Why is problem of the points so important? Hello, i’ve been told that the problem of the points is what ”started modern probability theory”. But I don’t understand why. Problem of the points is about finding the chance that one of two players wins, given a certain amount of required points, and how many points our player and his rivals have. I get that the big breakout, idea behind the solution is that you find the amount of possibilities where our player wins, and divide it by the total amount of possibilities. I understand why people hadn’t thought of this, because back then probability was more of a “spiritual” thing. But why couldn’t the same logic be used on a much simpler problem like: “What’s the chance of rolling 4 on a dice with 6 sides?”. To find the solution, $\frac{1}{6}$ we’d still use the logic of: $\frac{Possibilities-that-satisfy-requirement}{total-amount-of- possibilities}$. It’s a much simpler problem, and it uses dice which is the thing you’d mentally connect with randomness.
So, as my question: what’s special about the problem of points, and was the solution accepted even though people thought randomness was something decided by the gods?
The problem of points is important because it was in the area between simple and impossible reasoning about chance. It had defeated both Luca Pacioli and Niccolò Tartaglia, respectively the writer of the first vernacular algebra textbook and the solver of cubic equations.
What Blaise Pascal and Pierre de Fermat brought to the problem were reformulations of the question into solutions which were convincingly correct, respectively a careful translation from unequally probable outcomes to countable equally probable outcomes and the use of what became recurrence relations. Fermat's efforts developed into his triangle and thus the binomial distribution as an efficient way of calculating Pascal's counts.
Pascal and Fermat's work immediately triggered Christiaan Huygens's De ratiociniis in ludo aleae ("On Reasoning in Games of Chance") which in turn provoked a scientific approach to life assurance and annuities as well as the probability developments of Jacob Bernoulli and others.
So in this sense the problem of points and its correct solutions launched an explosion in probability theory that had previously been restricted to the most obvious gambling concepts often wrongly interpreted