THE PEOPLE,
Plaintiff and Respondent
v.
MALCOLM
RICARDO COLLINS, Defendant and Appellant
Docket No. Crim. 11176
Supreme Court of California
In Bank
March 11, 1968
68 Cal. 2d 319, 438 P.2d 33, 66 Cal.Rptr. 497 (1968)
APPEAL from a judgment of the Superior Court of Los
Angeles County. Maurice C. Sparling, Judge. Reversed.
Rex K. DeGeorge, under appointment by the Supreme
Court, for Defendant and Appellant.
Thomas C. Lynch, Attorney General, William E. James,
Assistant Attorney General, and Nicholas C. Yost, Deputy Attorney
General, for Plaintiff and Respondent.
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 incompatibility between the disciplines of law and
mathematics and intend no general disapproval or disparagement of the
latter as an auxiliary in the fact-finding processes of the former, we
cannot uphold the technique employed in the instant case. As we
explain in detail, infra, the testimony as to mathematical
probability infected the case with fatal error and distorted the
jury's traditional role of determining guilt or innocence according to
long-settled rules. Mathematics, a veritable sorcerer in our
computerized society, while assisting the trier of fact in the search
for truth, must not cast a spell over him. We conclude that on the
record before 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 (Pen. Code, §§ 211 Penal, 211a, 1157).
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
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.
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.
In the morning of June 22, Los Angeles Police
Officer Kinsey, who was investigating the robbery, went to defendants'
home. He saw a yellow Lincoln automobile with an off-white top in
front of the house. He talked with defendants. Janet, whose hair
appeared to be a dark blonde, was wearing it in a ponytail. Malcolm
did not have a beard. The officer explained to them that he was
investigating a robbery specifying the time and place; that the victim
had been knocked down and her purse snatched; and that the person
responsible was a female Caucasian with blonde hair in a ponytail who
had left the scene in a yellow car driven by a male Negro. He
requested that defendants accompany him to the police station at San
Pedro and they did so. There, in response to police inquiries as to
defendants' activities at the time of the robbery, Janet stated,
according to Officer Kinsey, that her husband had picked her up at her
place of employment at 1 p.m. and that they had then visited at the
home of friends in Los Angeles. Malcolm confirmed this. Defendants
were detained for an hour or two, were photographed but not booked,
and were eventually released and driven home by the police.
Late in the afternoon of the same day, Officer
Kinsey, while driving home from work in his own car, saw defendants
riding in their yellow Lincoln. Although the transcript fails to
disclose what prompted such action, Kinsey proceeded to place them
under surveillance and eventually followed them home. He called for
assistance and arranged to meet other
police officers in the vicinity of defendants' home.
Kinsey took a position in the rear of the premises. The other
officers, who were in uniform and had arrived in a marked police car,
approached defendants' front door. As they did so, Kinsey saw
defendant Malcolm Collins run out the back door toward a rear fence
and disappear behind a tree. Meanwhile the other officers emerged with
Janet Collins whom they had placed under arrest. A search was made for
Malcolm who was found in a closet of a neighboring home and also
arrested. Defendants were again taken to the police station, were kept
in custody for 48 hours, and were again released without any charges
being made against them.
Officer Kinsey interrogated defendants separately on
June 23 while they were in custody and testified to their statements
over defense counsel's objections based on the decision in Escobedo
and our first decision in Dorado.
According to the officer, Malcolm stated that he sometimes wore a
beard but that he did not wear a beard on June 18 (the day of the
robbery), having shaved it off on June 2, 1964.
He also explained two receipts for traffic fines totalling $35 paid on
June 19, which receipts had been found on his person, by saying that
he used funds won in a gambling game at a labor hall. Janet, on the
other hand, said that the $35 used to pay the fines had come from her
earnings.
On July 9, 1964, defendants were again arrested and
were booked for the first time. While they were in custody and
awaiting the preliminary hearing, Janet requested to talk with Officer
Kinsey. There followed a lengthy conversation during
the first part of which Malcolm was not present. During this time
Janet expressed concern about defendant and inquired as to what the
outcome would be if it appeared that she committed the crime
and Malcolm knew nothing about it. In general she indicated a wish
that defendant be released from any charges because of his prior
criminal record and that if someone must be held responsible, she
alone would bear the guilt. The officer told her that no assurances
could be given, that if she wanted to admit responsibility disposition
of the matter would be in the hands of the court and that if she
committed the crime and defendant knew nothing about it the only way
she could help him would be by telling the truth. Defendant was then
brought into the room and participated in the rest of the
conversation. The officer asked to hear defendant's version of the
matter, saying that he believed defendant was at the scene. However,
neither Janet nor defendant confessed or expressly made damaging
admissions although constantly urged by the investigating officer to
make truthful statements. On several occasions defendant denied that
he knew what had gone on in the alley. On the other hand, the whole
tone of the conversation evidenced a strong consciousness of guilt on
the part of both defendants who appeared to be seeking the most
advantageous way out. Over defense counsel's same objections based on
Escobedo and Dorado, some parts of the foregoing
conversation were testified to by Officer Kinsey and in addition a
tape recording of the entire conversation was introduced in evidence
and played to the jury.
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 blond
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.
Without presenting any statistical evidence whatsoever in
support of the probabilities for the factors selected, the
prosecutor then proceeded to have the witness assume
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.
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 something 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 re-occurring."
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.
Defendant makes two basic contentions before us:
First, that the admission in evidence of the statements made by
defendants while in custody on June 23 and July 9, 1964,
constitutes reversible error under the rules announced
in the Escobedo and Dorado decisions;
and second, that the introduction of evidence pertaining to the
mathematical theory of probability and the use of the same by the
prosecution during the trial was error prejudicial to defendant. We
consider the latter claim first.
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 defects — an 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 that any of the other individual probability
factors listed were even roughly accurate.
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,
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; Witkin,
Cal. Evidence (2d ed. 1966) § 771; McCormick on Evidence, pp. 19-20;
Evidence: Admission of Mathematical Probability Statistics
Held Erroneous for Want of Demonstration of Validity (1967)
Duke L.J. 665, 675-678, citing People v. Risley (1915)
214 N.Y. 75, 85 [108 N.E. 200, Ann. Cas. 1916 D 775]; 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." (See Note, supra, Duke
L.J. 665, 669-670, fn. 25.)
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),
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 defects — the inadequate evidentiary foundation and
the inadequate proof of statistical independence — the 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.
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
temptation to accord disproportionate weight to that index; only an
exceptional 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. (Pen. Code, § 1096 Penal.) 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 (see
People v. Jordan (1955) 45 Cal.2d 697, 707 [290 P.2d 484];
People v. Trujillo (1948) 32 Cal.2d 105, 109 [194 P.2d
681]; in a slightly differing context see Whitus v. Georgia
(1967) 385 U.S. 545, 552, fn. 2 [17 L.Ed.2d 599, 604, 87 S.Ct. 643];
Finkelstein, The Application of Statistical Decision Theory to
the Jury Discrimination Cases (1966) 80 Harv.L.Rev. 338,
338-340), 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. (See State v. Sneed, supra, 414
P.2d 858; Note, supra, Duke L.J. 665.) We feel that the
technique employed in the case before us falls into the latter
category.
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 case was apparently a close one. The jury began its
deliberations at 2:46 p.m. on November 24, 1964, and retired for the
night at 7:46 p.m.; the parties stipulated that a juror could be
excused for illness and that a verdict could be reached by the
remaining 11 jurors; the jury resumed deliberations the next morning
at 8:40 a.m. and returned verdicts at 11:58 a.m. after five ballots
had been taken. In the light of the closeness of the case, which as we
have said was a circumstantial one, there is a reasonable likelihood
that the result would have been more favorable to defendant if the
prosecution had not urged the jury to render a probabilistic verdict.
In any event, we 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. After an examination of the entire cause, including the
evidence, we are of the opinion that it is reasonably probable that a
result more favorable to defendant would have been
reached in the absence of the above error. (People v. Watson
(1956) 46 Cal.2d 818, 836 [299 P.2d 243].) The judgment against
defendant must therefore be reversed.
In view of the foregoing conclusion, we deem it
unnecessary to consider whether the admission of defendants'
extrajudicial statements constitutes error under the rules announced
in Escobedo and Dorado. Upon retrial, the admissibility
of these or any other extrajudicial statements sought to be introduced
by the prosecution must be determined in the light of the rules set
forth in Miranda v. Arizona (1966) 384 U.S. 436 [16
L.Ed.2d 694, 86 S.Ct. 1602, 10 A.L.R.3d 974]. (People v.
Doherty (1967) 67 Cal.2d 9, 12, 17-21 [59 Cal.Rptr. 857, 429 P.2d
177].) As we have pointed out, the trial herein took place between our
first and second Dorado decisions (see fn. 4, ante).
Although defense counsel was commendably alert in basing objections to
the admission of the statements upon the decisions in Escobedo
and Dorado, he of course did not have the benefit of our
numerous decisions beginning with the second Dorado decision
expounding various facets of the exclusionary rule. In the event any
extrajudicial statements made by defendant are offered in evidence on
retrial, the parties will have an opportunity to make a record on
pertinent issues subject to prior determination by the court in the
light of Miranda rules before such statements are received in
evidence. It would be fruitless for us to essay such a task at this
point when such record does not yet exist.
The judgment is reversed.
Traynor, C.J., Peters, J., Tobriner, J., Mosk, J.,
and Burke, J., concurred.
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]].
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 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)[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).)
