The Probability of the Improbable

"It is likely that unlikely things will happen."
-Aristotle

A recurrent conversation I seem to have with people is on the balance between analysis and intuition in daily life. I consider myself an incredibly analytical person, always taking in every scrap of information about something and viewing a problem from all angles before I arrive at a decision or a solution. Still, even though I find it difficult not to rationally observe all the information at my disposal, I actually have a rather embarrassing habit of impulsively choosing things on occasion. And it's not dependent on the size of the decision either: sometimes I agonize for several minutes in the morning over what pair of pants to wear, but have made decisions about trips to take and major purchases because they "felt" right. If this sounds ridiculous, it is; but I find it interesting that many, if not most people, make decisions this way. Perhaps even more interestingly is how good the record for making decisions this way is. Given this interpretation of human behavior, one thing that's especially humorous to me, is when something is especially counterintuitive. So, here's a little anecdote I ran across while recently perusing the book How to Think about Weird Things, which illustrates how intuition can sometimes be way off:

When we try to judge the probabilities involved in events, we're often wrong. Sometimes we're really wrong because the true probabilities are completely counter to our intuitive "feel" for the odds. Mathematician John Allen Paulos offers this surprising example of a counterintuitive probability:

"First, take a deep breath. Assume Shakespeare's account is accurate and Julius Caesar gasped 'You too, Brutus' before breathing his last. What are the chances that you just inhaled a molecule which Caesar exhaled in his dying breath? The surprising answer is that, with a probability better than 99 percent, you just did inhale such a molecule. For those who don't believe me: I am assuming 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. Given these reasonably valid assumptions, the problem of determining the relevant probability is straightforward. 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 probability 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 x 10²²), and N (about 10⁴⁴ molecules) are such that this probability is more than .99. It is intriguing that we're all, at least in this minimal sense, eventually part of one another."

I'd want to see a study on how long it takes molecules contained in 1/30th of a liter to diffuse into a body of gases the volume of the Earth's atmosphere as well as the likely amount of molecules that are not free in the atmosphere to be sure this is accurate. I guess that's just my analytical nature. However, I'm willing to give the study the benefit of the doubt. I guess that's just my intuitive nature.