Exercise 1.57 (Caesar's last breath)

Take a deep breath before attempting this problem. In the book Innumeracy [20], John Allen Paulos writes:

Now for better news of a kind of immortal persistence. First, take a deep breath. Assume Shakespeare’s account is accurate and Julius Caesar gasped [“Et tu, Brute!”] before breathing his last. What are the chances you just inhaled a molecule which Caesar exhaled in his dying breath?

Assume that one breath of air contains $10^{2}2$ molecules, and that there are $10^{4}4$ molecules in the atmosphere. (These are slightly simpler numbers than the estimates that Paulos gives; for the purposes of this problem, assume that these are exact. Of course, in reality there are many complications such as different types of molecules in the atmosphere, chemical reactions, variation in lung capacities, etc.)

Suppose that the molecules in the atmosphere now are the same as those in the at- mosphere when Caesar was alive, and that in the 2000 years or so since Caesar, these molecules have been scattered completely randomly through the atmosphere. Also as- sume that Caesar’s last breath was sampled without replacement but that your breathing is sampled with replacement (without replacement makes more sense but with replace- ment is easier to work with, and is a good approximation since the number of molecules in the atmosphere is so much larger than the number of molecules in one breath).

Find the probability that at least one molecule in the breath you just took was shared with Caesar’s last breath, and give a simple approximation in terms of $e$.