How can a random event be independent of the next and still converge to an expected value?

Question

Lets take roulette as an example. You can't tell anything about the outcome of the numbers where the ball will stay with just knowing the previous results. That means the random events are independant. (Lets forget the fact that the results may not be distributed equivalently.) Now all the individual results together still tend to settle on the expected distribution. For otherwise it wouldn't be profitable for casinos to operate the roulette tables. Doesn't this mean the previous results tend to affect the latter?

How is it possible that the events don't affect each other, but also do? In the end the results still converge to the expected value.

Answer 1

You asked essentially the same question two years ago. If the roulette wheel or the coin is really fair - so that the numbers on the wheel or the head on the coin appear with equal probability, then mathematicians have proved that under the assumption that the present is not influenced by the past the long run distribution of the numbers or heads will converge to the underlying probabilities. The central limit theorem even tells you about how far from the underlying probabilities you are likely to be.

There is no need for the wheel or the coin to "catch up". Believing that is the gambler's fallacy. You can lose money counting on it.

Answer 2

Let's say I am flipping an honest coin. It's possible (although not very probable) that, in the first $100$ flips, I obtain $75$ heads and $25$ tails. What is the proportion of heads? Well, is $0.75$. Way off the expected result.

Suppose that I continue flipping the coin more $1000000$ times. The coin is fair, and it obviously doesn't try to "correct" the difference between the number of heads and tails. So, let'say in the new flips, I obtain $500000$ heads and $500000$ tails. What is the proportion of heads now? Is $$\frac{500075}{1000100}=0.5000249975$$

Well, that is pretty close to $1/2$. Note that the absolute difference is still there (there is $25$ more heads than "should"). But this difference is relatively too small facing the number of flips.

As Ian Stewart says, the Law of Large Number does not eliminate the differences: it buries them.

(Sorry by the bad English, not a native speaker).