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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.

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?

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.

CO2 also gets sequestered in plant material. Further reducing the chance.
I think if quantum effects are taken into account, one can show there is almost zero probability that the atoms are the same.
I'm a physicist. Care to clarify exactly what you mean there?
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/

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.
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.

The atmosphere numbers seem correct: http://www.wolframalpha.com/input/?i=mass+air+earth+%2F+mass...

The breath numbers seem correct as well: http://www.wolframalpha.com/input/?i=volume+breath+human+*+d...

Sort of like Drake's Equation in that respect. Different assumptions for the variables -- even small differences -- can yield a maddeningly wide array of outcomes.
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.

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

Unfortunately similar calculations apply to Hitler's urine.
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.
I think you probably mean 'irreverant'...