The Incredibly Small Probability of Having a Beard

Estimated read time (minus contemplative pauses): 6 min.

Man Ray's Self-Portrait with Half a Beard
Man Ray (Half-Beard Self-Portrait)

People v. Collins is a court case commonly cited in discussions of probability (gone wrong) and the law.1 In the 1960s, a heterosexual couple matching the description of eye witness testimony was put on trial for a mugging in L.A. The relevant attributes were that she was was white with a blond ponytail, he was black with a mustache and beard, and they fled the scene in a yellow car.

According to the case record, a math teacher was brought in to testify on the probability of the tried couple being the wrong couple. He arrived at one in 12-million by multiplying the following probabilities, “Without presenting any statistical evidence whatsoever in support of the probabilities for the factors selected,” but rather the prosecutor intended the figures for “illustrative purposes” of how probabilities are computed (in particular, the “product rule”), while inviting the jury to “apply their own factors”:

A. Partly yellow automobile 1/10
B. Man with mustache 1/4
C. Girl with ponytail 1/10
D. Girl with blond hair 1/3
E. Negro man with beard 1/10
F. Interracial couple in car 1/10002

The most glaring problem here is that “mustache” and “beard” are treated as independent events! There’s also a deeper problem, connected to the fact that, if the crime was indeed committed by a couple matching that description, then the question shouldn’t be what the probability is that a randomly selected couple will have the above attributes, but rather what the probability is that a randomly selected couple bearing those attributes is the guilty couple. This in mind, were they the only such couple in L.A. (or, better, all of California), probability would be against the couple (randomly encountered or otherwise), but if we presume the existence of a few such couples, you’d need something more than the rareness of their descriptions to declare a given couple guilty (especially if encountered at random).

The tried couple was found guilty by a jury, but the case was reversed by the California Supreme Court: “We conclude that the court erred in admitting over defendant’s objection the evidence pertaining to the mathematical theory of probability and in denying defendant’s motion to strike such evidence.” The record goes on to state that “no mathematical formula could ever establish beyond a reasonable doubt that the prosecution’s witnesses correctly observed and accurately described the distinctive features which were employed to link defendants to the crime. (See 2 Wigmore on Evidence [3d ed. 1940] 478.) Conceivably, for example, the guilty couple might have included a light-skinned Negress with bleached hair rather than a Caucasian blonde; or the driver of the car might have been wearing a false beard as a disguise; or the prosecution’s witnesses might simply have been unreliable.”

There’s a lot to discuss here. For now, I just want to highlight the silliness of treating beards and mustaches as independent (with a nod to the reference class problem, more about which another day): What, then, is the probability of having half a beard on both the left and right side of one’s face?

Prosecutor: [In earnest, always.] We’ve heard today from eye-witnesses who saw a bearded man fleeing the scene of a robbery. Namely, running down an upward escalator. Half the witnesses were in a descending glass elevator to the left of the escalator, the other half were in an ascending elevator to the escalator’s right. So half the witnesses saw a man with a beard on the left side of his face, and half saw a right-face beard. My first question to you, as an expert, is: what is the probability that a man has a beard on the left side of his face.

Expert Witness: Just on the left side, but not on the right? Probably very small. I doubt it’s common.

Pro: Witnesses, all of who report a beard, only saw a given side of the fleeing man. So, I’m not asking about the probability of having a beard on one side and having no beard on the other. Right now I’m asking the probability of having a beard on the left at all.

ExpWit: According to your chart, one in ten men have a beard, so I’d estimate it’s also one in ten for the left side of a man’s face being bearded.

Pro: That’s 10%. And what is the probability of a man’s right face being bearded?

ExpWit: Same as on the left.

Pro: One in ten.

ExpWit: Yes.

Pro: And how does one find the probability of two events occurring together?

ExpWit: With the multiplication rule. The probability that two events occur together is the product of their individual probabilities.

Pro: So, if you would please demonstrate that rule: what is the probability of having both left- and right-face beards? Though I urge folks to remember that our starting number of one in ten is an estimate. Feel free, ladies and gentlemen, to use starting figures that strike you as more representative. The multiplication rule, however, is the mathematical operation to which your chosen starting figure should be subjected.

ExpWit: That’s one tenth squared. Or one out of a hundred.

Pro: Giving a starting figure of one in ten, one in a hundred men—or just 1%—have a beard on the left and right side! How rare. But let me ask you, what is the probability of having one half of a left-face beard?

ExpWit: Still 1/10th.

Pro: Interesting. And what is the probability of having both halves of a left beard, both halves of a right beard, and having a left-beard and a right-beard?

ExpWit: That’s one tenth to the sixth power. Or one in a million.

Pro: How rare! And if we cut everything in half again? And then again and again?

ExpWit: You’ll quickly pass one in a trillion… and get to basically zero. But not quite… well, depending on how finely we can split hairs. But it occurs to me that all this depends on events being independent. When a coin’s flipped, the probability of landing Heads doesn’t go from one half to one half squared just because two people are watching from different perspectives, or just because the probability is one half for Person A seeing it land Heads and one half for Person B seeing it land Heads. They’re both seeing the same coin toss.

Pro: Well, we don’t want to split hairs. [This gets a chuckle.] Leave that to the philosophers. Down here on Earth, witnesses saw two different things: a left-beard and a right-beard. It is possible for a person to have just one of those, to have some hair protruding only on one side of their face—while both of the people watching your coin flip see the entire face of the coin. Wouldn’t you say so, as an expert?

ExpWit: Expert?

Pro: In math and as someone with a beard.

ExpWit: Of course, but maybe the coin thing wasn’t the best example. Just something about this feels…

Pro: Right, the coin example is interesting, but, let’s agree, not relevant to today’s question. Now, as an expert, you said that the probability is practically zero that the man on trial today is innocent. Why not say precisely zero?

ExpWit: Precisely?

Pro: Well, if there are infinitely many things, and only some small portion of those things have beards, then the probability of a randomly chosen thing having a beard is zero.

ExpWit: That’s right. But…

Pro: Then this is easy. Are not many mathematicians and physicists and other esteemed folks of higher learning Platonists about numbers? Meaning they believe that real numbers are, well, real? [Blank stares.]

ExpWit: Or at least the whole numbers—there are different bents of mathematical Platonism…

Pro: So it is a somewhat commonly held view, especially among highly regarded professors and such?

ExpWit: Seems to be.

Pro: And it means that there are infinitely many things, if true. Correct?

ExpWit: Yes, but numbers can’t have beards.

Pro: I see what you mean. I stand corrected, and that’s why your the expert. Ladies and gentlemen, let it be understood that, according to our expert and mathematics itself, the probability of today’s defendant being the bearded man seen fleeing the robbery is 99.999-followed-by-many-more-nines-percent, or practically, if not precisely, 100%.


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Further Reading

Footnotes:

  1. See, for example, the article “A Bayesian Approach to Identification Evidence” by Michael O. Finkelstein and William B. Fairley (Harvard Law Review, Vol. 83, No. 3 (Jan., 1970), pp. 489-517); and the book The Drunkard’s Walk: How Randomness Rules Our Lives by Leonard Mladinow (2008).
  2. This table was found at a different, but harder to read, posting of the case file at the Harvard Law website: People v. Collins 68 Cal. 2d 319, 438 P.2d 33, 66 Cal. Rptr. 497 (1968)

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