I just found out about the “hot hand fallacy fallacy” (Dan Kahan, Andrew Gelman, Miller&Sanjuro paper) as a type of bias that more numerate people are likely more susceptible to, and for whom it's highly counterintuitive. It's described as a specific failure mode of the intuition used to get rid of the gambler's fallacy.

I understand the correct statement like this. Suppose we’re flipping a fair coin.

*If you're predicting future flips of the coin, the next flip is unaffected by the results of your previous flips, because the flips are independent. So far, so good.

*However, if you're predicting the next flip in a finite series of flips that has already occurred, it's actually more likely that you'll alternate between heads and tails.

The discussion is mostly about whether a streak of a given length will end or continue. This is for length of 1 and probability of 0.5. Another example is

...we can offer the following lottery at a $5 ticket price: a fair coin will be flipped 4 times. if the relative frequency of heads on flips that immediately follow a heads is greater than 0.5 then the ticket pays $10; if the relative frequency is less than 0.5 then the ticket pays $0; if the relative frequency is exactly equal to 0.5, or if no flip is immediately preceded by a heads, then a new sequence of 4 flips is generated. While, intuitively, it seems like the expected payout of this ticket is $0, it is actually $-0.71 (see Table 1). Curiously, this betting game may be more attractive to someone who believes in the independence of coin flips, rather than someone who holds the Gambler’s fallacy.