William Feller's book An Introduction to Probability Theory and Its Applications Volume I is more commonly and affectionately known as Feller Volume One. It is on many statistician's and mathematician's deserted (desert?) island book lists. The deserted island book list is your list of 10 (or so) books that you take with you to a deserted island to keep you mentally fit. Because, as is well known, there is so much food, shelter and water on any deserted island you will have plenty of time to read Feller Volume One and do the problems. I am assured there will be plenty of room to write solutions in the sand between low and high tide. I know this from watching reruns of Gilligan's island and the high-tech version of Gilligan's island, LOST. 

There is also Feller Volume 2, an excellent text on mathematical statistics. One of my favorite mathematical stories is from Volume II and involves the 'persistence of bad luck'. None of the following is mine, it is all rephrasing of Feller in Feller Volume II starting on page 15. 

Consider a waiting time $X$. I experience $X_0$ as my waiting time. What waiting time do you ask? Well, I usually think of this at Costco as I go to pick a checkout line to stand in. So $X_0$ is the waiting time until I get to the checker. 

Of course, this waiting time is much too long. For a spot of cruel fun, I get my countably infinite friends $i=1, 2, \ldots$ to visit Costco and stand in the same interminable line and experience waiting times $X_i$. How long until one of my friends $i$ experiences a waiting time $X_i$ longer than my waiting time? 

We are looking for the friend $i$ where $X_i > X_0$ but all previous friends, $j=1, \ldots, i-1$ have $X_j < X_0$. At friend $i$, the probability that the longest time is $X_i$ and the second longest is $X_0$ is $1/(i*(i+1))$. The random variable $i$ until friend $i$ waits longer than I do has an infinite mean! That shows that I was indeed very unlucky in how long I had to wait until I finally got to the front of the check out queue. 

Yes, there are a few conditions: the $X_0$ and $X_i$ need to be iid continuous random variables. My friends need to be numbered (You do have friends right? And you number your friends don't you?). As most people don't like the tattoo option, I've had to come up with a database of friends and their numbers. Also, every friend needs to have an infinitely accurate stop watch (available from Costco online!) and a Costco membership. You remember that quarter Costco had infinite profits? That was the year all my friends joined up.