An interesting problem is the optimization of a betting scheme based on the odds offered, and your probability of winning each bet. For a game like blackjack, you can analytically calculate your probability of winning a particular bet. For baseball, determination of your probability of winning a bet can take some sophisticated multivariate analysis, and still there is uncertainty on your probability of winning. I find this particular aspect of baseball data analysis to be engrossing.
A great paper on optimization of betting schemes was written in 1997 by Edward Thorpe, the king of card counting. You can find the paper here. The title of the paper might make it seem like it focusses completely on the Kelly Criterion (a betting scheme that can in practice be quite disasterous), but in fact the paper discusses the derivation of a number of different betting schemes.
