5 ideas
Bernoulli proves that as the number of independent trials grows, the probability that the observed ratio of outcomes lies within any chosen small margin of the true underlying ratio approaches certainty. Enough observation can therefore recover causes that cannot be seen directly, such as the true proportion of white to black stones in an urn.
Bernoulli defines moral certainty as a probability so close to 1 (he suggests 999/1000 or higher) that a reasonable person may act on it as if it were certain. This turns an unreachable demand for absolute proof into a workable, adjustable standard for decisions in law, commerce, and daily life.
Bernoulli contrasts games of chance, where probabilities can be counted in advance from equally likely cases, with matters such as disease, weather, and human longevity, where no such cases can be enumerated. For these he proposes seeking the probability a posteriori, from the outcomes of many similar past cases.
Bernoulli proposes treating each argument or piece of evidence as carrying a calculable share of proof, some pure (proving in some cases and saying nothing in others) and some mixed (proving in some cases and disproving in others). These shares are combined so that judgments in courts, business, and moral questions become graded probabilities rather than yes-or-no verdicts.
Applying his theorem to a 3:2 ratio within a margin of 1/50 at moral certainty, Bernoulli calculated that 25,550 trials would suffice. This is a sufficient bound from his method, not a proven minimum, but it shows that his guarantee, as he computed it, called for a very large number of observations.