The Drunkard's Walk is a fascinating book, but it is so maddening at times that I wished I had the author, Leonard Mlodinow, on the phone to elaborate on his conclusions and convince me that what he says is logically sound. As it is, I was on my own, which is unfortunate considering the great number of insightful ideas explored in the book.
The ultimate objective of The Drunkard's Walk is to bring to light the great effect that random chance (or luck) has on our everyday lives. We so often ignore or misunderstand this effect that Mlodinow's effort is vital. The issue of randomness isn't the sole domain of statisticians, physicists, and gamblers; we all have to make decisions based on our understanding of randomness and probability, and these decisions could come in life-or-death situations. Mlodinow asks us not to panic and points out that recognizing our prejudices or erroneous thinking is the first step in correcting it.
The Drunkard's Walk goes to great lengths to define these mistakes and to explain them. The biggest mistake we make comes from our overdeveloped sense of pattern recognition. Obviously, a strong sense of pattern recognition serves a strong evolutionary purpose; by learning which actions do or do not achieve a desired result, we succeed - or at least we should.
The Drunkard's Walk describes an experiment in which subjects were shown two sets of cards: red and green. Red cards show up roughly twice as often as green. The observer is then asked to predict, based on the previous sequence, what color will come up next. Most humans will try to figure out the pattern, even if there isn't one evident. Some will try to guess which is next based on the last few cards: if four reds show up in a row, then a green must be "due." Other people will act according to their understanding of probability: they'll guess red two-thirds of the time since it occurs two-thirds of the time.
The most successful strategy is to keep guessing red because regardless of the previous sequence, red will always be the most likely card to appear next. Rats, when confronted with a similar experiment, get it right more often than we do. Humans, with our amazing brainpower, are thus thwarted by an elementary tenet of probability.
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!