Question

Suppose Shakespeare’s account is accurate and Julius Caesar gasped “You too, Brutus” before breathing his last. What is the probability that you just inhaled a molecule that Julius Caesar exhaled in his dying breath?

Assume that after more than two thousand years the exhaled molecules are uniformly spread about the world and the vast majority are still free in the atmosphere. Assume further that there are 10⁴⁴ molecules of air in the world, and that your inhaled quantity and Caesar’s exhaled quantity were each about 2.2 x 10²² molecules.

Answer 1

. If there are N molecules of air in the world and Caesar exhaled A of them, then the probability that any given molecule you inhale is from Caesar is A/N. The probability that any given molecule you inhale is not from Caesar is thus 1 - A/N. By the multiplication principle, if you inhale three molecules, the prob-ability that none of these three is from Caesar is [1 - A/N]³. Similarly, if you inhale B molecules, the probability that none of them is from Caesar is approximately [1 - A/N]^B. Hence, the probability of the complementary event, of your inhaling at least one of his exhaled molecules, is

1 - [1 - A/N]^B.

A, B (each about 1/30th of a liter, or 2.2 × 10²²), and N (about 10⁴⁴ molecules)

are such that this probability is more than .99.