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If a coin comes up Heads 90 times out of the first 100 tosses, should one expect Tails to make a comeback over the next 100?

Reference from this page: https://www.financialwisdomforum.org/gummy-stuff/coin-tossing.htm

It is my understanding from the Law of Large numbers; that in the above case tails should make a comeback over the next 100. However arguments against this include that given they are independent events; tails is not likely to make any comeback and still be 50% over the next 100 tosses.

Let’s exaggerate it further; 1 million tosses; 900k heads; it is a fair coin; we are living in our reality (and not a computer simulation); would tails make a comeback to “even out/mean revert” the average to 50%?

Ehsan
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    You seem to be conflating mean reversion with the gambler's fallacy. They are not the same thing. – Glen_b Sep 13 '22 at 23:59
  • With a fair coin, the overall proportion of heads is expected to move towards 50% (it might increase from 90% but it is more likely to reduce), while the expected overall difference would remain at 80 (or 800000) and is equally likely to increase or decrease. If half your next 900 (or 9000000) tosses were heads, then the overall proportion would reduce to 54% while the overall difference would stay at 80 (or 800000) – Henry Sep 14 '22 at 00:16
  • @Glen_b What’s your definition of gambler’s fallacy? (Asking because I’ve seen slightly various definitions e.g. “only the next sample” or “that history determines outcome” – Ehsan Sep 14 '22 at 07:59
  • Thanks @Henry ; so just to be clear, would there be truth in “tails making a comeback”? – Ehsan Sep 14 '22 at 08:01
  • Thanks @dave ; your page implies coin to issuing does follow regression to mean (though that might not be the same as mean reversion; a term I think more used in finance). For clarity; would it be expected for “tails to make a comeback” over the next set of mass samples? – Ehsan Sep 14 '22 at 08:08
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    @Ehsan It depends what you mean by “tails to make a comeback”, but for a fair coin I would say that is a misleading comment: the overall difference is equally likely to increase or decrease, and the expectation (average future outcome) for the overall difference is that it stays the same - even though that would lead to the proportion tending towards 50%. – Henry Sep 14 '22 at 08:20
  • The gambler's fallacy is the belief that something happens to "compensate" for an excess / deficit of some outcome that subsequently moves things closer to their expected proportions. (It's called a fallacy because this is not what happens in situations like coin tossing, where trials are independent). An expectation that "Tails make a comeback" is an example of the gambler's fallacy. Mean reversion is different from "Tails make a comeback". – Glen_b Sep 14 '22 at 12:24
  • @Henry points to the key issue: what you might mean by "make a comeback." The textbook by Freedman, Pisani, & Purves has a nice discussion of this. There is no force in the universe that will cause tails suddenly to have a greater chance of appearing. However, laws of large numbers imply that the number of tails as a proportion of the total will eventually grow close to 50%, provided your assumption is correct that this is a fair coin-tossing process. – whuber Sep 14 '22 at 15:36

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