6
$\begingroup$

In my copy of An Introduction to Probability by William Feller (3rd ed, v.1), section I.2(b) begins as follows:

(b) Random placement of r balls in n cells. The more general case of [counting the number of ways to put] $r$ balls in $n$ cells can be studied in the same manner, except that the number of possible arrangements increases rapidly with $r$ and $n$. For $r=4$ balls in $n=3$ cells, the sample space contains already 64 points ...

This statement seems incorrect to me. I think there are $3^4 = 81$ ways to put 4 balls in 3 cells; you have to choose one of the three cells for each of the four balls. Feller's answer of 64 seems to come from $4^3$. It's clear that one of us has made a very simple mistake.

Who's right, me or Feller? I find it hard to believe the third edition of a universally-respected textbook contains such a simple mistake, on page 10 no less. Other possible explanations include:

(1) My copy, a cheap-o international student edition, is prone to such errors and the domestic printings don't contain this mistake.

(2) I'm misunderstanding the problem Feller was examining.

  • 2
    $3^4$ looks right. Imagine 4 balls in 1 cell: there is only one possible arrangement.2011-11-12
  • 0
    Volume $1$ seems to have the same typo, at the Amazon less cheapo price of $129.44$. Found a Canadian price of $189.95 (in Canadian dollars, which are roughly at par with the US dollar).2011-11-12
  • 0
    Are the balls distinguishable?2011-11-12
  • 0
    From the preface to the 1970 revised printing: "In contrast to the first edition, the third was marred by a disturbing number of errata. In the present revised printing, all discovered errata are corrected." However, this revised printing still contains the cited apparent error. ([At Amazon](http://www.amazon.com/Introduction-Probability-Theory-Applications-Vol/dp/0471257087), you can "Look Inside" to view this on pp 8-9.)2011-11-12
  • 0
    My el cheapo paperback edition (Wiley Eastern Limited) has $81$, not $64$. Perhaps different editions were typeset separately and proof-read by different people?2011-11-12
  • 0
    @NateEldredge I think so. But even if they weren't, you wouldn't get 64. I think I am correct and my **copy** of Feller is wrong. I think it's a good mental exercise to read textbooks that have errors in them, but I still expected correctness for my money.2011-11-12
  • 2
    Dimitrije: even not knowing the price you paid, I am sure the content of Feller volume I is worth it.2011-11-27
  • 0
    The error is also there in 2nd edition, 5th printing, June, 1960, page 10.2012-04-15
  • 0
    My edition in spanish (1973 - translated from 3rd edition) says 81.2012-04-15
  • 0
    @leonbloy, so 64 in Spanish is 81?2012-04-17
  • 0
    In the Second Edition (1957) I have: "(b) Distribution of r balls in n cells. The more general case .... For r = 3 balls and n = 4 cells, the sample space contains already 64 points, ...", so it seems it was introduced in updating to the 3rd Edition.2014-06-22

4 Answers 4