There are numerous examples of this sort in the book. Humans look for patterns when there are none; we think we've cracked the system. This mistake could be made by a dedicated gambler who's convinced he's found a "loose" slot machine. It could be your Uncle Joe who swears he has a "knack" for picking hot stocks. The error could even come from a baseball fan who thinks a player with a 20-game (or 30-game) hitting streak is "hot." The fact that mere luck is an explanation — in fact the most likely explanation — for these events rarely occurs to us. We humans will stubbornly resist admitting that randomness plays any part in our successes, although we'll easily attribute a series of terrible misjudgments to "bad luck."
The chapters spent addressing these questions almost make the book worthwhile, but there are still issues. One drawback is that Mlodinow addresses not only the theories underlying probability, but the people who developed them, their lives, and the relative importance of their discoveries within their discipline. That's not necessarily a bad thing, but unless you're already fascinated by the history of scientific or mathematical theory, a short biography of the life and works of Blaise Pascal or Jakob Bernoulli isn't likely to capture your interest.
Mlodinow does a brave job of bringing these half-familiar names out of the textbooks with a lively description, but one simply cannot get past the fact that as we focus on the struggles and paradoxes faced by these famous figures, any reader without an advanced degree in the subject will be either frustrated and befuddled or utterly lost. It's not Mlodinow's fault that the work of Alan Bayes is so counterintuitive as to make one's head spin, but the author simply cannot convey in one book what would take most of us a full semester (at least) to fully comprehend.
It's frustrating enough for a layman to wrap his mind around the practical applications of Pascal's triangle. Even more vexing is being forced to accept some of the proofs presented by Mlodinow. At one point, Mlodinow brings up the following problem: a pregnant woman is giving birth to fraternal twins and she wants to know the sex of her children. "What is the probability," he asks, "given that one of the children is a girl, that both children will be girls?" In my head, I answered 50%. All things being equal, there are two possible outcomes — male or female — and barring the birth of a transgender child, it will be one or the other. Right?
Article comments
1 - Probabilist
This may not apply to you, but a lot of people pick up a math book and get discouraged and give up when they realize they don't get it on the first reading. There's nothing wrong with you or with the book. Mathematical ideas are learned by doing, by sitting there with pen and paper and playing around with the ideas (why is this true? what would happen if it wasn't? can i think of a simplified case that might give me some intuition for why this is true?and so on). Most of the learning in a math class happens when doing the homework problems, not when sitting in class. And often the insight that comes from this doesn't translate well into language. For that twins problem, there really is little else to say. You just need to see it, and that takes a little time.
2 - Stuart Baker
Re Aaron Whitehead's review of 'The Drunkard's Walk' and the fraternal twin problem. It's not your fault, I think the book author has failed to explain how fraternal twins occur. They arise when 2 ova (eggs) are released at the same time - call them Ova A and Ova B, so there are 4 possible pairs - if Ova A becomes a boy then Ova B can become either a girl or a boy. Same for Ova B. That gives the 4 possibles and one (boy:boy) is ruled out, hence the author's answer. In my opinion, this is a continual problem with mathematicians in word problems - THEY DON'T USE WORDS PROPERLY, and so nearly always fail to explain the problem properly. Sorry I don't have a URL, just email, hope this works.
3 - Aaron Whitehead
Good points. In my previous math experience, I usually have to take the task into my own hands to finally get it.
I also feel that when you're presented with a problem where you think the answer is obvious, and it is in fact the opposite, the onus is on the teacher (or writer) to explain convincingly WHY I am wrong.
I can be very stubborn when I don't understand something, so this could be why I seem to have had a rougher go of it than others. I just wonder if other people will have similar problems.
4 - Mongo
You have to Mlodinow credit for his attempt at getting these very interesting and little-understood facts (outside of statisticians, etc.) into a palatable form. These facts have an impact on daily life, no doubt more than ever given how much computers (which calculate these things very easily) impact every aspect of peoples lives now.
One fact like this I learned a few years ago was the birthday paradox. Did you know that it only takes 23 people in a room for there to be better than a 50% chance that at least two people will have the same birthday?
5 - contrarian joe
Mlodinow is wrong. Describing the possibilities as "boy/boy, boy/girl, girl/boy and girl/girl" suggests that there are two ways you can have one of each gender. As if the first one can be a boy. But she cannot. We already know she's a girl. So there are only two possibilities. girl/boy and girl/girl. 50/50.
6 - Dane Wittrup
It's all in the words used in the problem definition. "What is the probability," he asks, "given that one of the children is a girl, that both children will be girls?"
A different problem would have a different answer. For example: "You meet one member of a pair of twins, and it is a girl. What is the probability that the other twin is a girl?" The answer to this problem is 50% because you have specified a particular twin being a girl in the problem statement (not that "one of the two" is a girl), and asked about the other twin.
The original problem statement is posed before you've met anybody, and so you have to count the possibility that you met either of the twins first, not a specific one.
7 - Eddington
I think the fraternal twin problem is difficult because of the counter-intuitive way it is posed, and nothing to do with the maths (that is, the authors fault). One immediately assumes that "Given that one of the twins is a girl" means that we know the gender of the first twin, and it's a girl.
(And hence we say the only remaining possibilities are Girl/Girl or Girl/Boy).
Rather it means the more unusual circumstance in which we know that at least one twin is a girl, but we don't know the gender of any particular twin (thus we get the extra possibility Boy/Girl).
I can certainly see that giving people not comfortable with maths an undeserved headache!