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This question is specifically in reference to how statistics apply to Catan.

For background, two fair dice are rolled and rolling a 7 is a bad event.

Given that the average number of turns for a game of catan is 80 and that means CLT can be applied, does it make sense to expect a 7 more and more as time goes on?

There's a similar question here (Does 10 heads in a row increase the chance of the next toss being a tail?) so my curiosity is based on a few assumptions.

Given that the distribution of 7 is the most likely, even though each roll is independent and not based on the history of the past rolls, one should expect a 7 to appear more often in the future rolls.

ex - if there are an average of 40 rolls in the game and ~6 of those rolls should be 7, if and it is currently roll 25, there are 6 7's expected in the remaining 15 turns. Thus there is a larger number of 7's to be rolled to maintain the distribution and given the limited 15 spaces, we should expect a 7 to be rolled before we eventually roll a non 7.

Is this an incorrect approach and if so why? It appears that this is a weird spin on the monty hall problem with a potential combination of gambler's fallacy?

As a follow up, can we used expected distribution to predict future events accurately when those events are independent?

wya
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    Re "to maintain the distribution:" this erroneous idea is often called the "Gambler's Fallacy." The universe does not work to balance out our expectations. Re: "predict future events accurately" using "expected distribution:" this is done with prediction intervals. – whuber Jun 06 '22 at 10:28
  • hmm i see - so we cannot expect a reversal but also cannot expect the trend of that is statistically unlikely to continue either? meaning that we're likely to get see more 7's in the future than in the past given low number of 7's in the past? – wya Jun 06 '22 at 10:43
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    Independence means the frequency of sevens you observe in the future will not depend on the frequency with which they have been observed in the past. There's another subtlety here: you can use the past frequency to estimate (that is, make a reasonable guess) about the frequency with which sevens are likely to appear in the future. But that still doesn't mean the future occurrences depend in any way on the past ones. It only means you assume the die continues behaving as it always did. – whuber Jun 06 '22 at 10:48
  • Imagine a game where you shuffle a deck of cards and grab the top one. You win if you get an ace of spades, and lose otherwise. The chances of winning are 1 in 52. If you lose 51 times, would you bet your life that the next game you will win because "the ace of spades is due"? Of course not, you could play a thousand games and lose them all. Or win in the first game. In order for there to be a sense of "relationship" between consecutive plays, a dependence most be imposed. For instance, if in each game we eliminate the card that appeared, if you lose 51 games, you will win the 52nd. – Jesús A. Piñera Jun 16 '22 at 02:05

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