Random numbers that aren’t

Originally published 24 March 1986

Thir­ty years ago, at about the time I began to study sci­ence, I came across a book called A Mil­lion Ran­dom Dig­its. Here, in a vol­ume as thick as the Boston tele­phone direc­to­ry, were page after page of num­bers with no appar­ent pat­tern. No mat­ter how long you stud­ied the num­bers, there was no way to pre­dict what dig­it would occur next.

Why would any­one spend good mon­ey on a book of ran­dom num­bers? I asked myself. I soon dis­cov­ered the extra­or­di­nary val­ue of such a tool.

Sci­en­tif­ic descrip­tions of com­plex phe­nom­e­na are often sta­tis­ti­cal: They make pre­dic­tions about trends or prob­a­bil­i­ties, not about spe­cif­ic events. For exam­ple, the out­come of the flip of a coin can­not be pre­dict­ed, but in a hun­dred flips the odds are high that you’ll get 50 heads. The book of ran­dom num­bers was sta­tis­ti­cal in the same way; you might not be able to pre­dict the next dig­it, but the odds are high that in a hun­dred dig­its half will be a dig­it less than five.

Sci­en­tists can cre­ate mod­els to illus­trate and study the effects of sta­tis­ti­cal laws by using ran­dom num­bers. The num­bers in the book could be used to sim­u­late the flip­ping of a coin (a dig­it less than five is a head, a dig­it five or greater is a tail). Biol­o­gists use ran­dom num­bers to mod­el the evo­lu­tion of pop­u­la­tions that are con­trolled by the sta­tis­ti­cal laws of genet­ics. Econ­o­mists use ran­dom num­bers to mod­el sta­tis­ti­cal fluc­tu­a­tions in prices and demand. The list of appli­ca­tions is endless.

Recipe for randomness

Today, every­one work­ing in sci­ence has access to a com­put­er that can gen­er­ate a hun­dred ran­dom dig­its in the blink of an eye. Unfor­tu­nate­ly, the “ran­dom num­bers” gen­er­at­ed by com­put­ers are not ran­dom at all. Com­put­ers use a math­e­mat­i­cal recipe to churn out num­bers that appear to be sta­tis­ti­cal­ly ran­dom, but in fact — if you knew the recipe — you could say with cer­tain­ty what dig­it would appear next.

Could we rec­og­nize a string of tru­ly ran­dom dig­its if we had one, dig­its that don’t fol­low from any recipe or rule? Sta­tis­ti­cians have labored might­i­ly to devise tests of ran­dom­ness. For exam­ple, in a string of a thou­sand ran­dom dig­its, half should be a dig­it less than five. But a string of dig­its that met this test could still hide some oth­er sub­tle lack of ran­dom­ness; for exam­ple, two’s could fol­low three­’s more often than any oth­er digit.

Because of the wide­spread use of ran­dom num­bers in sci­ence, it is impor­tant to have a sat­is­fac­to­ry test of ran­dom­ness. Per­si Dia­co­nis of Stan­ford Uni­ver­si­ty, a sta­tis­ti­cian who is present­ly teach­ing a course on ran­dom­ness at Har­vard, has been work­ing to devise a the­o­ry of randomness. 

But ran­dom­ness has proved to be splen­did­ly eva­sive. Says Dia­co­nis: “The more you think about ran­dom­ness, the less ran­dom things become.” He is wor­ried that sci­en­tists who are naive about the ran­dom­ness of com­put­er-gen­er­at­ed num­bers could end up with mis­lead­ing results.

By now, your head is prob­a­bly spin­ning like a roulette wheel. And no won­der: Not even sta­tis­ti­cians know how to rec­og­nize ran­dom num­bers. Ran­dom­ness is an idea that every­one thinks they under­stand, but it is dev­il­ish­ly dif­fi­cult to define.

There is more at stake here than sim­ply refin­ing math­e­mat­i­cal tech­niques. Philo­soph­i­cal ques­tions are at issue: Is the use­ful­ness of sta­tis­ti­cal laws in sci­ence a cov­er for our igno­rance of the detailed behav­ior of com­plex deter­mined sys­tems? Or, as the quan­tum physi­cists say, is there some­thing intrin­si­cal­ly prob­a­bilis­tic about the universe?

Bamboozling and Megabucks

I once heard a ran­dom num­ber gen­er­a­tor described as a “bam­boo­zling machine.” For exam­ple, the machine that pops out the num­bered balls for the win­ning Megabucks num­ber is sup­posed to be a bam­boo­zling machine; it is sup­posed to bam­boo­zle every­one but a few lucky mil­lion­aires. But it is almost cer­tain­ly not. If you knew the order in which the num­bered balls were put into the revolv­ing drum, and if you knew how many times the drum was turned, and if you knew the nature of the mech­a­nism that releas­es the balls from the drum, then you could build your own machine and, by test­ing it, have a bet­ter than ran­dom chance at win­ning Megabucks.

A per­fect bam­boo­zling machine would rule out the pos­si­bil­i­ty that any­one could ever know enough about the ini­tial con­di­tions of the machine to pre­dict the out­come. A per­fect bam­boo­zling machine would be fun­da­men­tal­ly and intrin­si­cal­ly random.

Com­plex­i­ty can make things look ran­dom that may not be. The “free will” of a human being looks like a bam­boo­zling machine to a philoso­pher. An amoe­ba looks like a bam­boo­zling machine to a physicist.

Is the uni­verse itself a bam­boo­zling machine? Is there a chaot­ic, hel­ter-skel­ter ele­ment to nature that will nev­er yield to deter­min­is­tic analy­sis? Is tomor­row a ran­dom pos­si­bil­i­ty, or is it deter­mined by today? The work of sta­tis­ti­cians like Dia­co­nis will not answer the ques­tion, but it will help us pose the ques­tion more precisely.