As others have said, maximize expected utility, not expected dollars. Money being roughly logarithmic in value works pretty well, and the common advice is to pick gambles that maximize your expected log-net-worth.

For a more specific recommendation, see http://en.wikipedia.org/wiki/Kelly_criterion

As to your final question, the answer is "yes". Probability can be applied to any unknown. A good description is in the middle of the quantum mechanics sequence: http://lesswrong.com/lw/oj/probability_is_in_the_mind/