Unfortunately, it is hard to achieve quantitative dexterity when you’re juggling VIIIs and XIVs. In the end, though Roman law had a certain legal rationality and coherence, it fell short of mathematical validity. In Roman law, for example, two half proofs constituted a complete proof. That might sound reasonable to a mind unaccustomed to quantitative thought, but with today’s familiarity with fractions it invites the question, if two half proofs equal a complete certainty, what do three half proofs make? According to the correct manner of compounding probabilities, not only do two half proofs yield less than a whole certainty, but no finite number of partial proofs will ever add up to a certainty because to compound probabilities, you don’t add them; you multiply.
That brings us to our next law, the rule for compounding probabilities: If two possible events, A and B, are independent, then the probability that both A and B will occur is equal to the product of their individual probabilities. Suppose a married person has on average roughly a 1 in 50 chance of getting divorced each year. On the other hand, a police officer has about a 1 in 5,000 chance each year of being killed on the job. What are the chances that a married police officer will be divorced and killed in the same year? According to the above principle, if those events were independent, the chances would be roughly 1⁄50 × 1⁄5,000, which equals 1⁄250,000. Of course, the events are not independent; they are linked: once you die, darn it, you can no longer get divorced. And so the chance of that much bad luck is actually a little less than 1 in 250,000.
Why multiply rather than add? Suppose you make a pack of trading cards out of the pictures of those 100 guys you’ve met so far through your Internet dating service, those men who in their Web site photos often look like Tom Cruise but in person more often resemble Danny DeVito. Suppose also that on the back of each card you list certain data about the men, such as honest (yes or no) and attractive (yes or no). Finally, suppose that 1 in 10 of the prospective soul mates rates a yes in each case. How many in your pack of 100 will pass the test on both counts? Let’s take honest as the first trait (we could equally well have taken attractive). Since 1 in 10 cards lists a yes under honest, 10 of the 100 cards will qualify. Of those 10, how many are attractive? Again, 1 in 10, so now you are left with 1 card. The first 1 in 10 cuts the possibilities down by 1⁄10, and so does the next 1 in 10, making the result 1 in 100. That’s why you multiply. And if you have more requirements than just honest and attractive, you have to keep multiplying, so . . . well, good luck.
Before we move on, it is worth paying attention to an important detail: the clause that reads if two possible events, A and B, are independent. Suppose an airline has 1 seat left on a flight and 2 passengers have yet to show up. Suppose that from experience the airline knows there is a 2 in 3 chance a passenger who books a seat will arrive to claim it. Employing the multiplication rule, the gate attendant can conclude there is a 2⁄3 × 2⁄3 or about a 44 percent chance she will have to deal with an unhappy customer. The chance that neither customer will show and the plane will have to fly with an empty seat, on the other hand, is 1⁄3 × 1⁄3, or only about 11 percent. But that assumes the passengers are independent. If, say, they are traveling together, then the above analysis is wrong. The chances that both will show up are 2 in 3, the same as the chances that one will show up. It is important to remember that you get the compound probability from the simple ones by multiplying only if the events are in no way contingent on each other.
Excerpted from ‘The Drunkard’s Walk – How Randomness Rules our Lives’ by Leonard Mlodinow