The Numbers Behind NUMB3RS

_*
We'll take a closer look at this formula in a moment.

^*
This was the main plot idea behind the 2002 blockbuster movie
Minority Report
, starring Tom Cruise. But that, of course, is fiction.

^*
An Application of Bayesian Networks to Antiterrorism Risk Management for Military Planners
, by Linwood D. Hudson, Bryan S. Ware, Suzanne M. Mahoney, and Kathryn Blackmond Laskey.

^*
The comparison is not made directly between the sequences of the four base letters, but on numerical counts of them. The “DNA profile” is actually a sequence of those counts. The distinction is not important for our account.

^*
Profile match probabilities are based on empirical studies of allele frequencies of large numbers of samples. The figure 1/10 used here is widely regarded as being a good representative figure.

^*
As will be explained later, care is required in interpreting this requirement in terms of exactly what numerical probability is to be computed.

^*
United States v. Sullivan
, 246 F. Supp. 2d 700, 704 (E.D. Ky. 2003).

^*
The FBI did consider using JPEG, but the special nature of fingerprint images—essentially narrowly separated, “black,” parallel, curved lines on a “white” background—meant that it was much more efficient to use a specially tailored system. For many images, such as a fairly uniform background, JPEG-2000 can achieve a compression rate of 200.

^*
It turns out that the actual numbers—one year, two years, five years—are not important, just the
comparisons between them,
but we'll stick with the figures we have.

*In
Statistical Methods in Counterterrorism
, Alyson G. Wilson, Gregory D. Wilson, David H. Olwell, editors (New York: Springer, 2006).

^*
Nevada is an exception. The lucrative gambling business in this otherwise fairly poor state enabled the casinos to exert pressure on the legislature to make card counting illegal.

^*
Actually, this is not entirely accurate. Because of inertial properties of a physical coin, there is a slight tendency for it to resist turning, with the result that, if a perfectly balanced coin is given a random initial flip, the probability that it will land the same way up as it started is about 0.51. But we will ignore this caveat in what follows.

*Some casinos use a so-called “soft 17 rule” that requires the dealer to hit when his or her total of 17 includes an ace counted as “11”.

^*
Edward O. Thorp,
Beat the Dealer: A Winning Strategy for the Game of Twenty-One
, Random House, New York, 1962.

^*
E. H. Simpson. “The Interpretation of Interaction in Contingency Tables,”
Journal of the Royal Statistical Society
, Ser. B, 13 (1951) 238–241.

^*
Sadly, despite many efforts to eliminate the problem, racial bias by police seems to be a persistent issue throughout the country. To cite just one recent report,
An Analysis of Traffic Stop Data in Riverside, California
, by Larry K. Gaines of the California State University in San Bernardino, published in
Police Quarterly
, 9, 2, June 2006, pp. 210–233: “The findings from racial profiling or traffic stop studies have been fairly consistent: Minorities, especially African Americans, are stopped, ticketed, and searched at a higher rate as compared to Whites. For example, Lamberth (cited in
State v. Pedro Soto
, 1996) found that the Maryland State Police stopped and searched African Americans at a higher rate as compared to their rate of speeding violations. Harris (1999) examined court records in Akron, Dayton, Toledo, and Columbus, Ohio, and found that African Americans were cited at a rate that surpassed their representation in the driving population. Cordner, Williams, and Zuniga (2000) and Cordner, Williams, and Velasco (2002) found similar trends in San Diego, California. Zingraff and his colleagues (2000) examined stops by the North Carolina Highway Patrol and found that African Americans were overrepresented in stops and searches.”

^*
The idea is based on several real-life projects to use the equations that describe fluid flows in order to analyze various kinds of crowd activity, including freeway traffic flow, spectators entering and leaving a large sports stadium, and emergency exits from burning buildings.

^*
Ref. R. Adderley and P. B. Musgrove, General Review of Police Crime Recording and Investigation Systems,
Policing: An International Journal of Police Strategies and Management
, 24 (1), 2001, pp.110–114.

^*
It's actually more accurate to say “can be regarded as” rather than “consists of,” since the entire “neural network” is simulated on a normal digital computer.

^*
After the riots, federal charges of civil rights violations were brought against the four officers. Sergeant Stacey Koon and Officer Laurence Powell were found guilty; the other two were acquitted.

^*
For those who know the lingo, the key idea is to use Euler-Lagrange PDE minimization, a calculus technique developed long before computers came onto the scene, on the total variation functional.

DR. KEITH DEVLIN
is executive director of Stanford University's Center for the Study of Language and Information and a consulting professor of mathematics at Stanford. Devlin has a B.Sc. degree in Mathematics from King's College London (1968) and a Ph.D. in Mathematics from the University of Bristol (1971). He is a fellow of the American Association for the Advancement of Science, a World Economic Forum fellow, and a former member of the Mathematical Sciences Education Board of the U.S. National Academy of Sciences. The author of twenty-five books, Devlin has been a regular contributor to National Public Radio's popular program
Weekend Edition
, where he is known as “the Math Guy” in his on-air conversations with host Scott Simon. His monthly column, “Devlin's Angle,” appears on Mathematical Association of America's web journal
MAA Online
.

 

DR. GARY LORDEN
is a professor in the mathematics department of the California Institute of Technology in Pasadena. He graduated from Caltech with a B.S. in mathematics in 1962, received his Ph.D. in mathematics from Cornell University in 1966, and taught at Northwestern University before returning to Caltech in 1968. A fellow of the Institute of Mathematical Statistics, Lorden has taught statistics, probability, and other mathematics at all levels from freshman to doctoral. Lorden has also been active as a consultant and expert witness in mathematics and statistics for government agencies and laboratories, private companies, and law firms. For many years he consulted for Caltech's Jet Propulsion Laboratory for their space exploration programs. He has participated in highly classified research projects aimed at enhancing the ability of government agencies (such as the NSA) to protect national security. Lorden is the chief mathematics consultant for the CBS TV series
NUMB3RS
.

Solving Crime with Mathematics

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Copyright © Keith Devlin and Gary Lorden, 2007
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LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA

Devlin, Keith J.
The numbers behind NUMB3RS: solving crime with mathematics/Keith Devlin, Gary Lorden.
p. cm.
ISBN: 978-1-1012-1346-9
1. Criminal investigation. 2. Mathematical statistics. 3. Criminal investigation—Data processing. I. Title: Numbers behind numbers. II. Lorden, Gary. III. Title.

HV8073.5.D485 2007
363.2501'5195—dc22

2007018115

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