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.