This is a must-read, but I'm not going to embarrass myself by trying to write a review. Instead, I'm going to post someone else's. This first review gives all the really good examples anyhow.
Here's the book on Amazon.com.
Playing the Odds
State lotteries, it’s sometimes said, are a tax on people who don’t understand mathematics. But there is no cause for anyone to feel smug. The brain, no matter how well schooled, is just plain bad at dealing with randomness and probability. Confronted with situations that require an intuitive grasp of the odds, even the best mathematicians and scientists can find themselves floundering.
Suppose you want to calculate the likelihood of tossing two coins and coming up with one head. The great 18th-century mathematician Jean Le Rond d’Alembert thought the answer was obvious: there are three possibilities, zero, one or two heads. So the odds for any one of those happening must be one in three.
But as Leonard Mlodinow explains in “The Drunkard’s Walk: How Randomness Rules Our Lives,” there are, in fact, four possible outcomes: heads-heads, heads-tails, tails-heads and tails-tails. So there is a 25 percent chance of throwing zero or two heads and a 50 percent chance of throwing just one. In the long run, anyone offering d’Alembert’s odds in a coin-flipping contest would lose his shirt.
The key to puzzles like this, Mlodinow writes, is Cardano’s method, named for Gerolamo Cardano, author of the 16th-century “Book on Games of Chance.” To lay the odds for even the simplest-seeming event, one constructs a table, or “sample space,” of all the ways Fortuna’s dice might fall. Trust your instincts instead and you’re bound to go wrong.
If a woman has two children and one is a girl, the chance that the other child is also female has to be 50-50, right? But it’s not. Cardano again: The possibilities are girl-girl, girl-boy and boy-girl. So the chance that both children are girls is 33 percent. Once we are told that one child is female, this extra information constrains the odds. (Even weirder, and I’m still not sure I believe this, the author demonstrates that the odds change again if we’re told that one of the girls is named Florida.)
Mlodinow — the author of “Feynman’s Rainbow,” “Euclid’s Window” and, with Stephen Hawking, “A Briefer History of Time” — writes in a breezy style, interspersing probabilistic mind-benders with portraits of theorists like Jakob Bernoulli, Blaise Pascal, Carl Friedrich Gauss, Pierre-Simon de Laplace and Thomas Bayes. The result is a readable crash course in randomness and statistics that includes the clearest explanation I’ve encountered of the Monty Hall problem (named for the M.C. of the old TV game show “Let’s Make a Deal”).
You are presented with three doors, behind one of which is a new car. You take your pick, but before your fate is revealed, the M.C. swings open one of the other doors, revealing a booby prize.
So far, so good, but now comes the big decision. Do you stay with your original choice or switch to the other unopened door? Even the great mathematician Paul Erdos was dumbfounded to realize that this is not a 50-50 proposition. If you lay out the possibilities on Cardano’s grid, you will see that your odds actually improve if you change doors.
The key to this puzzle is that the door the M.C. opens is not chosen at random. (He’s not going to ruin the game by prematurely revealing the car.) As in the two-daughter problem, the additional information skews the odds, and with Cardano’s method you can make a rational, though counterintuitive, decision. [continue]
Here's a second review:
Win some, lose some
A chance is what you take when you cannot calculate the odds. If the odds are in your favour, then in the long run, you'll win. What are the chances that you could flip a coin 10 million times and get heads every time? Very high, according to probability theory. Go on flipping and, over a period almost indistinguishable from eternity, you'd get myriad uninterrupted stretches of heads or tails. The catch is you'd never know whether you were in a stretch of 10 million consecutive wins or losses until after the event.
That is the second lesson of this delightful book: risky ventures, long shots and random outcomes have a way of looking like good bets, but only after the event. Almost everything that happens in life is contingent upon a series of unconscious gambles: of turnings taken, of chance encounters and unconsidered choices - in short, the drunkard's walk of the title. After the Japanese attack on Pearl Harbor in December 1941, it was easy enough to track back to the warning signs and condemn the high command for not having read them correctly. But this was to impose a selected pattern on what - before the bombs began to fall - would have been a bewildering array of conflicting intelligence amassed over many months from listening posts around the globe. Pearl Harbor wasn't a random event - somebody planned it - but until it had happened, such an attack could have been predicted in many places, or not at all. The dilemma for all gamblers is: just because such a thing is probable, does that mean it is going to happen this time?
The answer depends on how you calculate the odds. The odds at the start of the OJ Simpson trial certainly seemed against an acquittal, but a clever defence lawyer called Alan Dershowitz helped the jury to think constructively about probabilities. Around four million American women were beaten up by their husbands or boyfriends each year, he argued, but in 1992 only 1,432 had been murdered by their partners. So the probability that the African-American sports star turned actor had committed the fatal assault was actually one in 2,500. The jury were impressed by the calculation, but, as Mlodinow points out, it was the wrong calculation. In fact, the prosecution should have demonstrated that the same data told another story: that since Nicole Brown Simpson had already been murdered, the probabilities had to be considered differently. Of all those battered wives and girlfriends who had been murdered in the US, 90% had been murdered by their abusers. [continue]
I've never liked the phrasing "If a woman has two children and one is a girl, what is the probability that the other is a girl?" It's ambiguous. It can be interpreted as "If a woman has two children and at least one is a girl, what is the probability that both are girls?" Or it can be interpreted as "If a woman has two children and a specified one of them is a girl, what is the probability that both are girls?" The former probability is 1/3. The latter probability is 1/2. My suspicion is that most people, upon hearing the question, interpret it the second way: specify a child as a girl and then look at the probability that the other is a girl.
There are lots of problems having such ambiguous phrasings in probability courses, and it takes practice to get used to what is intended by the authors of books on probability. But just because the authors of books on probability have silently agreed to use one interpretation doesn't mean that their phrasings are not ambiguous to anyone who has not also agreed to that interpretation.
My suspicion is that most people, upon hearing the question, interpret it the second way: specify a child as a girl and then look at the probability that the other is a girl.
Without realizing it, you pretty much summed up the book. Mlodinow did not write the book in order to teach us why the answer to that question is 1 in 3. He wrote it in order to teach us why we so often get it wrong.
In other words, the problem is not that the question is ambiguous. The problem is that we don't realize it's ambiguous to begin with.
But we only "get it wrong" if we interpret it the first way but still give the answer 1/2. If we interpret it the second way, then we are not getting it wrong when we give the answer 1/2. An answer is only right or wrong *given an interpretation of the question*.
Yes, but in order to get those separate answers, you changed the structure of the sentence. Meaning that it matters how questions are framed, and that simple questions can be ambiguous without our realizing it.
Again, I think that's his point--that what we intuit is not always right.
I didn't change the structure of the sentence. Rather, I interpreted it two different ways. I think both interpretations are reasonable.
But I agree that it matters how questions are framed and that simple questions can be ambiguous without its being realized.