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.
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?
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.
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.
P.S. - Reminded me of one of those questions they ask at McKinsey when recruiting new analysts. E.g. How many golf balls are in the United States?
(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 is b
>>> a is c
>>> a, b, c
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.)
[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 breath numbers seem correct as well: http://www.wolframalpha.com/input/?i=volume+breath+human+*+d...
> 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.