The Law of Large Numbers

Jacob Bernoulli proved in 1713 that as the number of trials of a random experiment increases, the observed average converges to the true probability. Flip a fair coin 10 times — you might get 7 heads. Flip it 10,000 times — you will get very close to 5,000 heads. This is the Law of Large Numbers: individual randomness is tamed by repetition. Casinos are profitable precisely because of this law — the 'house edge' of a few percent guarantees profit over millions of bets, even though any single bet is unpredictable.