Gamblers fallacy, fairness and surprise rationality

Question

Regarding the gamblers fallacy I have 2 questions:

1. Is it rational for getting more surprised after each time for example a six side dice comes 6?
2. How is it that a dice rolling 6 ten times changes the probability of dice fairness to us?

Let me clear a bit each question:

Surprise rationality:

The probability of a dice rolling 6 three times is $(1/6)^3$ as we know the probability of a dice rolling 6 on third roll after rolling 6 in the first and second is $1/6$, so its not more than the first time. Is it rational to be more surprised when we see a 6 for third time in a row? Also note that the probability of anything that you chose for these three rolls is $(1/6)^3$ for example the dice rolling 1 then 4 then 2.

The math I think is showing that the increase of surprise is not rational but still my intuition is that its rational.

Dice fairness

In gamblers fallacy wiki page under "Reverse position" section it comes:

After a consistent tendency towards tails, a gambler may also decide that tails has become a more likely outcome. This is a rational and Bayesian conclusion, bearing in mind the possibility that the coin may not be fair; it is not a fallacy.

I think this is related to my first question somehow, here I can see the math of Bayesian conclusion that we should change the probability of dice fairness that we suppose, But its not intuitive. The probability of 10 times rolling a dice and getting 6 is $(1/6)^{10}$ also the probability of rolling dice any other 10 numbers is still $(1/6)^{10}$. How come that the first result changes our supposition about the dice fairness but not the second one while they are both equally probable.

Answer 1

If we assume a fair dice process, this includes that each roll is independent of others. This means we can ignore past history when considering the next roll outcome. The fallacy arises when past history is used to guide future actions.

In contrast, if past history exposes a pattern that leads you to question the fairness of the dice then you can test that by collecting (a lot of) new data (making sure not to include the original dice history).

So there are two separate situations here. The answers depend on which applies.

Answer 2

This "reverse position" from the gambler's fallacy was examined in a series of papers looking at binomial prediction under the Bayesian paradigm (see O'Neill and Puza 2005; O'Neill 2012; O'Neill 2015). These papers argue in favour of your view shown in the quoted section in your question --- that observing more tails in a series of coin-flips should shift your belief somewhat towards having more tails in the future. You might also be interested in a related paper looking at the relationship between exchangeability and correlation in a classical or Bayesian context (O'Neill 2009).