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In general, I am trying to figure out what equation or process one could use to mathematically determine the probability of certain hands based on the cards on the board.
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Jon,
The only way to be mathematical about it is to assume that your opponents are dealt random hands out of the unseen cards. This method is not too helpful, though, as it ignores important facts, like the meaning of your oppo's previous bets and the fact that even poor players throw many hands away (like 72o) preflop.
However, if you are interested in how to solve the problem assuming your opponents cards are dealt randomly out of the unseen cards, I solved a similar problem here:
link
See my last response, especially, for the kind of methods you use. I urge you not to try to learn this stuff to improve your poker game, though -- there are much better ways.
If you are still really interested in seeing the specific problem posted solved, let me know and I'll do it. Again, the only real use of this exercise is to improve your probability skills in a way that will not translate directly to the poker table.
HTH,
gm