That's a nice way to look at it. In an example, for someone who was as confused as I: If you're playing a perfect good blackjack strategy, even without countin

That's a nice way to look at it. In an example, for someone who was as confused as I: If you're playing a perfect good blackjack strategy, even without counting cards, you have a small edge over the house (if I remember my Thorpe correctly.) Let's call it 0.5 %. So if you play 100 hands, you can expect to win 51 of them. However, sometimes you might win 59, or 63, sometimes only 45 or 37. But you'll be reasonably close to 51. If you won fewer than 36 hands or more than 66 I'd be surprised. So in the worst case you win 15 hands fewer than your expectation, and in the best case you win 15 hands more. If you instead sat day and night and played 10,000 hands, you'll still be expected to win the same fraction: 5,050. But you might win 5,073, or 5,168. Or win 5,012, or win 4,931. With this large number of hands, you might well win 150 hands fewer than you expected to, or 150 hands more! Relatively speaking, you're closer to the expectation with the additional trials. With 100 hands, you might win anywhere between 36 % and 66 % of them. With 10,000 hands, you'll probably win between 49 % and 52 % of them. But when you're playing 100 hands, the difference between 36 and 51 % is only 20 hands. When you're playing 10,000 hands, the difference between 49 % and 51 % is 200 hands! In terms of actual absolute money won or lost, you're more likely to end up farther away. ---- And the reason, of course, is that while the expectation grows linearly when you increase the number of hands, the variation shrinks with the square root of the number of hands. The reason they converge in the law of large numbers is not that the variation shrinks as you play more hands -- the reason they converge is that the expectation outgrows the variation, which makes the latter seem small relative to the former. This is also why the distance traveled in a random walk is dominated by variation in the short term and bias in the long term. When you have a small enough time step, the square root is greater than the line.