What's the chance that the breath you just inhaled contains at least one air molecule that was in Julius Caesar's last breath--the one in which he said (according to Shakespeare) "Et tu Brute? Then die Caesar"?
Assume that the more than two thousand years that have passed have been enough time for all the molecules in Caesar's last breath to mix evenly in the atmosphere, and that only a trivial amount of the molecules have leaked out into the oceans or the ground. Assume further that there are about 1044 molecules of air, and about 2 x 1022 molecules in each breath--yours or Caesar's.
That gives a chance of 2 x 1022/1044 = 2x 10-22 that any one particular molecule you breathe in came from Caesar's last breath. This means that the probability that any one particular molecule did not come from Caesar's last breath is [1-2x10-22]. But we want the probability that the first molecule did not come from Caesar's last breath and that the second molecule and that the third molecule and so forth. To determine the probability of not just one thing but of a whole bunch of things that are causally unconnected happening together, we multiply the individual probabilities. Since there are 2x10-22 different molecules, and since each has the same [1-2x10-22] chance of not coming from Caesar's last breath, we need to multiply the probability of any single event--[1-2x10-22]--by itself 2x1022 times. That gives us:
as the probability that none of the molecules in the breath you just inhaled (assuming you are still holding out) came from Julius Caesar's last breath.
How to evaluate this? Recall that if a number x is small, then (1-x) is approximately equal to e-x, where e=2.718281828... is the so-called base of the natural logarithms. So we can rewrite the equation above as:
And remember that when we raise numbers with exponents to further exponents, we simply multiply the exponents together. In this case, one exponent (the chance that a molecule came from Caesar) is very small, and the other (the number of molecules in a breath) is very large. When we multiply them together, we get: [-2x10(-22)] x [2x10(22)] = -4. e-4 is approximately 1/(2.72 x 2.72 x 2.72 x 2.72) = 1/54.7 = 0.018.
Thus there is a 1.8% chance that none of the molecules you are (still) holding in your lungs came from Caesar's last breath. And there is a 98.2% chance that at least one of the molecules in your lungs came from Caesar's last breath.
From John Allen Paulos's Innumeracy.