Here is an interesting probability problem who recently generated long discussions in our team:

Say that you have an array of N boolean values, with all values initially set to FALSE. At each iteration step, you arbitrary pick an element in the array and set it to TRUE. What is the average number of elements set to TRUE after K iteration steps?

By the way, as an anecdotic fact, this problem came from analyzing paged pool memory consumption when you randomly access a large volume over time. I will not go in more details, but I just wanted to mention this problem as a particular application of theoretical statistics in the design of operating system components...