After figuring out the best possible hand I can make from the 2 known cards in my hand and the 5 known cards on the board, how would I calculate the odds that my best hand is better than any hand my opponent can make with their 2 unknown cards and the 5 cards already on the board?
You can use a poker calculator. A very good one in my opinion is the following:
You can fill in the particular board and your particular hand and simulate what your winning chances are against a random hand. However you can do much more than that. Go check out the site and you would be surprised.
They will not let you use a poker calculator at a live table.
Even on line many sites restrict the levels of calculators.
There are 990 two card hands after you take out your two cards and the board. Just count up the number of hands that beat you and divide by 990. This would be for a random hand but at the river they are not on a random hand.
Add up all the cards your opponent could have based on action and your opponent. If you beat 2/3 of them then that is your odds. There are only 6 ways to make a pair. If you are worried about a set there are only 3 ways. Suited connector 1 way. Unsuited connector 16 ways. If there is a clock it is hard to get done.
If you have a good hand just count the number of hands that beat you. If you have the low end of straight then 16 hands can beat you. Can you put your opponent on 16 hands that don't beat you.
First to act
Are ahead is not really the issue. Are there hands they could call with that you beat. So you need a pretty dominant hand like a straight that could get called by a trips or two pair. If the board is paired for sure you are getting called by a boat.
Example You have a set of tens there are two over cards and a flush draw that filled in on the river. You did not bet very hard so they could put you on top pair. If you add it up it is going to be about 50 50. So you it will come down can he bluff. If he missed does he have it in him to fire a bluff.