I was watching the “Deadwood” HBO series on DVD the other night and in watching the gold miners play poker I wondered how early the application of mathematical probabilities to poker was implemented.
I don’t know anything for a fact, but I do know that Pascal (1623-1662) is often credited with the originating probability theory. Once that concept was out there, I feel sure that the probabilities of poker hands soon followed; the hands can be counted using fairly elementary methods involving binomial coefficients (which, while I don’t know if he’s actually credited with originating them, Pascal studied a lot).
In fact, I wouldn’t be at all surprised if the numbers of various poker hands had been counted far in advance of the 17th century.
Actually, I should probably amend that comment about Pascal originating probability theory to include Fermat as well:
There’s a bit in my copy of ‘scarne on cards’ saying that the first poker probability charts published in a book on card games appeared “about a hundred years ago.” (this was an edition last updated in the 1950’s,) and contained numerous errors, which were copied into other books and occasionally altered slightly.
He also took the opportunity to get on his soapbox a little and rail about the meaninglessness of so many books on card games being called Hoyle 
Only if they were calculated by Nostradamus. First you need poker before you can calculate the odds.
True, but poker rules might have evolved over time, being adapted to mathematical progress. It is often said that poker derived from a Persian game called as nas; I don’t seem to find a website with more info on this, but I guess as nas predates Blaise Pascal.
When I learned poker (I’m still pretty poor at it regarding winning, but at least I know the rules of several of the most important variations), I was astonished to see how differentiated the hierarchy of ranks is. I think the difference in probability between a four of a kind and a straight flush isn’t obvious enough to know freehand which one will occur more often; nor are they frequent enough for players to learn that through empiry alone. I think the designers of the rules must have had some knowledge about probability to determine that a straight flush beats a four of a kind.
The hierarchy of hands, of course.