Well, I finished the first draft of the big Monty Hall book this past week. Still need to make some diagrams, and there's probably a fair amount of rewriting in my future, but the “words from nothing” phase is now over. Yay!
If anyone would care to give me some feedback, here is the first chapter. And also the bibliography for the book, to make the citations work out properly. I'm already aware of a number of typos, but don't hesitate to point them out anyway. I'm more interested in what people think of the tone and the style. Or anything else it occurs to you to comment on.
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Just downloaded it, and before I even start reading, you need to *quickly* get a copyright statement on that document.
Well, one quick comment: section 1.1 comes across a bit arrogant in spots, at least in my opinion. For example, the penultimate sentence of the first paragraph. I would probably tone that down to something like, "I resist the temptation to say something impolite and instead say, 'Well, maybe you just never had the right teacher.'"
And two paragraphs later, I'd omit the bit about giving the student the "smack in the face he so richly deserves."
The tone for the rest of Ch.1 seems very agreeable to me. Just those bits in 1.1 seemed a bit off.
That's my $0.02, anyway.
Good job so far I think -- and I'm one who has been perplexed by the MHP for 40 years. (Not the result. I'm not a denialist -- I'll switch, I promise -- but the "whys" and "how comes" continue to puzzle me.)
Just a couple of comments:
"We stipulate that if Monty has a choice of doors
to open, then he chooses randomly from among his options." --How about, "If both of Monty's doors conceal goats, he chooses which one he opens randomly."
"To this point we have behaved as though the point of probability was to discover the properties of certain real world objects. Assigning a probability of 1 2 to the result of a coin toss was viewed as a statement about coins, for example. More specically, it was a description of something coins tend to do when they are flipped a large number of times. This, however, is not the only way of viewing things. We might also think of probability assignments as representing our degree of belief in a given proposition. In this view, the assignment of 1 2 to each possible result of a coin toss means that we have no basis for believing that one outcome is more likely than another. It is a statement about our beliefs, as opposed to a statement about coins.
We now ask for the variables affecting how we update our degree of belief."
-- This is not clear to me. I think it's not clear to me why these two are not just different descriptions of the same thing.
"Scientists have a saying that extraordinary claims require extraordinary evidence. This captures the insight that if we initially view a proposition as exceedingly unlikely, it will take impressive evidence indeed to make it suddenly seem likely."
-- I dislike this slogan, unless it is limited to making a purely "folk psychological" point.
[Regarding the Bertrand box} "Since the boxes are identical they have equal probabilities of being chosen."
-- Why does the identicalness of the boxes matter? If the boxes were different, wouldn't there still be a 1-in-3 chance of any given outcome?
What qetzal said. Also, in the paragraph which begins "Bayes occupies a curious position in the history of mathematics", "can not" should be a single word, and the reference to Todhunter's book is broken (BibTeX error of some kind).
Congratulations. I think it is great and I found the whole thing very clear. Only one meaningless nit to pick here and it's not relevant to your arguments at all. On pages 9/10 the sentence, "Imagine that instead of using fifty-two card decks, you instead remove the ace of spades and a joker from each deck" jumped out at me, since, of course, there is no joker in a 52 card deck. Adding a joker makes it a 53 card deck. As I said it's not relevant to the example but if it were me I would choose any random card instead. Anyway, I think it's great and I know 2 or 3 people who will certainly get it for Xmas.
In your discussion of the Bertrand box, I believe you use "drawer" in one instance instead of "box".
Finally, the bit about the binomial coefficients seemed to jump out at the reader. If I didn't already know what these objects were, I'm not sure I could follow the discussion in that passage.
This chapter is great and very well written! It definitely draws the reader in, and I'm looking forward to the final product. I think that overall, the tone is very inviting and conversational, while the mathematical exposition is very clear and understandable.
I do, however, have to agree with what qetzal said. As a math student, I actually enjoyed section 1.1 very much (haha), but I definitely felt that it might come off as rather abrasive to the majority of your intended audience.
One grammatical note: In the section on the Birthday Problem, just before you introduce the equation for the probability of 2 out of n people sharing a birthday, you have the sentence:
"The probability P that in a roomful of n people no two of them will have the same birthday is obtained by multiplying these numbers together, we obtain the formula..."
I think this sentence needs tidying...perhaps inserting the word "hence" after the comma.
My intent in downloading the chapter was simply to verify that you were indeed writing a book about Monty Hall's game show. Instead, I got sucked in and read the whole darn thing!
If picking nits is of value, mine would be to find a sharper typeface for text. I have geezer eyes and this fuzzy, shadowed stuff was harder to read than most books, even when enlarged. Of course, there's a 2/3 chance that the viewing problem is with my monitor...
If nothing else, your timing for interest in the subject must be at a high for some reason unknown to me. I just saw the movie 21 & the Monty Hall problem (although it wasn't specifically named) was featured near the beginning. Other than the Monty Hall problem I can't say that I could recommend the movie, but it might be worth it just for that.
Not sure I understand why there is an apostrophe in "What Bayes' Wrought".
Would it be possible to split the chapter up, or leave some of it out? Looking at a couple of similar science-for-the-layman books on my shelf, chapters run from 6-20 pages or so. Absolute maximum is about 30. How lomg is the whole book? It might be more appealing if it's shorter.
On p. 13, you write: "And if the question had been, \How many people do you need before obtaining a probability greater than one half that someone has the same birthday as you?" then 183 would be the correct answer."
In fact, it's not. The correct answer to that problem is 253. Let's say you stand in front of a group of N people; the probability that a given person has a different birthday than you is 364/365, and birthdays have been assumed independent, so the probability that none of the N people share your birthday is (364/365)N. To get this to be less than one half one needs N = 253.
On page 46, you accuse vos Savant of shifting from a conditional problem to an unconditional problem and say the conditional problem was "as posed by her correspondent (in which it is stipulated that the contestant always chooses door one and the host always opens door three)" [emphasis mine]. However, I don't see that in Whittaker's letter on page 37, in which he writes, "You pick a door, say number 1, and the host, who knows what's behind the doors, opens another door, say number 3, which has a goat." It seems to me that Whittaker is giving doors one and three as examples for the sake of argument rather than as conditions.
In section 1.8 you write
"Monty now opens a door he knows to conceal a goat",
I don't think you make it clear that Monty chooses a door
other than the door chosen by the contestant.
Enjoying the read so far!
Out of interest, who is your intended audience? This might help us in our critique/advice of your chapter.
Thanks for all the comments!
You're probably right that I overdid things in the first section. I'll tone it down in the second draft.
The first draft came to almost exactly 250 pages (just under 80,000 words). There are eight chapters in total. There is also an appendix listing the specific variations of the Monty Hall problem discussed in the text. The final chapter is a short concluding chapter, so really there are seven substantive chapters. They're all quite long (though I'm pretty sure Chapter One is the longest), but they're broken up into so many sections that I was hoping that wouldn't be too burdensome to the reader.
The level of the book is another sticking point. Large segments of the book contain little mathematics, and should be comprehensible to anyone. For example, Chapter Six addresses some of the psychological literature on the problem, and while I wouldn't call it light reading, I don't think it will be leaving anyone in the dust either. All of the probability theory you need to understand what I am saying is developed from scratch in the text. So for the most part the intended audience is anyone who likes mathematics and isn't scared off by a few equations. Say, typical undergraduate math majors, for example. However, Chapter Four, which discusses an elaborate multi-door version of the problem, inevitably got rather complex. The level of the mathematics is still just elementary probability and recurrence relations, but the equations get a bit thick. It's just the nature of the beast.
So let's say the book is intended for undergraduate math majors, but they might have to work a bit to get through certain portions.
Concerning the binomial coefficients, I debated with myself whether to include an explanation of what they were and decided that would take too much space. Perhaps the thing to do is simply to take out that part, and then merge sections 1.4 and 1.5 into section. That will take out the incongruous mathematics in what is otherwise a math-free chapter, and will also shorten a lengthy chapter. Sound good?
As for the nit picks, keep them coming! They are precisely the sort of thing I'm likely to overlook no matter how many times I proofread the darn thing.
I've got another couple of comments. These are attempts to highlight areas in which I have been puzzled about the MHP. I recognize the arrogance of this. I'm sure you're not writing the book just for me. However, I'm assuming that one purpose you have is to produce a "guide for the perplexed," and I'm also assuming that my particular perplexities are not unique.
I hope you grasp that for someone like me who doesn't really understand why the "3-door problem" doesn't "reset" to become a "2-door problem" once Monty opens his door and reveals a goat, starting with a "million door problem" is not helpful. If I think that the problem is a "2-door problem" when only two doors remain unopened, it doesn't matter how many doors you start with (since I want to "reset" the problem based on however many unopened doors remain).
The UFO example (p. 44) highlights this and raises another, underlying perplexity. For the "little green woman" who comes on the scene after Monty's opened his door, it's a "2-door problem," but it remains a "3-door problem" for a contestant who has been there from the beginning. The reason, we're told, is that the contestant started with three doors and this has not changed just because Monty opened one of them. But the perplexity is why the opened door remains relevant. The "seems like" error appears to be if the contestant thinks that once that door is opened it "might as well be" a two door problem. It "seems like" the choice is between the two remaining unopened doors. You can accurately call it a "3-door problem," but that "seems" to permit the contestant to choose the opened door and no one would do that.
The UFO example also highlights a point which I have never quite grasped conceptually: The "little green woman's" odds are different from the contestant's odds. To extend this a bit, the contestant's odds are different from "the" odds -- or, perhaps, there is no such thing as "the" odds? This is somehow contrary to my colloquial understanding of what "odds" are. I somehow want there to be only one set of odds -- a nice, objective mathematical thing, you know, the same for everyone. When I'm told that odds vary from person to person it's disconcerting (and a little mystical, which is not what I want my mathematics to be).
In section 1.3.2, sentence 4 reads: "But 5% of the people who test positive for the disease are in reality disease-free."
I think you mean "But 5% of the people who are in reality disease-free test positive for the disease."
Two very different statements, and the second is what you seem to be using later.
Misspelled first author in reference 16.
A few things to note here:
Minor first issue -- the question of how long we expect to wait for the ace of spades has two different answers depending on our interpretation. The expected value is a million trials, but fewer than a million trials are needed in order to have a 50% chance of seeing the ace. (Around 700,000 trials, I think. You can probably figure out the exact number in your sleep.)
Second issue -- I'm sure that my interpretation of the last sentence in the above quote doesn't match your intent, but it sounds to me like a case of the Gambler's Fallacy.
Also, in the same paragraph, "no to happen again" should be "not to happen again."
Sounds like a fun book, and I love it when writers of pop science/math have real expertise in the subject that they write about.
In Section 1.9 (p.33), in the first line of indented quotation, Monty is spelled "Monte."
Also, just a few lines below the typo I just pointed out, Monty says "I'll offer you $ for the box" with no number next to the dollar sign.
Sorry for the sporadic posts regarding lines which are so close to each other. I'm reading chunks of the chapter in between homework problems...
Craig is right about section 1.3.2. You want to say that 5% of the people who are disease-free test positive, and in fact you use the information in this way later in the paragraph.
That's as far as I've gotten so far, but I have to say I like the discussion at the beginning about your interactions with nonmathematicians and how they misunderstand what math is, as I've encountered the same kinds of situations. It sounds like your intended audience is laymen and interested mathematicians, and I think both groups will appreciate it.
page 6, first paragraph: parenthetical section needs a closing quote
throughout: is numeric footnote style (e.g. [77]) used in popular books, or is it primarily from academic and technical papers?
section 1.5: "What Bayes' Wrought" does not need the apostrophe; "Bayes'" is the correct possessive, but not proper elsewhere
page 30, second full paragraph: "prisoner's" has extra apostrophe
Overall, though, quite nice. Is the Richard Bedient mentioned in your sources the same one who co-wrote a differential equations text with Rainville? I remember that one from undergrad physics.
Congratulations on reaching this milestone! Do you think you'll be able to get Monty Hall himself to write an introductory comment for the book, or at least a cover blurb?
~David D.G.
I don't agree that the first section is too arrogant at all. In fact I enjoyed it BECAUSE you dare to tell it like it is (and because it was funny). But perhaps I'm just an intellectual elitist.
Out of curiosity, did you ever see Proof?
Skimmed your chapter without reading carefully, so I have no editorial comments. I do want to thank you, though, for giving me the inspiration to finally work out the correct answer for myself; I have bumped into the problem many times but never "got it."
Here's what seems to me to be the crux, expressed non-mathematically:
You (the contestant) make your choice. Then Monty opens a goat-door. Now, there are two possibilities: your choice is correct (the car) or incorrect (the other goat). If you were correct in the first place, then switching would always be a mistake because you'd get the other goat. However, if your initial choice was wrong, then Monty has just shown you the other goat, so switching pays off every time.
Of course, you don't know whether your initial choice is correct or not, but you know the odds: one in three. Therefore it will pay to switch 67% of the time.
So thanks again; it finally makes sense.
pg. 25-26: Clarify the statement of the problem. It's not clear at first that each box has two drawers. You randomly choose one of the 6 drawers. Also change "chests" to "boxes".
pg. 26: end parantheses in second paragraph
pg. 37: line -4, Erd\"os
pg. 48: line 6, misquoted, "judgment" should be "insight", line 7, member(s)
pg. 50: line 2 of example, LET"S should be LET'S?
pg. 55: [24] Conditional
pg. 57: [49] Language
pg. 59: [67] Paradoxes, Company, [68] Considerations
pg. 61: [91] The
pg. 22: line -9, It (is) also
pg. 26: third paragraph. "only three possibilities:..." You only describe in words the first two possibilities.
pg. 28: line -13, coin(,)
pg. 36: missing citation
pg. 38: line 2, from NYT quote, "one of (the) many"
pg. 43: line 13, "than" -> "then"
pg. 46: line 8, from TAS quote, "letters(,) nearly"
pg. 47: from TAS quote, line 4, "that (this) reflects", line 9, "out of (the) many"
I love the intro. Hooked! I will have to finish all. Of course I am evolved a half a chromosone pair away from the Cajun swamp world. Sure miss those shrimp. BTW I like the physics geek show on CBS too.
I don't think that the first section was too arrogant; I really enjoyed it. But then again, I really identified with it--even though I'm not even a PhD (but all my friends know me as a math guy).
page 10, paragraph 2 "no to happen again" should probably be "not to happen again."
page 43 You start off with a blockquote from Savant using peas and shells, and then in the next paragraph you're talking about cars and goats again. It took me a few seconds to figure out that you had gone back to the original example. I think you could use a short segue (maybe not even a sentence long) to alert the reader that you're done with the shells.
This was a very entertaining chapter. I'll be passing it along to some friends and family that also enjoy math puzzles and brain teasers.
Thanks is My Friend
"...instead remove the ace of spades and joker from each deck. Now we have two small decks of two cards each ..." Another poster already pointed out the lack of jokers to remove; I'd like to point out that removing two cards from a 52-card deck leaves a 50-card deck. Took me a few seconds to figure out what you meant there.
Starting around "Let us try a more rigorous argument" on p. 10, the tone changes from storytelling to mathematics textbook.
1.6 seems to use Bgg, Bgs, and Bss just for the fun of it; it doesn't really serve any purpose. Gold bold, silver box, and mixed box could be used just as well, and would make the section more accessible.
1.6 again, end of page 26. Was something about stopping on the first head omitted? Because, otherwise, d'Alembert comes across as an idiot arguing HH is not possible.
(Hey, but after that, we lose the mathematics textbook feel)
p35, "Considering the venue, a high level journal read primarily by professional statisticians, you would have expected a raised eyebrow or two and little more. But this is the Monty Hall problem we are discussing, and it has the power to make otherwise intelligent people take leave of their senses."... LOL. Love that pair of sentences.
1.10, opening: Is that inspired by the Frantics? How many beater bars did the vacuum have at the time?
"we shall have more to say about it in a later chapter." I shall have more to say about this line in a later post. [ok, not really, but that phrase is in the chapter far too much; or at least it feels that way on p41]
An appendix to an introduction?! I think some cutting is required. Hey, at least some of it will be easy "These are just a few representative citations. There are many others." becomes "These are representative few of many citations", saving 23 characters.
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