Drunkard's Walk - Leonard Mlodinow [56]
That might all seem obvious. Feeling cocky, you may think you could have figured it out without the help of dear Reverend Bayes and vow to grab a different book to read the next time you step into the bathtub. So before we proceed, let’s try a slight variant on the two-daughter problem, one whose resolution may be a bit more shocking.2
The variant is this: in a family with two children, what are the chances, if one of the children is a girl named Florida, that both children are girls? Yes, I said a girl named Florida. The name might sound random, but it is not, for in addition to being the name of a state known for Cuban immigrants, oranges, and old people who traded their large homes up north for the joys of palm trees and organized bingo, it is a real name. In fact, it was in the top 1,000 female American names for the first thirty or so years of the last century. I picked it rather carefully, because part of the riddle is the question, what, if anything, about the name Florida affects the odds? But I am getting ahead of myself. Before we move on, please consider this question: in the girl-named-Florida problem, are the chances of two girls still 1 in 3 (as they are in the two-daughter problem)?
I will shortly show that the answer is no. The fact that one of the girls is named Florida changes the chances to 1 in 2: Don’t worry if that is difficult to imagine. The key to understanding randomness and all of mathematics is not being able to intuit the answer to every problem immediately but merely having the tools to figure out the answer.
THOSE WHO DOUBTED Bayes’s existence were right about one thing: he never published a single scientific paper. We know little of his life, but he probably pursued his work for his own pleasure and did not feel much need to communicate it. In that and other respects he and Jakob Bernoulli were opposites. For Bernoulli resisted the study of theology, whereas Bayes embraced it. And Bernoulli sought fame, whereas Bayes showed no interest in it. Finally, Bernoulli’s theorem concerns how many heads to expect if, say, you plan to conduct many tosses of a balanced coin, whereas Bayes investigated Bernoulli’s original goal, the issue of how certain you can be that a coin is balanced if you observe a certain number of heads.
The theory for which Bayes is known today came to light on December 23, 1763, when another chaplain and mathematician, Richard Price, read a paper to the Royal Society, Britain’s national academy of science. The paper, by Bayes, was titled “An Essay toward Solving a Problem in the Doctrine of Chances” and was published in the Royal Society’s Philosophical Transactions in 1764. Bayes had left Price the article in his will, along with £100. Referring to Price as “I suppose a preacher at Newington Green,” Bayes died four months after writing his will.3
Despite Bayes’s casual reference, Richard Price was not just another obscure preacher. He was a well-known advocate of freedom of religion, a friend of Benjamin Franklin’s, a man entrusted by Adam Smith to critique parts of a draft of The Wealth of Nations, and a well-known mathematician. He is also credited with founding actuary science, a field he developed when, in 1765, three men from an insurance company, the Equitable Society, requested his assistance. Six years after that encounter he published his work in a book titled Observations on Reversionary Payments. Though the book served as a bible for actuaries well into the nineteenth century, because of some poor data and estimation methods, he appears to have underestimated life expectancies. The resulting inflated life insurance premiums enriched his pals at the Equitable Society. The hapless British government, on the other hand, based annuity payments on Price’s tables and took a bath when the pensioners