations of probabilities.
One US professor challenges his students to write down a sequence of 36 heads and tails, using any rule whatsoever. The professor claims to be able to tell which sequences are truly random, and which have been artificially contrived. The students are challenged to defeat him. There are many rules he might use. For example, he might count the number of heads, and decide that the sequence is random whenever this number lies between 15 and 21.