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What did Jakob Bernoulli prove about the proportion of successes in repeated trials?

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sample space for a pair of dice — probability theory Sample space for a pair of dice.

In the context of probability theory, Jakob Bernoulli proved what is now known as the law of large numbers in his posthumously published work, Ars Conjectandi (1713).123

Bernoulli considered an endless sequence of independent repetitions of a game of pure chance with two possible outcomes, such as winning or losing, where the probability of a win is denoted as p. He proved that as the number of repetitions increases indefinitely, the probability that the empirical proportion of wins (the fraction of times the game is won) remains within any prespecified distance from the theoretical probability p approaches 1.13

Essentially, this theorem demonstrates that the observed relative frequency of an event will converge to its theoretical probability as the number of trials grows, providing a mathematical foundation for the relative frequency interpretation of probability. Although Bernoulli's original proof was mathematically imprecise compared to later developments, it remains a fundamental pillar of modern sampling theory.132