<span>Let </span>N<span><span>= 10^</span><span>44 </span><span>and </span>n</span><span><span>= 10^</span>22</span>.
Each molecule that we breathed
in has probability (N-n)<span><span>/N </span><span>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:</span></span>
<span> 1</span><span><span>-[</span>(</span>N-n<span><span>)^</span><span>n/</span><span>N^</span><span>n] </span>= </span>
= 1-[(10^44-10^22)^(10^22) /
10^44^(10^22)]
Simplifying will result in:
= 1- (1/e)
<span>This is about a 63% chance!</span>