## Devlin's Angle |

It's the Law of Small Errors: As the number of words in a manuscript tends to infinity, the probability of a significant error making it into the final version approaches 1. As with any self-respecting infinite sequence in an undergraduate calculus class, "approaching infinity" here means "a tractable bunch" -- the 100,000 words of the average book is definitely big enough.

I've written 23 books and several dozen long research papers in my
career, and the same thing has happened every time. So I suppose I
should have been prepared for it to happen again, when the first copy
of my new book *The Maths Gene* arrived on my desk earlier this
week. (This is the British edition. The American translation -- *The
Math Gene* (no "s") -- won't come out until August. Publishing is like
that. Don't ask why. As the theatrical producer kept remarking in the
movie *Shakespeare in Love,* it's a mystery.) But, experienced
as I was, that first glaring slip still caught me off guard. As I was idly
flicking through the book for the very first time, the following passage
leapt out at me and grabbed me by the throat:

*A classic example is the birthday problem, which asks how many
people you need to have at a party so that there is a better-than-even
chance that two of them will share the same birthday. Most people
think the answer is 183, the smallest whole number larger than 365/2.
In fact, you need just 23. The answer 183 is the correct answer to a
very different question: How many people do you need to have at a
party so that there is a better-than-even chance that one of them will
share your birthday? If there is no restriction on which two people will
share a birthday, it makes an enormous difference.*

Now the birthday problem is one of my favorite mathematical examples. I must have written and talked about it dozens of times. I could probably give a short lecture on it in my sleep. So how come that above passage slipped into print? For it's quite wrong. Not the 23 part. Surprising though that number is, you really do need just 23 people at the party to have a better than 0.5 probability that two people will share the same birthday. But the number of people you need to have present for there to be a better-than-evens chance of someone sharing your birthday is not 183, but the much larger 254. (Yes, really, 254, including yourself.)

How did such a howler find its way into the text and go unnoticed? Before I try to answer that, here are the relevant computations to establish those two answers of 23 and 254.

First, the coincidence of two birthdays. It turns out to be easier to compute the probability that no two people at the party have the same birthday, and then subtract the answer from 1 to obtain the probability that two people will share a birthday. For simplicity, let's ignore leap years. Thus, there are 365 possible birthdays to consider.

Imagine the people entering the room one-by-one. When the second person enters the room, there are 364 possible days for her to have a birthday that differs from the first person. So the probability that she will have a different birthday from the first person is 364/365. When the third person enters, there are 363 possibilities of him having a birthday different from both of the first two, so the probability that all three will have different birthdays is 364/365 x 363/365. When the fourth person enters, the probability of all four having different birthdays is 364/365 x 363/365 x 362/365. Continuing in this way, when 23 people are in the room, the probability of all of them having different birthdays is

364/365 x 363/365 x 362/365 x . . . x 343/365.

This works out to be 0.492. (It is when you have 23 people that the above product first drops below 0.5.) Thus, the probability that at least two of the 23 have the same birthday is 1 - 0.492 = 0.508, better than even.

Now for the problem of the birthdays different from yours. Pick any
person at the party. The probability of that person having a birthday
different from you is 364/365. (Again, I'm ignoring leap years, for
simplicity.) Thus, if there are n people at the party besides yourself,
the
probability that they all have a different birthday from you is
(364/365)^{n}. (Since we don't have to worry whether their
birthdays coincide or not, we don't have to count down 364, 363, 362,
etc. as we did last time.) The first value of n for which the number
(364/365) ^{n} falls below 0.5 is n = 253. And that's all
there is to it.

That was all pretty easy. A far more difficult question is: How did that
error find its way into my book? The answer is, I don't really know. A
manuscript for a trade book (i.e., a book which the publisher thinks will
have general appeal) goes through several stages of editing before it
goes to press. In addition to the author's own editing, the
commisioning editor for the publisher usually goes through it carefully,
generally making changes on each page to improve the wording, and
then the copy editor goes through it looking for any remaining
grammatical or stylistic errors. Everyone involved tries to shorten and
simplify the text to make it as accessible as possible, cutting out any
redundant prose. Having written several trade books now, and gone
through this process each time, I try to preempt as many editing
changes as possible by doing my own editing before sending the
manuscript off to the publisher, particularly as production schedules
usually mean that the author has only a few days to check over the
changes suggested by the two editors. In the case of my book
*Mathematics: The New Golden Age,* I did have to hold
up publication for a week or so when I discovered that a particularly
keen copy editor, unfamiliar with mathematics, had replaced all my
carefully crafted statements of mathematical results with more
colloquial forms that, while I agree they read much more smoothly,
were logically incorrect. In the case of *The Maths Gene,* however,
I suspect that the original error with the birthday paradox was mine, as
I cut down what had started as a rather lengthy discussion of the problem
to just a couple of sentences. (I was faced with reducing a 120,000 word
manuscript to the 100,000 words I had contracted for with the publisher.
To be honest, rough as this process might seem at time, it beats
coalmining as a way to make a living.) Looking back now, I see that
an earlier version contained not only discussions of the numbers
23 and 254, but also the crucial passage:

*The answer 183 is the correct answer to a very different question:
How many different birthdays do you need to have represented at a
party so that there is a better-than-even chance that one of them
will be your birthday?*

This is correct. Somehow, in the heat of the editing, by the time I got to page 271 (the very last page of the main text), either I was too liberal with my red pen or else I marked up the cuts too scrappily for the typesetter to make sense of. In any event, the error snuck in. The really annoying thing is that I passed over this error on the two or three subsequent times I checked over the manuscript. The law of small errors had struck again. The only redeeming factor on this occasion is that the US edition of the book is not due out until August, so I can correct the error before it has an opportunity to confuse or mislead American readers.

Let me leave you with another probability question. This one is for anyone who sets out to write a book. What do you think is the probability that the published version will contain at least one error? I know what the answer is in my case. It's unity.

Devlin's Angle is updated at the beginning of each month.