We've been talking a lot about probability and poker at lunch. I'm sure somewhere someone has done the calculation for the probability of winning given a certain (partial) hand. It may be related to the number of players. Anyway, I found a great starting place. Interestingly, the probability of getting one pair is about 42.3%. So it is appropriate to feel bad when you don't even have a pair. Incidentally, there are 2,598,960 possible hands (52C5). That's not actually that many - though you can imagine the math can get pretty crazy when there are multiple players. A lot of factors need to be computed to really give the probability of winning. And then there is the chance of switching cards...
Some Poker Odds Calculator
BTW, use at your own risk - by no means am I advocating gambling... this is just for education. Also, be careful what you click.