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In a roulette game that continues indefinitely, is it correct to say that achieving a sequence of 100 consecutive reds will occur far less frequently (perhaps once in every million spins) compared to a sequence of 10 consecutive reds?

If the probability of getting a long sequence of reds decreases as the sequence gets longer, why is it considered a gambler's fallacy to believe that black is more likely after a long sequence of reds? If a sequence of 101 reds is less likely than a sequence of 100 reds, why is it incorrect to assume that the current sequence is more likely to be 100 reds long and therefore the next spin will be black?

Edit:

I'm aware that those are independent events and the probability has no memory and thus must stay the same. I'm asking how is that fact compatible with stating that longer sequences occur less often.

Moreover, I'm not saying that according to the law of large numbers it should balance out and therefore it is more likely to be black on the next spin. I am saying that it is more likely that we are on a squence of 100 reds than a 101 sequence, since it occurs at a higher rate, and if it is more likely to assume that now it is the shorter sequence rather than the longer one, then doesn't it logically follows that the next spin is more likely to be black?

LDBT
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  • Your idea of rare isn't rare enough. 10 reds in a row happens about one in every 1000 spins. 20 in a row is about one in a million. 100 in a row is one in $1000^{10}$, so 1 followed by 30 zeros. – Ethan Bolker Sep 04 '24 at 00:56
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    The spins are independent. They don't remember any of the previous spins so the probability of getting red or black can't change. – CyclotomicField Sep 04 '24 at 00:58
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    This question is similar to: Gambler's fallacy and the Law of large numbers. If you believe it’s different, please edit the question, make it clear how it’s different and/or how the answers on that question are not helpful for your problem. (You can understand how the answer there answers your question without having to understand anything about the Law of Large Numbers.) – Ethan Bolker Sep 04 '24 at 02:13

1 Answers1

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Consider this:

The chances of 100 Reds then 1 Black are the same as 101 Reds in a row.

So no matter the sequence length of either color (assuming only two colors each with equal likelihood) the probability of the next spin being red remains at one half.

The fallacy comes from the very unlikely chance of getting those 10 (or 100, or however many) results in a sequence.

And to answer your edit
A sequence of $n-1$ of the same color can be expected to occur at exactly twice the rate of a sequence of $n$ of the same color so there is no (Mathematical) problem here.

But when we start comparing 10 reds in a row to 100 reds in a row, the frequencies of these will differ by $2^{90}$. There are of course 90 different places the run of reds can end though, so the gamblers fallacy of assuming any one spin is more or less likely to end this run is part of the problem.

Red Five
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  • Please check my edit – LDBT Sep 04 '24 at 01:38
  • I see (I think). There is no issue here because it is just a matter of language. A sequence of $n$ reds in a row will occur with half the frequency of $n-1$ reds in a row as a consequence of this being a binomial distribution. – Red Five Sep 04 '24 at 01:56
  • Doesn't 00 enter into the picture ? – true blue anil Sep 04 '24 at 02:57
  • If we are talking about a typical roulette game, yes $0$ and $00$ should be considered. In my answer I stated an assumption that only Red and Black were outcomes. Some tables around the world only have one green although two seems more common. – Red Five Sep 04 '24 at 03:00