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My favorite fact about the law of large numbers is that the more you play the higher expected distance from the expectation is. This sounds counterintuitive to

My favorite fact about the law of large numbers is that the more you play the higher expected distance from the expectation is. This sounds counterintuitive to many so it's worth considering for a bit. The law of large numbers says that the number of successes divided by the number of tries converges. It doesn't mean the number of successes converges to expectation - quite the contrary. For example when you flip a coin and go up a unit every time it's heads and down one unit every time it's tails your expected distance to 0 is sqrt(n) where n is the number of flips. Similarly when you play poker your expected distance from real EV to what you actually get is going to increase the more you play. With this mind it's not correct to say the amount is already close to calculated results with more tries as the article says. Try it, you will get numbers farer away from the expectation the more times you play!

kqr 5y ago
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.

kqr 5y ago

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.