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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.

elFreak
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  • What do you mean by "tend to settle on the expected distribution"? What you would rather consider as a casino owner is the expected gain/loss over a large number of games (see Central Limit Theorem and Law of Large Numbers). – Jfischer Mar 31 '20 at 16:52
  • Jep I mean the Law of Large Numbers. Doesn't this law implicit state how the future events will be? – elFreak Mar 31 '20 at 16:57
  • The convergence comes because you divide by the number of trials. – Brady Gilg Mar 31 '20 at 18:20
  • "Doesn't this law implicit state how the future events will be?" Absolutely not. If we throw a coin 10 times, then:

    "It's more probable to have 5 heads and 5 tails than to have 6 heads and 4 tails" : TRUE "Suppose that we have thrown 9 coins, and got 5 heads and 4 tail; then it's more probable that the next coin is tail" : FALSE

     – leonbloy Mar 31 '20 at 20:19

2 Answers2

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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.

Ethan Bolker
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  • I'm trying to ask a bit a different question. Maybe I couldn't formulate it quite right... I understand that you can't say anything about the result based on the previous results of independant events. But I don't get how this doesn't get in conflict with the Law of Large Numbers? – elFreak Mar 31 '20 at 17:02
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    @elFreak Read the other answer too. It doesn't (necessarily) catch up in the sense of "difference" but it does (with probability $1$) catch up in "proportion," because catching up in proportion does NOT depend on knowing what happened before. Catching up in "difference" WOULD so depend, but that's not what the theorems say. – Ned Mar 31 '20 at 20:02
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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).

Célio Augusto
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