People v. Collins
68 Cal. 2d 319, 438 P.2d 33, 66 Cal. Rptr. 497 (1968)

SULLIVAN, J. We deal here with the novel question whether evidence of mathematical probability has been properly introduced and used by the prosecution in a criminal case. While we discern no inherent incombefore us defendant should not have had his guilt determined by the odds and that he is entitled to a new trial. We reverse the judgment.

A jury found defendant Malcolm Ricardo Collins and his wife defendant Janet Louise Collins guilty of second degree robbery. Malcolm appeals from the judgment of conviction. Janet has not appealed.

On June 18, 1964, about 11:30 a.m. Mrs. Juanita Brooks, who had been shopping, was walking home along an alley in the San Pedro area of the City of Los Angeles. She was pulling behind her a wicker basket carryall containing groceries and had her purse on top of the packages. She was using a cane. As she stooped down to pick up an empty carton, she was suddenly pushed to the ground by a person whom she neither saw nor heard approach. She was stunned by the fall and felt some pain. She managed to look up and saw a young woman running from the scene. According to Mrs. Brooks the latter appeared to weigh about 145 pounds, was wearing "something dark,'' and had hair "between a dark blond and a light blond,'' but lighter than the color of defendant Janet Collins' hair as it appeared at trial. Immediately after the incident, Mrs. Brooks discovered that her purse, containing between $35 and $40 was missing.

About the same time as the robbery, John Bass, who lived on the street at the end of the alley, was in front of his house watering his lawn. His attention was attracted by "a lot of crying and screaming'' coming from the alley. As he looked in that direction, he saw a woman run out of the alley and enter a yellow automobile parked across the street from him. He was unable to give the make of the car. The car started off immediately and pulled wide around another parked vehicle so that in the narrow street it passed within 6 feet of Bass. The latter then saw that it was being driven by a male Negro, wearing a mustache and beard. At the trial Bass identified defendant as the driver of the yellow automobile. However, an attempt was made to impeach his identification by his admission that at the preliminary hearing he testified to an uncertain identification at the police lineup shortly after the attack on Mrs. Brooks, when defendant was beardless.

In his testimony Bass described the woman who ran from the alley as a Caucasian, slightly over 5 feet tall, of ordinary build, with her hair in a dark blonde ponytail, and wearing dark clothing. He further testified that her ponytail was "just like'' one which Janet had in a police photograph taken on June 22, 1964.

On the day of the robbery, Janet was employed as a housemaid in San Pedro. Her employer testified that she had arrived for work at 8:50 a.m. and that defendant had picked her up in a light yellow car(1)2 about 11:30 a.m. On that day, according to the witness, Janet was wearing her hair in a blonde ponytail but lighter in color than it appeared at trial.(2)


There was evidence from which it could be inferred that defendants had ample time to drive from Janet's place of employment and participate in the robbery. Defendants testified, however, that they went directly from her employer's house to the home of friends, where they remained for several hours....

At the seven-day trial the prosecution experienced some difficulty in establishing the identities of the perpetrators of the crime. The victim could not identify Janet and had never seen defendant. The identification by the witness Bass, who observed the girl run out of the alley and get into the automobile, was incomplete as to Janet and may have been weakened as to defendant. There was also evidence, introduced by the defense, that Janet had worn light-colored clothing on the day in question, but both the victim and Bass testified that the girl they observed had worn dark clothing.

In an apparent attempt to bolster the identifications, the prosecutor called an instructor of mathematics at a state college. Through this witness he sought to establish that, assuming the robbery was committed by a Caucasian woman with a blonde ponytail who left the scene accompanied by a Negro with a beard and mustache, there was an overwhelming probability that the crime was committed by any couple answering such distinctive characteristics. The witness testified, in substance, to the "product rule,'' which states that the probability of the joint occurrence of a number of mutually independent events is equal to the product of the individual probabilities that each of the events will occur.(3)8 Without presenting any statistical evidence whatsoever in support of the probabilities for the factors selected, the prosecutor then proceeded to have the witness assume(4)

9 probability factors for the various characteristics which he deemed to be shared by the guilty couple and all other couples answering to such distinctive characteristics.(5)10

Applying the product rule to his own factors the prosecutor arrived at a probability that there was but one chance in 12 million that any couple possessed the distinctive characteristics of the defendants. Accordingly, under this theory, it was to be inferred that there could be but one chance in 12 million that defendants were innocent and that another equally distinctive couple actually committed the robbery. Expanding on what he had thus purported to suggest as a hypothesis, the prosecutor offered the completely unfounded and improper testimonial assertion that, in his opinion, the factors he had assigned were "conservative estimates'' and that, in reality, "the chances of anyone else besides these defendants being there, ... having every similarity, ... is somewhat like one in a billion.''

Objections were timely made to the mathematician's testimony on the grounds that it was immaterial, that it invaded the province of the jury, and that it was based on unfounded assumptions. The objections were "temporarily overruled'' and the evidence admitted subject to a motion to strike. When that motion was made at the conclusion of the direct examination, the court denied it, stating that the testimony had been received only for the "purpose of illustrating the mathematical probabilities of various matters, the possibilities for them occurring or reoccurring.''

Both defendants took the stand in their own behalf. They denied any knowledge of or participation in the crime and stated that after Malcolm called for Janet at her employer's house they went directly to a friend's house in Los Angeles where they remained for some time. According to this testimony defendants were not near the scene of the robbery when it occurred. Defendants' friend testified to a visit by them "in the middle of June'' although she could not recall the precise date. Janet further testified that certain inducements were held out to her during the July 9 interrogation on condition that she confess her participation....

As we shall explain, the prosecution's introduction and use of mathematical probability statistics injected two fundamental prejudicial errors into the case: (1) The testimony itself lacked an adequate foundation both in evidence and in statistical theory; and (2) the testimony and the manner in which the prosecution used it distracted the jury from its proper and requisite function of weighing the evidence on the issue of guilt, encouraged the jurors to rely upon an engaging but logically irrelevant expert demonstration, foreclosed the possibility of an effective defense by an attorney apparently unschooled in mathematical refinements, and placed the jurors and defense counsel at a disadvantage in sifting relevant fact from inapplicable theory.

We initially consider the defects in the testimony itself. As we have indicated, the specific technique presented through the mathematician's testimony and advanced by the prosecutor to measure the probabilities in question suffered from two basic and pervasive defectsan inadequate evidentiary foundation and an inadequate proof of statistical independence. First, as to the foundational requirement, we find the record devoid of any evidence relating to any of the six individual probability factors used by the prosecutor and ascribed by him to the six characteristics as we have set them out in footnote 10, ante. To put it another way, the prosecution produced no evidence whatsoever showing, or from which it could be in any way inferred, that only one out of every ten cars which might have been at the scene of the robbery was partly yellow, that only one out of every four men who might have been there wore a mustache, that only one out of every ten girls who might have been there wore a ponytail, or tha any of the other individual probability factors listed were even roughly accurate.(6)12

The bare, inescapable fact is that the prosecution made no attempt to offer any such evidence. Instead, through leading questions having perfunctorily elicited from the witness the response that the latter could not assign a probability factor for the characteristics involved,(7)13 the prosecutor himself suggested what the various probabilities should be and these became the basis of the witness' testimony (see fn. 10, ante). It is a curious circumstance of this adventure in proof that the prosecutor not only made his own assertions of these factors in the hope that they were "conservative'' but also in later argument to the jury invited the jurors to substitute their "estimates'' should they wish to do so. We can hardly conceive of a more fatal gap in the prosecution's scheme of proof. A foundation for the admissibility of the witness' testimony was never even attempted to be laid, let alone established. His testimony was neither made to rest on his own testimonial knowledge nor presented by proper hypothetical questions based upon valid data in the record. (See generally: 2 Wigmore on Evidence (3d ed. 1940) 478, 650-652, 657, 659, 672-684; McCormick on Evidence, pp.19-20; State v. Sneed (1966) 76 N.M. 349, 414 P.2d 858.) In the Sneed case, the court reversed a conviction based on probabilistic evidence, stating: "We hold that mathematical odds are not admissible as evidence to identify a defendant in a criminal proceeding so long as the odds are based on estimates, the validity of which have [sic] not been demonstrated.'' (Italics added.) (414 P.2d at p.862.)

But, as we have indicated, there was another glaring defect in the prosecution's technique, namely an inadequate proof of the statistical independence of the six factors. No proof was presented that the characteristics selected were mutually independent, even though the witness himself acknowledged that such condition was essential to the proper application of the "product rule'' or "multiplication rule.'' To the extent that the traits or characteristics were not mutually independent (e.g., To the extent that the traits or characteristics were not mutually independent (e.g., Negroes with beards and men with mustaches obviously represent overlapping categories(8)15), the "product rule'' would inevitably yield a wholly erroneous and exaggerated result even if all of the individual components had been determined with precision. (Siegel, Nonparametric Statistics for the Behavioral Sciences (1956) 19; see generally Harmon, Modern Factor Analysis (1960).)

In the instant case, therefore, because of the aforementioned two defectsthe inadequate evidentiary foundation and the inadequate proof of statistical independencethe technique employed by the prosecutor could only lead to wild conjecture without demonstrated relevancy to the issues presented. It acquired no redeeming quality from the prosecutor's statement that it was being used only "for illustrative purposes'' since, as we shall point out, the prosecutor's subsequent utilization of the mathematical testimony was not confined within such limits.

We now turn to the second fundamental error caused by the probability testimony. Quite apart from our foregoing objections to the specific technique employed by the prosecution to estimate the probability in question, we think that the entire enterprise upon which the prosecution embarked, and which was directed to the objective of measuring the likelihood of a random couple possessing the characteristics allegedly distinguishing the robbers, was gravely misguided. At best, it might yield an estimate as to how infrequently bearded Negroes drive yellow cars in the company of blonde females with ponytails.

The prosecution's approach, however, could furnish the jury with absolutely no guidance on the crucial issue: Of the admittedly few such couples, which one, if any, was guilty of committing this robbery? Probability theory necessarily remains silent on that question, since no mathematical equation can prove beyond a reasonable doubt (1) that the guilty couple in fact possessed the characteristics described by the People's witnesses, or even (2) that only one couple possessing those distinctive characteristics could be found in the entire Los Angeles area.

As to the first inherent failing we observe that the prosecution's theory of probability rested on the assumption that the witnesses called by the People had conclusively established that the guilty couple possessed the precise characteristics relied upon by the prosecution. But 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.(9)16

The foregoing risks of error permeate the prosecution's circumstantial case. Traditionally, the jury weighs such risks in evaluating the credibility and probative value of trial testimony, but the likelihood of human error or of falsification obviously cannot be quantified; that likelihood must therefore be excluded from any effort to assign a number to the probability of guilt or innocence. Confronted with an equation which purports to yield a numerical index of probable guilt, few juries could resist the tional juror, and indeed only a defense attorney schooled in mathematics, could successfully keep in mind the fact that the probability computed by the prosecution can represent, at best, the likelihood that a random couple would share the characteristics testified to by the People's witnesses--not necessarily the characteristics of the actually guilty couple.

As to the second inherent failing in the prosecution's approach, even assuming that the first failing could be discounted, the most a mathematical computation could ever yield would be a measure of the probability that a random couple would possess the distinctive features in question. In the present case, for example, the prosecution attempted to compute the probability that a random couple would include a bearded Negro, a blonde girl with a ponytail, and a partly yellow car; the prosecution urged that this probability was no more than one in 12 million. Even accepting this conclusion as arithmetically accurate, however, one still could not conclude that the Collinses were probably the guilty couple. On the contrary, as we explain in the Appendix, the prosecution's figures actually imply a likelihood of over 40 percent that the Collinses could be "duplicated'' by at least one other couple who might equally have committed the San Pedro robbery. Urging that the Collinses be convicted on the basis of evidence which logically establishes no more than this seems as indefensible as arguing for the conviction of X on the ground that a witness saw either X or X's twin commit the crime.

Again, few defense attorneys, and certainly few jurors, could be expected to comprehend this basic flaw in the prosecution's analysis. Conceivably even the prosecutor erroneously believed that his equation established a high probability that no other bearded Negro in the Los Angeles area drove a yellow car accompanied by a ponytailed blonde. In any event, although his technique could demonstrate no such thing, he solemnly told the jury that he had supplied mathematical proof of guilt.

Sensing the novelty of that notion, the prosecutor told the jurors that the traditional idea of proof beyond a reasonable doubt represented "the most hackneyed, stereotyped, trite, misunderstood concept in criminal law.'' He sought to reconcile the jury to the risk that, under his "new math'' approach to criminal jurisprudence, "on some rare occasion ... an innocent person may be convicted.'' "Without taking that risk,'' the prosecution continued, "life would be intolerable ... because ... there would be immunity for the Collinses, for people who chose not to be employed to go down and push old ladies down and take their money and be immune because how could we ever be sure they are the ones who did it?''

In essence this argument of the prosecutor was calculated to persuade the jury to convict defendants whether or not they were convinced of their guilt to a moral certainty and beyond a reasonable doubt. Undoubtedly the jurors were unduly impressed by the mystique of the mathematical demonstration but were unable to assess its relevancy or value. Although we make no appraisal of the proper applications of mathematical techniques in the proof of facts, we have strong feelings that such applications, particularly in a criminal case, must be critically examined in view of the substantial unfairness to a defendant which may result from ill conceived techniques with which the trier of fact is not technically equipped to cope. We feel that the technique employed in the case before us falls into the latter category....

[W]e think that under the circumstances the "trial by mathematics'' so distorted the role of the jury and so disadvantaged counsel for the defense, as to constitute in itself a miscarriage of justice.... The judgment against defendant must therefore be reversed....


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].

Given 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-1), 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)N-1].

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-1]. Dividing this difference by the probability that C will occur in at least one couple (i.e., dividing the difference by [1 - (1 - Pr)N]) 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 ponytailed 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 conculsion 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)N-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 - 1/12,000,000)N] - [(N) X (1/12,000,000) X (1 - 1/12,000,000)N-1], and B designates the denominator [1 - (1 - 1/12,000,000)N].

N, 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 (3d ed. 1962); Hodges & Leymann, Basic Concepts of Probability and Statistics (1964); Lindgren & McElrath, Introduction to Probability and Statistics (1959).)

Do you agree with the result in Collins? With all the reasons the court gives for reversing the conviction? Would the same circumstantial evidence as was introduced in Collins, but without the probability testimony and argumentation, be sufficient to sustain defendant's conviction? Should probability evidence of the sort introduced in Collins and Sneed be admitted if the foundational problemsthe size of the sample, the establishment of realistic individual probabilities, and the certainty of mutual independencecan be overcome? Can these problems ever be overcome? Could the court take judicial notice of individual probabilities?

For more on the use of probability theory in trialsand in the Sneed and Collins cases in particularsee P. Tillers & E. Green, Probability and Inference in the Law of Evidence (1988); Finkelstein & Fairley, A Bayesian Approach to Identification Evidence, 83 Harv. L. Rev. 489 (1970); Tribe, Trial by Mathematics: Precision and Ritual in the Legal Process, 84 Harv. L. Rev. 1329 (1971); Finkelstein & Fairley, The Continuing Debate over Mathematics in the Law of Evidence: A Comment on "Trial by Mathematics,'' 84 Harv. L. Rev. 1801 (1971); Tribe, A Further Critique of Mathematical Proof, 84 Harv. L. Rev. 1810 (1971); Note, Evidence, etc., 1967 Duke L.J. 665. See also Kaplan, Decision Theory and the Factfinding Process, 20 Stan. L. Rev. 1065 (1968).

1. 2. Other witnesses variously described the car as yellow, as yellow with an off-white top, and yellow with an egg-shell white top. The car was also described as being medium to large in size. Defendant drove a car at or near the times in question which was a Lincoln with a yellow body and a white top.

2. 3. There are inferences which may be drawn from the evidence that Janet attempted to alter the appearance of her hair after June 18. Janet denies that she cut, colored or bleached her hair at any time after June 18, and a number of witnesses supported her testimony.

3. 8. In the example employed for illustrative purposes at the trial, the probability of rolling one die and coming up with a "2" is 1/6, that is, any one of the six faces of a die has one chance in six of landing face up on any particular roll. The probability of rolling two "2's" in succession is 1/6 X 1/6, or 1/36, that is, on only one occasion out of 36 double rolls (or the roll of two dice), will the selected number land face up on each roll or die.

4. 9. His argument to the jury was based on the same gratuitous assumptions or on similar assumptions which he invited the jury to make.

5. 10. Although the prosecutor insisted that the factors he used were only for illustrative purposesto demonstrate how the probability of the occurrence of mutually independent factors affected the probability that they would occur togetherhe nevertheless attempted to use factors which he personally related to the distinctive characteristics of defendants. In his argument to the jury he invited the jurors to apply their own factors, and asked defense counsel to suggest what the latter would deem as reasonable. The prosecutor himself proposed the individual probabilities set out in the table below. Although the transcript of the examination of the mathematics instructor and the information volunteered by the prosecutor at that time create some uncertainty as to precisely which of the characteristics the prosecutor assigned to the individual probabilities, he restated in his argument to the jury that they should be as follows:

Characteristic Individual Probability

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 1/1000

In his brief on appeal defendant agrees that the foregoing appeared on a table presented in the trial court.

6. 12. We seriously doubt that such evidence could ever be compiled since no statistician could possibly determine after the fact which cars, or which individuals, ``might'' have been present at the scene of the robbery; certainly there is no reason to suppose that the human and automotive populations of San Pedro, California, include all potential culpritsor, conversely, that all members of these populations are proper candidates for inclusion. Thus the sample from which the relevant probabilities would have to be derived is itself undeterminable. (See generally, Yaman, Statistics, An Introductory Analysis (1964), ch. I.)

7. 13. The prosecutor asked the mathematics instructor: ``Now, let me see if you can be of some help to us with some independent factors, and you have some paper you may use. Your specialty does not equip you, I suppose, to give us some probability of such things as a yellow car as contrasted with any other kind of car, does it? ... I appreciate the fact that you can't assign a probability for a car being yellow as contrasted to some other car, can you?'' A. ``No, I couldn't.''

8. 15. Assuming arguendo that factors B and E (see fn. 10, ante), were correctly estimated, nevertheless it is still arguable that most Negro men with beards also have mustaches (exhibit 3 herein, for instance, shows defendant with both a mustache and a beard, indeed in a hirsute continuum); if so, there is no basis for multiplying 1/4 by 1/10 to estimate the proportion of Negroes who wear beards and mustaches. Again, the prosecution's technique could never be meaningfully applied, since its accurate use would call for information as to the degree of interdependence among the six individual factors. Such information cannot be compiled, however, since the relevant sample necessarily remains unknown.

9. 16. In the instant case, for instance, the victim could not state whether the girl had a ponytail, although the victim observed the girl as she ran away. The witness Bass, on the other hand, was sure that the girl whom he saw had a ponytail. The demonstration engaged in by the prosecutor also leaves no room for the possibility, although perhaps a small one, that the girl whom the victim and the witness observed was, in fact, not the same girl.


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