While I can't speak much to its affect on the game of poker, I can lend a little insight into the origins of probability theory. Poker's inception is generally reported as early 19th century, with first documentation coming from 1829 New Orleans. Probability theory was developed in the mid-17th century in a fascinating correspondence between two legendary mathematicians, Pierre Fermat (he of Fermat's Last Theorem) and Blaise Pascal (he of Pascal's wager). I'm going to use a quote from Tom Apostol's Calculus to sum up the origin:

"A gambler's dispute in 1654 led to the creation of a mathematical theory of probability by two famous French mathematicians, Blaise Pascal and Pierre de Fermat. Antoine Gombaud, Chevalier de Méré, a French nobleman with an interest in gaming and gambling questions, called Pascal's attention to an apparent contradiction concerning a popular dice game. The game consisted in throwing a pair of dice 24 times; the problem was to decide whether or not to bet even money on the occurrence of at least one "double six" during the 24 throws. A seemingly well-established gambling rule led de Méré to believe that betting on a double six in 24 throws would be profitable, but his own calculations indicated just the opposite.

Pascal wrote to Fermat about it and the two began a long correspondence concerning the process of calculating the odds of each dice throw. This series of letters between the two great minds resulted in a detailed documentation of probability theorem, and it really hasn't changed much since then. For an in-depth analysis of the correspondence which is accessible to the non-mathematicians among us, I recommend the excellent book The Unfinished Game by Keith Devlin.

So probability theory as we know it now was about 150 years old, well-developed, and well-understood in mathematical circles at the time of poker's birth. Whether or not the first poker players were aware of it is not a question I can answer.

EDIT: omitted link

At least from the 18th century on, there was already a solid mathematical understanding of probability and chance, as seen by Abraham de Moivre's "The Doctrine of Chances". Before the Age of Reason, the ancient Greek philosophers discussed chance but in non-quantitative means. Arguably, Epicurus in 300 BC proposed the most advanced idea in probability theory yet when he suggested that the the invisible movement of atoms caused randomness of larger events at the macro-level. Moreover, the Romans attributed chance to the goddess Fortuna Privata, who symbolized specifically individual luck.

But for the purposes of your question, it's pretty clear that by the Enlightenment, Europeans, who would be playing card games with a French deck, had a good understanding of the principles behind chance; they would be able to create games with balanced rules. To answer your question more fully, it would be good to know exactly what type of poker game you are inquiring about. For example, if you're wondering about Texas Hold'em, Wikipedia provides that the game was established sometime in the 1900s where people would understand completely the probability of permutations.

Edit: Terminology

Not an answer to your question, but I've always found it fascinating that the rules of blackjack work out to an ideal edge for the house with perfect player strategy, even though the rules were created long before the technology and existed to correctly prove the correct strategy or to prove the house edge. I'd love it if anyone could shed light on that too.

This 1868 edition gives the rankings of the hands in draw poker. A few unique differences though. For one thing, the jack was yclept knave. The Modern Hoyle

As a follow up question, throughout the 1800's were card games such as poker actually considered to be "gambling" vs a game of skill either generally or by the skilled and successful players of the time?

Building on this, what is the history of card games and playing cards in general? Whenabouts would poker and other modern games have become commonplace?