we’re more surprised when we see the former than the latter
I don’t think this is actually true. If MileyCyrus successfully predicted the exact sequence of coinflips HTHTTHTHTTTHHTH, wouldn’t you be more surprised than if it were HHHHHHHHHHHHHHH?
Of course. When I said “we’re more surprised” I was referring to the typical person who hasn’t read this discussion thread. In the absence of the above prediction, I would be far more surprised to see HHHHHHHHHHHHHHH than HTHTTHTHTTTHHTH. Once the prediction is made, I become extremely surprised if either sequence appears, but somewhat more surprised by HTHTTHTHTTTHHTH.
Oh, I see. In the case of the typical person, the answer is even easier: Lack of understanding of the conjunction rule of probability. HTHTTHTHTTTHHTH feels more representative of a random series of coin flips, so it is intuitively judged as more probable than HHHHHHHHHHHHHHH.
First reaction: I don’t know about “far” more probable. What’s the prior that a coin is rigged? I would have said less than 1⁄32768, but low confidence on that.
According to this, you can’t rig a coin to do that, which increases my confidence.
But you can rig your tossing, even by mistake; if it lands heads, and you balance it to flip with heads up again, then it’s slightly more likely to land heads. I remember hearing a figure of 51% for that; in which case H*15 has probability 1/24331 instead of 1⁄32768; about a third more probable. But that scenario (fifteen times) is itself unlikely… if we estimate P(next is heads | last was heads) = 0.505 (corresponding to keeping the same side up 3⁄4 of the time, I still feel that’s an overestimate), we get 1/28204, 16% more likely.
If we switched to dice, I would agree that 666666666666666 is far more probable than 136112642345553.
I suppose that isn’t all that unintuitive (though does this actually work if you start with a uniform prior over weights and do the math?). But does your intuitive model also predict the fact that HTHTHTHTHT is more probable than HTHHTHTHTT? :D
Well, it is the case that all the random sequences together have much larger probability than HHHHHHHHHHHH , and so we should expect the sequence to be one among the random sequences.
edit: interesting issue: suppose you assign some prior probability to each possible sequence. Upon seeing the actual sequence, with probability that your eyes deceived you 0.0001, how are you to update the probability of this particular sequence? Why would we assume sensory failure (or a biased coin) when we observe hundred heads, but not something random-looking? It should have to do with the sensory failure being much less likely for something random looking.
If you flip a coin 15 times, this result:
HHHHHHHHHHHHHHH
is far more probable than this:
HTHTTHTHTTTHHTH
That’s because some coins are rigged, and it’s much easier to rig a coin to conform the first pattern than the second.
This is true, but doesn’t explain why we’re more surprised when we see the former than the latter.
I don’t think this is actually true. If MileyCyrus successfully predicted the exact sequence of coinflips HTHTTHTHTTTHHTH, wouldn’t you be more surprised than if it were HHHHHHHHHHHHHHH?
Of course. When I said “we’re more surprised” I was referring to the typical person who hasn’t read this discussion thread. In the absence of the above prediction, I would be far more surprised to see HHHHHHHHHHHHHHH than HTHTTHTHTTTHHTH. Once the prediction is made, I become extremely surprised if either sequence appears, but somewhat more surprised by HTHTTHTHTTTHHTH.
Oh, I see. In the case of the typical person, the answer is even easier: Lack of understanding of the conjunction rule of probability. HTHTTHTHTTTHHTH feels more representative of a random series of coin flips, so it is intuitively judged as more probable than HHHHHHHHHHHHHHH.
First reaction: I don’t know about “far” more probable. What’s the prior that a coin is rigged? I would have said less than 1⁄32768, but low confidence on that.
According to this, you can’t rig a coin to do that, which increases my confidence.
But you can rig your tossing, even by mistake; if it lands heads, and you balance it to flip with heads up again, then it’s slightly more likely to land heads. I remember hearing a figure of 51% for that; in which case H*15 has probability 1/24331 instead of 1⁄32768; about a third more probable. But that scenario (fifteen times) is itself unlikely… if we estimate P(next is heads | last was heads) = 0.505 (corresponding to keeping the same side up 3⁄4 of the time, I still feel that’s an overestimate), we get 1/28204, 16% more likely.
If we switched to dice, I would agree that 666666666666666 is far more probable than 136112642345553.
I suppose that isn’t all that unintuitive (though does this actually work if you start with a uniform prior over weights and do the math?). But does your intuitive model also predict the fact that HTHTHTHTHT is more probable than HTHHTHTHTT? :D
Well, it is the case that all the random sequences together have much larger probability than HHHHHHHHHHHH , and so we should expect the sequence to be one among the random sequences.
edit: interesting issue: suppose you assign some prior probability to each possible sequence. Upon seeing the actual sequence, with probability that your eyes deceived you 0.0001, how are you to update the probability of this particular sequence? Why would we assume sensory failure (or a biased coin) when we observe hundred heads, but not something random-looking? It should have to do with the sensory failure being much less likely for something random looking.