Court Opinion

ID: 9562202
Source: CourtListenerOpinion
Date Created: 2023-08-21 18:23:33.888622+00
Date Added: 2024-06-11T09:17:15.015951
License: Public Domain

McCOMB, J.
I dissent. I would affirm the judgment in its entirety.
Appendix
If “Pr” represents the probability that a certain distinctive combination of characteristics, hereinafter designated “C,” will occur jointly in a random couple, then the probability that C will not occur in a random couple is (1 — Pr). Applying the product rule (see fn. 8, ante), the probability that C will occur in none of N couples chosen at random is (1 — Pr)N, so that the probability of C occurring in at least one of N random couples is [1— (1 — Pr)N].
*334Given a particular couple selected from a random set of N, the probability of C occurring in that couple (i.e., Pr), multiplied by the probability of C occurring in none of the remaining N — 1 couples (i.e., (1 — pr)N—yields the probability that C will occur in the selected couple and in no other. Thus the probability of C occurring in any particular couple, and in that couple alone, is [ (Pr) X (1—Pr)N—•1]. Since this is true for each of the N couples, the probability that C will occur in precisely one of the N couples, without regard to which one, is [(Pr) X (1 — Pr)N—1] added N times, because the probability of the occurrence of one of several mutually exclusive events is equal to the sum of the individual probabilities. Thus the probability of C occurring in exactly one of N random couples {any one, but only one) is [ (N) X (Pr) X (1 —Pr)»-*].
By subtracting the probability that C will occur in exactly one couple from the probability that C will occur in at least one couple, one obtains the probability that C will occur in more than one couple: [1—(1 — Pr)N] — [(N) X (Pr) X (1 — Pr)N—*]. Dividing this difference by the probability that C will occur in at least one couple (i.e., dividing the difference by [1— (1 — Pr)11]) then yields the probability that C will occur more than once in a group of N couples in which C occurs at least once.
Turning to the case in which C represents the characteristics which distinguish a bearded Negro accompanied by a pony-tailed blonde in a yellow car, the prosecution sought to establish that the probability of C occurring in a random couple was 1/12,000,000—i.e., that Pr = 1/12,000,000, Treating this conclusion as accurate, it follows that, in a population of N random couples, the probability of C occurring exactly once is [(N) X (1/12,000,000) X (1 —1/12,000,000)k-1]. Subtracting this product from [1 — (1 —1/12,000,000)N], the probability of C occurring in at least one couple, and dividing the resulting difference by [1 — (1 —1/12,000,000)N], the probability that C will occur in at least one couple, yields the probability that C will occur more than once in a group of N random couples of which at least one couple (namely, the one seen by the witnesses) possesses characteristics C. In other words, the probability of another such couple in a population of N is the quotient A/B, where A designates the numerator [1 — (1 —1/12,000,0001N] — [(N) X (1/12,000,000) X (1 — l/12,000.000)li—2]. and B designates the denominator [1 — (1 — 1/12,000,000)N],
*335N, which represents the total number of all couples who might conceivably have been at the scene of the San Pedro robbery, is not determinable, a fact which suggests yet another basic difficulty with the use of probability theory in establishing identity. One of the imponderables in determining N may well be the number of N-type couples in which a single person may participate. Such considerations make it evident that N, in the area adjoining the robbery, is in excess of several million; as N assumes values of such magnitude, the quotient A/B computed as above, representing the probability of a second couple as distinctive as the one described by the prosecution’s witnesses, soon exceeds 4/10. Indeed, as N approaches 12 million, this probability quotient rises to approximately 41 percent. We note parenthetically that if 1/N = Pr, then as N increases indefinitely, the quotient in question approaches a limit of (e — 2)/(e — 1), where “e” represents the transcendental number (approximately 2.71828) familiar in mathematics and physics.
Hence, even if we should accept the prosecution's figures without question, we would derive a probability of over 40 percent that the couple observed by the witnesses could be “duplicated” by at least one other equally distinctive interracial couple in the area, including a Negro with a beard and mustache, driving a partly yellow car in the company of a blonde with a ponytail. Thus the prosecution’s computations, far from establishing beyond a reasonable doubt that the Collinses were the couple described by the prosecution’s witnesses, imply a very substantial likelihood that the area contained more than one such couple, and that a couple other than the Collinses was the one observed at the scene of the robbery. (See generally: Hoel, Introduction to Mathematical Statistics f3d ed. 1962); Hodges & Leymann, Basic Concepts of Probability and Statistics (1964) ; Lindgren & McElrath, Introduc tion to Probability and Statistics (1959).)