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.
I think this is not quite right, and it's not-quite-right in an important way. It really isn't true in any sense that "it's more likely that you'll alternate between heads and tails". This is a Simpson's-paradox-y thing where "the average of the averages doesn't equal the average".
Suppose you flip a coin four times, and you do this 16 times, and happen to get each possible outcome once: TTTT TTTH TTHT TTHH THTT THTH THHT THHH HTTT HTTH HTHT HTHH HHTT HHTH HHHT HHHH.
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