

Julius Caesar's Last Breath - signa11
http://econ161.berkeley.edu/movable_type/archives/001392.html

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hugh3
That's great, but one problem: molecules don't stay molecules over two
thousand years. Certainly not oxygen, which is extremely chemically active. N2
somewhat less so (but <http://en.wikipedia.org/wiki/Nitrogen_cycle>), but the
chances of any particular nitrogen molecule retaining its identity over two
thousand years is incredibly low.

Redo this calculation with atoms, and I might believe you. But it'll need to
be a bit more complicated, since I don't think the amount of oxygen and
nitrogen getting sequestered in the ocean or the soil is actually "trivial" as
stated.

~~~
anxrn
As long as there is some component of air that stays the same: both in
identity and proportion, the calculation should hold. Is there such a
component?

Simplifying assumptions, even those seemingly false, are common in fun math
problems. The point is just that, the math. I'm sure the traveling salesman
had issues to think about other than the classic math of the problem.

Just for my information, any references to strengthen your last statement?

~~~
hugh3
_As long as there is some component of air that stays the same: both in
identity and proportion, the calculation should hold. Is there such a
component?_

Atoms, as long as the amount being sequestered in the water or the soil isn't
significant. Oxygen is probably a write-off, since there's far more oxygen
atoms in the oceans than in the atmosphere, and since O_2 to H_2O is part of
animal respiration. Nitrogen, perhaps, might be more constant, but honestly I
just don't know enough about the nitrogen cycle to have a good idea.

Ah, but the third most common component of the air is argon, which is
deliciously chemically inactive. You could at least compute the probability
that you're breathing in some of Caesar's argon.

 _Simplifying assumptions, even those seemingly false, are common in fun math
problems. The point is just that, the math. I'm sure the traveling salesman
had issues to think about other than the classic math of the problem._

Of course. And the other thing that's common in fun math problems is that as
soon as you're done someone's gonna say "That's great, but..." and point out
something you've ignored. It's all part of the game, and I'm just playing
along, not being critical.

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MBlume
This is a _really_ cute little Fermi calculation, and so I hesitate to say
this...

(I also hesitate because it will kick off a storm of People on the Internet
Arguing about Physics...)

but this is just wrong. It is irreparably wrong. The idea that a given oxygen
molecule (or an oxygen atom, or an electron) in BCE 44 can be identified with
an oxygen molecule (etc.) in the present day runs fundamentally counter to the
way the universe works.

Put it this way. In python, we have mutable variables, which have identity. so
we can say

>>> a=b=[]

>>> c=[]

>>> a is b

True

>>> a is c

False

>>> a.append(5)

>>> a, b, c

([5],[5],[])

Starting out, a and b are _the same_ empty list, and c is a _different_ empty
list. It seems naively that we could say the same of particles or atoms. That
though we couldn't see it, or hope to trace its history, there existed some
electron in 44 BCE that "was the same electron as" some electron today. But
that is not how the universe is _implemented_. Every electron is the same as
every other electron. Think immutable, not mutable variables. The state in 44
BCE is not "electron #4892489 is here, and electron #4892490 is there", it is
"there exist electrons here, here, here (etc.)" (and they have thus-and-such
spins, momenta, etc. etc.)

Edit: <http://lesswrong.com/lw/pl/no_individual_particles/>

~~~
Dylan16807
While that is an useful idea, I don't see how it applies to something as large
as a molecule. Something that size does not coalesce, does not overlap. You
can bombard it with light and track its progress indefinitely. It won't have
the 'same' electrons at the end of the day but those are no more than the
sails on our ship of Theseus.

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dpbrane
What is more mathematically interesting about this is that if the
"assumptions" are off by a factor of 10 or so (say, the atmosphere actually
has more like 10^45 molecules instead of 10^44, and a breath contains 1x10^22
molecules, not 2x10^22), the result is reversed:

[1-10^-23]^[10^22] ~ [e^(10^-23)x(10^22)] = e^(0.1) ~ 0.9

-> 90% chance that any given breath contains none of Caesar's last.

~~~
ars
The atmosphere numbers seem correct:
[http://www.wolframalpha.com/input/?i=mass+air+earth+%2F+mass...](http://www.wolframalpha.com/input/?i=mass+air+earth+%2F+mass+nitrogen)

The breath numbers seem correct as well:
[http://www.wolframalpha.com/input/?i=volume+breath+human+*+d...](http://www.wolframalpha.com/input/?i=volume+breath+human+*+density+air+%2F+mass+nitrogen)

------
raquo
This assumption alone holds OP's argument from folding:

> To determine the probability of not just one thing but of a whole bunch of
> things that are causally unconnected happening together, we multiply the
> individual probabilities

You can multiply probabilities of individual events only if you know they are
independent, i.e. their correlation is zero (as opposed to anything involving
causation). I am not convinced that the molecules that were once in one place
would get distributed to zero correlation even after 2000+ years. Well, this
at least requires an analysis on its own.

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huhtenberg
Anyone liking this sort of thing should have a look at _A Mathematicians
Miscellany_ by Littlewood [1]. It is an exceptionally enjoyable collection of
mathematical anecdotes and some such. It covers Caesar's Last Breath topic,
but it also goes over the probabilities of highly unlikely events such as an
upright drumstick not falling over during a long train ride.

[1] <http://www.archive.org/details/mathematiciansmi033496mbp>

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indrax
Unfortunately similar calculations apply to Hitler's urine.

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lurker19
This puzzle was printed in college application brochure for (I think it was)
Princeton in the 1990s. It was an example of the stimulating and irrelevant
academic university culture, or something like this.

~~~
apetresc
I think you probably mean 'irreverant'...

