Somewhere on my shelves, there is a math book with a page very like the following:
I know this because I remember seeing it (or at least I think I do), but I can’t quite remember which book it’s in.
Fortunately, I didn’t try to steal this joke for Can You Outsmart an Economist?, because it turns out there’s an actual erratum in the main text. It comes in Chapter 16 where a series of problems leads the reader to discover the basics of option pricing. The arithmetic in those problems is all correct, except for one thing: In order to keep the math easy, I assumed an interest rate of 50%. But with an interest rate that high (and given the other assumptions in the chapter), nobody would be mucking around with buying options in the first place; we’d all just be putting our money in the bank and getting rich in a hurry.
So what I should have done is assumed an interest rate of 20%. If you’ve got a copy of the book, you should pencil in the following changes:
- On page 262, five lines from the bottom, the assumption that “The going interest rate is 50% per year” becomes “The going interest rate is 20% per year”.
- On page 263, first line: “This morning, Jeter borrowed $40, withdrew $32 from his bank…” becomes “This morning, Jeter borrowed $50, withdrew $22 from his bank….”
- On page 264, line 2, “Jeter has just borrowed $40 withdrawn $32 from his savings…” becomes “Jeter has just borrowed $50, withdrawn $22 from his savings…”
- On page 265, first full paragraph: The number $32 appears three times; all three of these become $22.
Again, the original example works perfectly well as long as you don’t notice that nobody should be buying options in the first place when the interest rate is so high. But I’d prefer that you pencil in the corrections.
The good news is that in the days of e-books and on-demand printing it’s easy to fix these things; the correction will soon appear automatically in the e-version and then in the next round of printed books. (So of course dedicated collectors should rush out and buy the current printing so as to eventually own both versions!)
A tip of the hat to Keith C. Brown of the University of Texas for catching this.
Ok, I have one or two minor errata!
“If that happens, there are no girls, so the ratio of boys to girls is infinite. To get the expected ratio, we have to average over all possible ratios, including infinity. That average is infinity. If you said that the answer was ½, you were infinitely wrong.”
Landsburg, Steven E.. Can You Outsmart an Economist?: 100+ Puzzles to Train Your Brain (Kindle Locations 2453-2457). Houghton Mifflin Harcourt. Kindle Edition.
I agree that if you said that “the answer was ½, you were infinitely wrong.” But why would anyone say that? The incorrect argument you criticize would suggest that the ratio would be 1, not 1/2.
“Answer: Then there are infinitely many girls and infinitely many boys, giving a fraction of infinity over infinity, which is not a number at all, and certainly not ½.”
Landsburg, Steven E.. Can You Outsmart an Economist?: 100+ Puzzles to Train Your Brain (Kindle Locations 2469-2470). Houghton Mifflin Harcourt. Kindle Edition.
First, this repeats the same problematic 1/2. More importantly, “giving a fraction of infinity over infinity, which is not a number at all” is not something somebody as well trained in mathematics as the author should say. The most plausible way to talk about a country is to talk about the limit, if it exists, of the ratio as population goes to infinity. And I think we’d agree that this ratio is a meaningful number and that this number is 1. Moreover, that number is a pretty good approximation for real country sizes with large, but finite, populations
“Step 1: Either I’m at First Street or I’m at Second, with probabilities ⅔ and ⅓.”
Landsburg, Steven E.. Can You Outsmart an Economist?: 100+ Puzzles to Train Your Brain (Kindle Locations 3709-3710). Houghton Mifflin Harcourt. Kindle Edition.
It’s nothing serious but this reader’s hackles were raised until the next page were it is stated correctly that:
“Step 1 assumes that the current intersection is First Street with probability ⅔. Albert got that from the solution to problem 1, which assumed that in the past Albert has always flipped coin F.
Landsburg, Steven E.. Can You Outsmart an Economist?: 100+ Puzzles to Train Your Brain (Kindle Locations 3719-3720). Houghton Mifflin Harcourt. Kindle Edition.
Overall, loved the book and will post an Amazon review shortly!
“The arithmetic in those problems is all correct, except for one thing: In order to keep the math easy, I assumed an interest rate of 50%. But with an interest rate that high (and given the other assumptions in the chapter), nobody would be mucking around with buying options in the first place; we’d all just be putting our money in the bank and getting rich in a hurry.”
I’m not convinced of this. Is there a proof?
I am not convinced because you can usually plug any interest rate into the option pricing formulas and get an option price at which a rational investor would buy or sell the option. Implicit in this is that buying or selling the option is not dominated by investing in the bank.
Has Sub Specie identified the problem, i.e., that the numbers blow up and we get a bunch of infinities? If so, then it doesn’t seem right to say that investing in the bank is better because nothing makes any sense anymore.
Sub Specie: I am inclined to quibble with you a bit, but not nearly so much as I am inclined to thank you for the (prospective) Amazon review. I hope you inspire others.
I might be back to do that quibbling.
I remember the boy/girl ratio posts, at least vaguely. Did Lubos Motl ever take you up on the bet invigilated by a statistician? I presume not, but I recall he was adamant the answer was 1/2. There was much jolly discussion of “the extra 1/2 boy” and such things. What fun. I have not yet obtained the book, so I am not sure what SSAE is arguing about in #1, but I don’t recall anyone arguing for a ratio of 1.
@Harold : It is a really simple thing, so much so that it likely slipped beneath your attention.
If x boys and y girls are born, the simple, wrong theory (equal number of boys and girls under all circumstances) predicts that the ratio of boy births to all births (i.e., x/(x+y)) is 1/2.
Similarly, for ratio of girl births to all births.
But if you talk about the ratio of boy births to girl births (only!) or vice versa, the simple, wrong theory predicts 1.
It is all simple to convert and entirely outside the scope of the larger debate. But I think when the text clearly refers to one kind of ratio or the other, you should use the right wrong number.
#5 – oh I see what you are talking about.
A possible bug in the book in The Obedient Prisoners
Based on the numbers given: 40 total, 25 reds (i.e. 15 blues) and assuming all of them are completely rational, the reasoning should be symmetric wrt red/blue replacement.
So the correct answer according to the same logic should be:
in 15 days the blues figure out their colors and leave.
The next day the reds understand their colors and leave too.
No one comes for the Christmas.
Did I miss some symmetry breaker that makes the blues wait for the reds and not leave on the 15th midnight?
Sorry, my mistake. I missed the rule that says only reds are permitted to leave.
Withdrawn.
The errata/erratum quote at the start of the OP is mentioned in Metamathematical Themas by Douglas Hosfstadter, which is a collection of his columns from Scientific American in the early 80s.
In a column from January 1982 about self reference and the Epimenides Paradox he attributes it to a Deborah Rowe who describes it as a possible errata section in a hypothetical book.
This doesn’t mean, of course, that it hasn’t subsequently been used as a genuine errata section in a book.
Eric: It seems very likely that I saw this in the Hofstadter book and misremembered it as a page in a book on my shelf. This would explain why I can’t seem to find it.
Vic: Sorry I was out of town when you posted and didn’t have an opportunity to respond. But yes, I see that you managed without me: The asymmetry is that only reds are allowed to leave.
@Eric Thank you for reminding me of Douglas Hofstadter. I had not thought about that name for years, but when I was a teenager I carried Goedel Escher Bach everywhere with me for many months. It was one of a relatively small number of books I’ve read that literally rewired my brain, changing how I think about many important and interesting things in the decades since.
Ditto Sub.
I read GEB, and ‘Metamagical Themas’, and ‘The Mind’s I’ many dozens of times as a teenager/young man. And even though it’s been many years since I looked at them, as soon as I read Steve’s post, I knew exactly where I could find that quote.
Just reading the book now, and noticed another one. Page 76 “all tend to deteriorate at the time”. The word “same” got dropped