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.

Let N = 10^ 44 and n = 10^ 22.

Each molecule that we breathed in has probability (N-n) / N of not being from Caesar’s last breath. Also, the molecules breathed in are independent if it is assumed that sampling with replacement is performed. So the probability of at least one molecule being shared with Caesar’s last breath is:

1 - [ (N-n) ^ n / N^ n] =

= 1-[ (10^44-10^22) ^ (10^22) / 10^44^ (10^22) ]

Simplifying will result in:

= 1 - (1/e)

This is about a 63% chance!