Drunkard's Walk - Leonard Mlodinow [18]
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.
The rule we just applied could be applied to the Roman rule of half proofs: the chances of two independent half proofs’ being wrong are 1 in 4, so two half proofs constitute three-fourths of a proof, not a whole proof. The Romans added where they should have multiplied.
There are situations in which probabilities should be added, and that is our next law. It arises when we want to know the chances of either one event or another occurring, as opposed to the earlier situation, in which we wanted to know the chance of one event and another event both happening. The law is this: If an event can have a number of different and distinct possible outcomes, A, B, C, and so on, then the probability that either A or B will occur is equal to the sum of the individual probabilities of A and B, and the sum of the probabilities of all the possible outcomes (A, B, C, and so on) is 1 (that is, 100 percent). When you want to know the chances that two independent events, A and B, will both occur, you multiply; if you want to know the chances that either of two mutually exclusive events, A or B, will occur, you add. Back to our airline: when should the gate attendant add the probabilities instead of multiplying them? Suppose she wants to know the chances that either both passengers or neither passenger will