
Why Throwing 92 Heads in a Row is not Surprising [pdf] - tr352
http://www.marcsandersfoundation.org/wp-content/uploads/Stoppard_final.pdf
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hprotagonist
_It’s not only when we repeatedly flip coins that something unlikely is bound
to happen – something unlikely is bound to happen with every intake of breath,
every heartbeat, every step . If I’m surprised by throwing 92 consecutive
heads, based just on its low probability, then I should be in a state of
constant amazement ._

Or, from Terry Pratchett:

 _People have reality-dampers. It is a popular fact that nine-tenths of the
brain is not used and, like most popular facts, it is wrong. Not even the most
stupid Creator would go to the trouble of making the human head carry around
several pounds of unnecessary gray goo if its only real purpose was, for
example, to serve as a delicacy for certain remote tribesmen in unexplored
valleys. It is used. And one of its functions is to make the miraculous seem
ordinary and turn the unusual into the usual. Because if this was not the
case, then human beings, faced with the daily wondrousness of everything,
would go around wearing big stupid grins, similar to those worn by certain
remote tribesmen who occasionally get raided by the authorities and have the
contents of their plastic greenhouses very seriously inspected. They 'd say
"Wow!" a lot. And no one would do much work._

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oconnor663
It's very surprising for 92 coin flips to have a tiny Kolmogorov complexity :)

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dzdt
What is surprising about 92 heads in a row is the low entropy of the outcome,
not the sheer unlikeliness of it. Yes every other outcome is just as unlikely.
But almost very other outcome has MUCH higher entropy.

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extra__tofu
If we are flipping a fair coin, the "surprise" we have at the result of any
individual flip should be the same for all individual flips. Therefore, the
entropy of the event, call it "A", of 92 fair coin tosses each resulting in
heads is the same as the entropy of any other 92 fair coin tosses, "B". The
sum of the individual event entropys must be the same so entropy(A) =
entropy(B).

If you are talking about information theory entropy and saying a message of a
billion bits composed of all "ones" has lower entropy than a "random" billion
bit message, then sure. This is like saying we can compress the billion bits
of ones and send less bits but the same amount of information. But I don't
think this is synonymous with the above. It would be like saying for event A,
each proceeding individual event has less entropy than the previous -- we
aren't dealing with a fair coin anymore.

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szemet
It is true, if you do not have memory (memoryless chanel with a memoryless
observer;)

But if you do have memory, then you are free to chose to see "92 consecutive
throws with a fair coin" as one event.

In that case the Kolmogorov complexity describes perfectly well why you should
be suprised at low entropy outcomes - simply because those are rare events
(using our new definition of 'event').

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kthejoker2
Isn't it part of the issue that he conflates sequences with totals?

As he notes there are trillions of ways to get between 40 and 50 heads. Our
surprise isn't in the sequence but the distribution of outcomes?

I would absolutely be "surprised" if someone accurately predicted a coin flip
92 times regardless of the distribution of heads and tails.

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gr3yh47
i disagree with the author's logic here - yes any individual result is
disconnected, yes the gambler's fallacy is wrong, but if we simply expand it
to say, a 100 sided die landing on 1, 92 times in a row, then we can see the
flaw here.

or to put it another way - expand his argument to 1 million coin flips, and
all of them heads.

at some point probabilities should be evinced in results. it is surprising
when they are not.

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pcmonk
He addresses this in the article if you read a little further down. Briefly,
_any_ sequence of 1 million coin flips is equally unlikely, so why are most of
them not surprising?

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ythn
He addresses this in the article if you read a little further down. Briefly,
any sequence of 1 million coin flips is equally unlikely, but because most
fall within our expectations, they are not surprising. 1 million heads in a
row does not fall within our expectations _and_ is unlikely, and hence is
surprising.

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babygoat
Surprising is a poor word choice.

A: I just threw 92 heads in a row.

B: I’m surprised!

A: No you’re not.

B: The hell you say?

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pcmonk
"Surprising" is fine. The title isn't "Why you're not surprised at throwing 92
heads in a row". There's no agent in the title, so he's referring to an
objective idea of an event being surprising. I'm not sure I agree with his
concept of "surprising", but it's at least objective.

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sonofgod
Does the author actually define surprising anywhere other than "George
Shackle's definition"?

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pcmonk
Yes, in the last section:

> Roughly speaking, it’s rational to be surprised by an event if and only if
> that event requires investigation and explanation.

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lokedhs
If that happened to you, wouldn't you investigate it and try to explain it?

In practice, if you saw this happen you'd have probably notice that something
was off. The coin is probably weighted, or some sleight of hand tricked you.

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vorotato
Well yeah because it's a weighted coin.

