How Not to Be Wrong: The Power of Mathematical Thinking

by Jordan Ellenberg

When you try to think carefully about what probability means, you get a little woozy. When we say, “The probability that a flipped coin will land heads is 1/2,” we’re invoking the Law of Large Numbers from chapter 4, which says that if you flip the coin many, many times, the proportion of heads will almost inevitably approach 1/2, as if constrained by a narrowing channel. This is what’s called the frequentist view of probability. But what can we mean when we say, “The probability that it will rain tomorrow is 20%”? Tomorrow only happens once; it’s not an experiment we can repeat like a coin flip again and again. With some effort, we can shoehorn the weather into the frequentist model; maybe we mean that among some large population of days with conditions similar to this one, the following day was rainy 20% of the time. But then you’re stuck when asked, “What’s the probability that the human race will go extinct in the next thousand years?” This is, almost by definition, an experiment you can’t repeat.