There are branches of probability theory that deal with these questions rigorously. https://en.m.wikipedia.org/wiki/Large_deviations_theory https://en.m.wikipe

There are branches of probability theory that deal with these questions rigorously. https://en.m.wikipedia.org/wiki/Large_deviations_theory https://en.m.wikipedia.org/wiki/Large_deviations_theory https://en.m.wikipedia.org/wiki/Extreme_value_theory https://en.m.wikipedia.org/wiki/Extreme_value_theory