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The hot hand fallacy: seeing data that is typical for independent coin flips as evidence for correlation between adjacent flips.

The hot hand fallacy fallacy (Miller, Sanjurjo 2018): Not correcting for the fact that amongst random length-k (k>2) sequences of independent coin tosses with at least one heads before toss k, the expected proportion of (heads after heads)/(tosses after heads) is less than 1/2.

The hot hand fallacy fallacy fallacy: Misinterpreting the above observation as a claim that under some weird conditioning, the probability of Heads given you have just seen Heads is less than 1/2 for independent coin tosses.

amongst random length-k (k>2) sequences of independent coin tosses with at least one heads before toss k, the expected proportion of (heads after heads)/(tosses after heads) is less than 1/2.

Does this need to be k>3? Checking this for k=3 yields 6 sequences in which there is at least one head before toss 3. In these sequences there are 4 heads-after-heads out of 8 tosses-after-heads, which is exactly 1/2.

Edit: Ah, I see this is more like a game score than a proportion. Two "scores" of 1 and one "score" of 1/2 out of the 6 equally likely conditional sequences.

Random thought after reading "A model of UDT with a halting oracle": imagine there are two super-intelligent AIs A and B, suitably modified to have access to their own and each other's source codes. They are both competing to submit a python code of length at most N which prints the larger number, then halts (where N is orders of magnitude larger than the code lengths of A and B). A can try to "cheat" by submitting something like exec(run B on the query "submit a code of length N that prints a large number, then halts") then print(0), but B can do this as well. Supposing they must submit to a halting oracle that will punish any AI that submits a non-halting program, what might A and B do?

submit busy beaver

edit: wait nevermind, busy beaver itself takes a halting oracle to implement

Has to be a python code; allowing arbitrary non-computable natural language descriptions gets hairy fast