After reading (and responding) to the "3 door problem" (revisited yet again), I was reminded of a poker problem I encountered several years ago. I was retained to design a slot machine which was based on poker, but included (by nature) replacement and also included wild cards.
I wrote a program to calculate the frequencies, and soon after realized the simulation would take 2 months to run. I built a K7 dual 3G processor box and knocked it off in 2 days -- but I still wanted to see the calculations which matched the simulation. Being a commercial endeavor, I sent the question to Google answers to save time and money. You can read the thread at
http://answers.google.com/answers/threadview?id=147900
I was astounded by both my errors and the errors made by very skilled mathematicians. Poker problems employ a different calculus at every level, I had made errors and so had they.
In the end I confirmed results both by equations and simulations. This post is offered because I find that many of the calculations posted on this site disagree with my simulations. Please don't ask for specifics. I am interested in your approach to problems, and whether you use simulations to confirm calculations.
SheetWise