Drunkard's Walk - Leonard Mlodinow [37]
As I said, the same reasoning applies to more than games, gambling, and sports. For example, it shows that if two companies compete head-to-head or two employees within a company compete, though there may be a winner and a loser each quarter or each year, to get a reliable answer regarding which company or which employee is superior by simply tallying who beats whom, you’d have to make the comparison over decades or centuries. If, for instance, employee A is truly superior and would in the long run win a performance comparison with employee B on 60 out of 100 occasions, in a simple best-of-5 series of comparisons the weaker employee will still win almost one-third of the time. It is dangerous to judge ability by short-term results.
The counting in all these problems has been simple enough to carry out without much effort. But when the numbers are higher, the counting becomes difficult. Consider, for example, this problem: You are arranging a wedding reception for 100 guests, and each table seats 10. You can’t sit your cousin Rod with your friend Amy because eight years ago they had an affair and she dumped him. On the other hand, both Amy and Leticia want to sit next to your buff cousin Bobby, and your aunt Ruth had better be at a table out of earshot or the dueling flirtations will be gossip fodder for holiday dinners for the next five years. You carefully consider the possibilities. Take just the first table. How many ways are there to choose 10 people from a group of 100? That’s the same question as, in how many ways can you apportion 10 investments among 100 mutual funds or 10 germanium atoms among 100 locations in a silicon crystal? It’s the type of problem that comes up repeatedly in the theory of randomness, and not only in the problem of points. But with larger numbers it is tedious or impossible to count the possibilities by listing them explicitly. That was Pascal’s real accomplishment: a generally applicable and systematic approach to counting that allows you to calculate the answer from a formula or read it off a chart. It is based on a curious arrangement of numbers in the shape of a triangle.
THE COMPUTATIONAL METHOD at the heart of Pascal’s work was actually discovered by a Chinese mathematician named Jia Xian around 1050, published by another Chinese mathematician, Zhu Shijie, in 1303, discussed in a work by Cardano in 1570, and plugged into the greater whole of probability theory by Pascal, who ended up getting most of the credit.10 But the prior work didn’t bother Pascal. “Let no one say I have said nothing new,” Pascal argued in his autobiography. “The arrangement of the subject is new. When we play tennis, we both play with the same ball, but one of us places it better.”11 The graphic invention employed by Pascal, given below, is thus called Pascal’s triangle. In the figure, I have truncated Pascal’s triangle at the tenth row, but it can be continued downward indefinitely. In fact, it is easy to continue the triangle, for with the exception of the 1 at the apex, each number is the sum of the number in the line above it to the left and the number in the line above it to the right (add a 0 if there is no number in the line above it to the left or to the right).
Pascal’s triangle
Pascal’s triangle is useful any time you need to know the number of ways in which you can choose some number of objects from a collection that has an equal or greater number. Here is how it works in the case of the wedding guests: To find the