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How to Count

Brian Hayes

Counting is something we learn so early in life that we tend to dismiss it as a trivial skill, beneath the notice of mathematics. The recent U.S. presidential election suggests otherwise. Although most of the vote-counting controversies last fall concerned what to count rather than how to count, the counting process itself also proved to be imprecise and unreliable. Counting and recounting the same batch of ballots seldom gave the same total twice. Evidently, counting is not the utterly deterministic procedure we take it to be. There is some wiggle and wobble in it.

And ballots are not the only things we can lose count of. The Census Bureau has reported the U.S. population as 281,421,906, but no one believes this "actual enumeration" of the people is exact; the Bureau will deserve congratulations if even the first two of those nine digits are correct. Similarly, a U.S. Treasury web site displays "the public debt to the penny"; the last time I checked, the amount shown was $5,719,452,925,490.54, but again it's hard to believe that the accuracy of this figure matches its precision. Other purveyors of large numbers are more modest in their claims to exactitude. The New York Stock Exchange reports daily trading volume only to the nearest 100 shares. And the sign in front of my local McDonald's is stuck on "Over 99 billion served."

If counting is not a process we can count on, it seems prudent to look into the various ways it might go wrong. A theory of counting errors would describe the relation between the true number of things counted, which I'll designate N, and the number of counts actually registered, R. If you knew enough about the errors, you could predict R for any N. The inverse problem is harder: Given an observed count R, can you estimate the true N? The answer will not settle any election controversies, but it does lead into some curious byways of mathematics and computation.

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