I have seen this argument on LessWrong before, and don’t think the other explanations are as clear as they can be. They are correct though, so my apologies if this just clutters up the thread.
The Bayesian way of looking at this is clear: the prior probability of any particular sequence is 1/2^[large number]. Alice sees this sequence and reports it to Bob. Presumably Alice intends on telling Bob the truth about what she saw, so let’s say that there’s a 90% chance that she will not make a mistake during the reporting. The other 10% will cover all cases ranging from misremembering/misreading a flip to outright lying. The point is that if Alice is lying, this 10% has to be divided up between the other 2^[large number]-1 other possible sequences—if Alice is going to lie, any particular sequence is very unlikely to be presented by her as the true sequence, since there are a lot of ways for her to lie. So, assuming that Alice was intending to speak the truth, her giving that sequence is very strong (in my example 9*(2^[large number]-1):1) evidence that that particular sequence was indeed the true one over any specific other sequence - ‘coincidentally’ precisely strong enough to turn the posterior belief of Bob that that sequence is correct to 90%.
A fun side remark is that the above also clearly shows why Bob should be more skeptical when Alice presents sequences like HHHHHHHHHH or HTHTHTHTHTHT—if Alice were planning on lying these are exactly the sequences that she might pick with a greater than uniform probabilty out of all the sequences that were not thrown, and therefore each possible actual sequence contributes a higher-than-average amount of probability that Alice would present one of these special sequences, so the fact that Alice informs Bob of such a sequence is weaker evidence for this particular sequence over any other one than it would be in the regular case, and Bob ends up with a lower posterior that the sequence is actually correct.
I have seen this argument on LessWrong before, and don’t think the other explanations are as clear as they can be. They are correct though, so my apologies if this just clutters up the thread.
The Bayesian way of looking at this is clear: the prior probability of any particular sequence is 1/2^[large number]. Alice sees this sequence and reports it to Bob. Presumably Alice intends on telling Bob the truth about what she saw, so let’s say that there’s a 90% chance that she will not make a mistake during the reporting. The other 10% will cover all cases ranging from misremembering/misreading a flip to outright lying. The point is that if Alice is lying, this 10% has to be divided up between the other 2^[large number]-1 other possible sequences—if Alice is going to lie, any particular sequence is very unlikely to be presented by her as the true sequence, since there are a lot of ways for her to lie. So, assuming that Alice was intending to speak the truth, her giving that sequence is very strong (in my example 9*(2^[large number]-1):1) evidence that that particular sequence was indeed the true one over any specific other sequence - ‘coincidentally’ precisely strong enough to turn the posterior belief of Bob that that sequence is correct to 90%.
A fun side remark is that the above also clearly shows why Bob should be more skeptical when Alice presents sequences like HHHHHHHHHH or HTHTHTHTHTHT—if Alice were planning on lying these are exactly the sequences that she might pick with a greater than uniform probabilty out of all the sequences that were not thrown, and therefore each possible actual sequence contributes a higher-than-average amount of probability that Alice would present one of these special sequences, so the fact that Alice informs Bob of such a sequence is weaker evidence for this particular sequence over any other one than it would be in the regular case, and Bob ends up with a lower posterior that the sequence is actually correct.