Whilst under house arrest in the sixteenth century, Galileo Galilee wrote "Dialogues Concerning Two New Sciences", wherein the two new sciences demonstrated were "On the movement of objects" (physics) and "On the hardness of materials" (chemistry). Within this text there is also a diagram of a bird along with measurements, our very first example of (anatomy with numbers in it).

There was a big shift in mathematics during the renaissance. Algebra flooded into Europe quite all at once and an age of constant innovation began then which has not stopped. Numbers changed shape and were applied to many new things as they never had before.

The Greeks had famously rigid ideas about what numbers were, and for all their masterful mayhematical achievements which were sometimes invented by top natural philosophers, nobody applied numbers to the real world as Galileo would. Greek mathematical works have few numbers and rather more variables, which does not translate easily to probability. Moreover, the paradox of traversing infinitely-many points was a well-known and early, formative result to Greek mathematicians... it characterises the belief that numbers are incongruous with time and motion. Clocks weren't so good that you could measure increments of time, and we plainly don't find events quantified. Every calendar in Greece was local to a small region, so that the date would not seem objective.

Hindu mathematicians had been much quicker to apply mathematics to time, as exemplified by the Yajur Veda's (~1,000BC) enumeration of transinfinite eschatological timeframes. Ancient Hindu mathematics really exclusively comes from religious texts by polymaths, to whom the study of chance will have been contrary to sensibilities. Chance hardly exists outside of games of chance, and not before we have quantified it so we can see any matter in comparison to rolling a dice.

Theological concerns and those of piety also preclude the pre-reformation Europe from studies of chance. If the Almighty controls everything, there is no chance. Our mathematical texts from the days when scribes were clergy are all religious texts too, where there would be no place for discussion of gambling and indeterminate futures. This stands for muslim mathematics too.

It is not long after clocks became common in Europe that we have the first investigation of probability.

In 1494 Luca Pacioli described the game of stakes in "Everything of Arithmetic, Geometry, Proportion, and Proportionality". An ante is continually increased by two players, each contributing, until one player wins a certain amount of times. He gives a method to split the ante if the game ends early, simply divide by the proportion of wins each player has had.

This solution is reasonable, but you might have been one win away from winning the whole thing when the game ends. It is equally skew that you could win the whole ante with just one win, after one round, when the game was supposed to go on for perhaps 100 rounds.

These contentions were raised by Nicolo Tartaglia in the 16th century who proposed methods to get around them, by dividing also by the proportion of rounds played "so far" when the game stops. This has us quantifying non-astronomical temporal events in abstract and that was brand new.

Tartaglia's paper and question was picked up by Blaise Pascal and Pierre Fermat in the mid 17th Century who worked on it more, and that's where the modern theory of probability begins, where it is loosed from the specific question of Pacioli's game of stakes.

It is relevant to this question that when probability theory did start up, it was in the context of enlightenment age academics sharing letters and keeping up a contemporary mathematics of the time, something quite new and also cliquey.

The game of stakes problem cam be solved in several ways, any of which can be deemed the most fair or best applicable. This is still a feature of statistical science, that one does not produce an answer without first assuming a model. A lot of conceptual prerequisites had to be fulfilled, and once they were, this science required a bit of group consensus to establish that models are fair and applicable.

With these sorts of concerns in mind, the mid-17th century and in Europe looks and was ripe for the canonical birth of probability theory. Theological concerns were gone, taboos about gambling gone, ontology of number in a fully transient state about to settle on something wonderful, and men sitting around, talking about games, agreeing with each other.

Measuring arbitrary time slots was a big deal, it was done piece by piece. Galileo lived between the first discussion of the game of stakes and the second, and quantified the natural world. Probability is part of that same era and thrust of quantification.

There is another lineage going back through Fibonacci, arriving at Blaise Pascal's work on combinatorics. The non-stochastic bit of Probability Theory is combinatorics, and so there is also this dependency. You needed combinatorics for Probability Theory, and combinatorics was another enlightenment thing.

Perhaps because humans have a very poor instinctive understanding of chance, so they will play games they are likely to lose. Consider a few examples:

There's the Gambler's Fallacy- that somehow past events will influence probability in the future. So, if a coin is flipped 50 times and comes up heads each time, humans will want to believe that the next flip will almost certainly be tails- when, the actual chance for that flip will still be even for either heads or tails ( if it's not a trick coin).

Similarly, humans would like to think a significant event has to have a significant cause, can't be simply chance. If I win the lottery, for example, the odds in my favor are so small that there must be some trick- a rigged lottery, or supernatural forces at work. When, in reality, someone inevitably has to win. This lottery fallacy has been seen quite a bit in various attempts to re-write history. Surely, the pyramids could not have been built by someone so backward as the early Egyptians. Surely, a moderately skilled gunman could not have pulled off the feat of shooting J. F. Kennedy in a moving car from that great a distance.

In the 18th c. there was a simple betting strategy called the Martingale. In a simple series of coin tosses, where you lose a bet when a coin is heads, win the bet when tails, you would double your bet with every heads. The popularity is easy to see: the first win should wipe out the first loss and pay for the original stake. However, if it is used for long, the chances are equal that the gambler will eventually have a catastrophic failure or a huge win. Only if the gambler has infinite funds does the Martingale system actually work...and gamblers never have infinite funds. The Martingale still had adherents who felt it should work.

Mansuy, Roger (2009) The Origins of the Word "Martingale. Electronic Journ@l for Probability and Statistics