Only a breath away aka "Caesar's last breath" 

As you watch a sunset, you breath some atoms exhaled by each human who has ever watched a sunset.
As you read this page, you will breath atoms breathed by Franklin when he read the final draft of the Declaration of Independence for the first time.
As you head out the door, you share breath with a furclad traveler leaving home in Europe 10 thousand years ago. As you drive to work, with his struggles across the Alps. As you return home, with his/her return.
A few deep breaths get you atoms from your mother giving birth, from yourself taking your first breaths, and from... a Ceasar's last breath (the common name of this observation).
Your breath is about a liter of air, and thus about 10^{22} atoms. The atmosphere is about 10^{44} atoms. The breath's atoms will be well mixed back into the atmosphere after something like 10 years. Then, any randomly selected atmospheric atom has about a 10^{22} chance (10^{22}/10^{44}) of being from the your breath. Suppose you take another breath, again of 10^{22} atoms. You can expect something like one of them, on average, to be from your original breath. The exact number will vary. It is not a direct hop from one breath to the next  the atom may have been breathed by others (or by yourself), in between the two breaths.
Notes
Which atoms to use?  Nitrogen vs Argon
Which atoms to use? There are two factors  how common, and how complex.
The atmosphere's most common components as N_{2},
O_{2}, Ar, and CO_{2}, at roughly 0.8, 0.2, 0.01,
0.0004 (by number of molecules, though at this roughness, `by mass' is
about the same). Ar is even worse if you consider atoms, for with a
foo_{2} molecule you get two foo atoms. So N atoms
are something like 5000 times more common than Ar. (Averaged over the
atmosphere  Ar is slightly heavier than N_{2}, so it will tend
to hang out near the ground, but I dont expect this to make even a
× 10 difference.) However, while Ar basically just hangs out in
the atmosphere, N_{2}, O_{2}, and CO_{2} are
part of biological cycles. An N atoms spends some of its time in the
atmosphere, but also some time in seawater, and buried as dead ocean
critters. Argon is thus the choice of simplicity  you don't have to
consider the bio part. But Nitrogen is so much more common, it is
worth using if we can.
Using Nitrogen  the bio doesn't matter.
The atmosphere has about 4 × 10^{18} kg(N). 3.5 ×
10^{14} kg inorganically in ocean, 3 × 10^{14} as
dead critters in ocean, 1.5 × 10^{14} inorganically in
soil, 1 × 10^{14} as dead critters in soil. Basically,
the atmosphere more than 1000 times bigger than the rest of the
surface stores. And for a people related timescale (say a million
years), my guess is removal, by seafloor crust being subducted down
into the mantel, is trivially small. Thus, the atmosphere has so much
more N than anyplace else, we can ignore the rest.
Atmosphere numbers
The atmosphere is 5.14 × 10^{18} kg, and has 0.755 kg(N)
/ 1 kg(air). So there are 3.88 × 10^{18} kg(N). N's
average atomic mass is 14.01 (almost all N's are ^{14}N), so
with 1.66 × 10^{27} kg/nucleon, thats 23.3 ×
10^{27} kg/atom(N). So the atmosphere has about 1.7 ×
10^{44} Nitrogen atoms. (10^{18} / 10^{26} =
10^{18  26} = 10^{44}).
Breath numbers
Lung volume is around 4 or 5 liter for an adult female or male. Lets
just take a breath volume of 0.5 l. STP air density is 1.3
kg/m^{3}. A liter is 10^{3} m^{3}. So
that's 0.65 × 10^{3} kg(air). N mass fraction is 0.75,
so that's 0.49 × 10^{3} kg(N). At 23.3 ×
10^{27} kg/atom(N), that's 2.1 × 10^{22} Nitrogen
atoms.
Mixing breath back into atmosphere
[Cow p237] gives mixing times of
vertically in troposphere  order 10 days 
between hemispheres  order 1 year 
with stratosphere  order 10 years 
The result
A "marked" breath of 2 × 10^{22} atoms(N) after 10 years
is well mixed with the atmosphere's 2 × 10^{44} atoms(N).
So the odds of any randomly sampled atom(N) being "marked" is
10^{22}. So a sampling breath, again of 2 ×
10^{22} atoms(N), can be expected to have on average 2 or 3
"marked" atoms.
(One can also bypass the atom mass calculation  the atmospheric
3.88 × 10^{18} kg(N) divided by the breath 0.49 ×
10^{3} kg(N) gives 7.9 × 10^{21}. Remember that
the breath volume might vary by ×2, so great precision isn't
useful here.)
I get nervous when long calculations with big rough numbers come down to tight tolerances around one. So let's escape. The easiest way is to add breaths. At say 4 sec/breath, we get ×10 in a minute, ×100 in 10 minutes, and ×1000 in an hour. As long as the accumulated breaths are roughly independent. And any time the breather isn't in an enclosed space with still air, this seems a good bet (though it would be nice to run the numbers to make sure). One can play this game at both or either end. If the Phoenician sailor journeyed for days, even a tiny sip of your breath is going to comfortably intercept some of the atoms. If the sailor "huff"ed once jumping off the boat at journeys end, a few hours of your breathing should be sufficient. Also, breathing deeply rather than shallowly gives almost ×10.
Links
Moles for Noles and its solution.[links broken] A backoftheenvelope calc using Ar.
Comments encouraged.  Mitchell N Charity <mcharity@lcs.mit.edu> 
Notes: The current "clean" state of the above notes doesn't reflect the flailing that I went through first. Thus misleading and potentially dispiriting to others trying something similar for the first time. So I note  the first draft was a mess. Doables: Improve intro  it's still rough and doesn't capture the concept. Encourage use, describe as tool. Give use parameters up front rather than buried in notes. Probability of direct hop atoms. Better mixing time. Atom's perspective, lung hopping. Pretty this up trip, hand, signing, sunset, sail, personally applicable, trip begin/end, Phon. sailor, Rocky trade, Alps cross, sistine, franklin, magna carta, how to choose? Where does this belong on the site? Is link to `Deep understanding' really appropriate? Discuss as example of big numbers interacting. Mention von C's `Fermi Solution'. Did Paulos also have something like this?
History: 2003Feb03 Repaired links  2 flagged. 1998.May.11 Cleaned up. 1998.May.09 Initial draft.