The defendant was convicted by a jury in the Superior Court of unarmed robbery of a person sixty years of age or older, in violation of G. L. c. 265, § 19 (a), and was sentenced to life in prison after the trial judge found him guilty of being a habitual offender, in violation of G. L. c. 279, § 25. The Appeals Court affirmed the conviction, Commonwealth v. Ferreira, 77 Mass. App. Ct. 675, 682 (2010), and we granted the defendant’s application for further appellate review. The issue presented on appeal is whether the prosecutor’s mathematical analysis in his closing argument of the probability that the lone eyewitness’s identification of the defendant was accurate created *782a substantial risk of a miscarriage of justice where the analysis was not supported by expert testimony, was fundamentally flawed, and equated proof beyond a reasonable doubt with a statistical probability. We conclude that where, as here, the prosecution’s case rested solely on an uncertain eyewitness identification and the defendant’s association with an admitted perpetrator of the crime, it did. We therefore reverse the judgment and remand the case to the Superior Court for a new trial.
Background. Because we evaluate whether the prosecutor’s error resulted in a substantial risk of a miscarriage of justice, we summarize the evidence at trial in some detail. On November 5, 2006, someone telephoned a pizza parlor and ordered a pizza for delivery to 356 June Street in Fall River. The victim delivery man, sixty-one year old Allan Course, left his vehicle between 7 and 7:30 p.m. on June Street with the pizza, and searched on foot for the delivery address. He heard someone say, “There he is,” and saw two men emerge from between two houses about twenty-five feet away and begin to walk quickly toward him. The victim noted that one man, who was approximately five feet, ten inches tall, and approximately 150 to 160 pounds, wore a hooded sweatshirt while the other man, who was taller (approximately five feet, ten or eleven inches) and stockier (approximately 175 to 190 pounds) wore a jacket.
When the men came within an arm’s length of the victim, the hooded man pushed the victim to the ground, placed his knee in the victim’s back, and said they wanted the victim’s money. The man wearing the jacket, who was standing behind the victim, then said that they did not want to hurt him and that he should just give them what was in his pockets and they would let him go. The hooded man showed the victim a paring knife and then took the victim’s wallet and money. After telling the victim, “Stay on the ground a couple of minutes,” the two men fled the scene. The victim waited a moment, got up, and ran to a nearby house to telephone the police.
Officer Brett Kimball responded, and the victim provided him with physical descriptions of the two men who had just committed the robbery. Officer Kimball put these descriptions into “our computer database,” and produced two photographic arrays of potential suspects, each consisting of six photographs, *783with one photograph per page.1 Later that evening, Officer Kimball showed the arrays to the victim at the pizza parlor. The officer instructed the victim to look through the entire array and then return to the first photograph and look through a second time before “making a selection.”2 Looking at the first array, the victim disregarded the officer’s instruction and, before even seeing the sixth photograph, selected the fifth photograph, later identified as Shawn Pacheco, as the “male with the gray hooded sweatshirt” who had “grabbed” him. The victim then looked at the second array, this time following the officer’s instruction, and selected the third photograph, which depicted the defendant, and identified him as “the one standing behind the guy with the sweatshirt.”3 At trial, the victim said he was “eighty per cent” sure of the second identification. However, when asked whether he could look around the court room “and say for sure whether or not the stockier one is here today,” the victim said he was “not sure” and did not identify the defendant.4
Pacheco and the defendant were friends who saw each other three or four times per week. The defendant dated Pacheco’s sister and was the father of her child. Officer Frederick M. Mello of the Fall River police department saw them together twelve times over a six-month period before November, 2006. Officer David Gouveia saw the defendant on March 2, 2006, carry Pacheco, who was unconscious, to an ambulance.
Pacheco, who was called to the witness stand by the prosecutor under an order of immunity, testified that he committed the *784robbery with Robert Dias, not the defendant. However, Pacheco had earlier pleaded guilty to the robbery and, during the plea colloquy, admitted the facts recited by the prosecutor in presenting the factual basis for the plea, which included the prosecutor’s assertion that the defendant was with Pacheco when they approached the victim and demanded his money.5
The defendant called Dias to the witness stand, who testified, without a grant of immunity, that he lived on June Street on November 5, 2006, and committed the robbery with Pacheco; he said the defendant was not with them.6 The defendant also called Kristen Bennett, who testified that the defendant was at her home, a “good eight minutes” drive from June Street, with his girl friend from 11:30 a.m. on November 5, 2006, until the next morning, baby-sitting Bennett’s children while she recuperated from a medical procedure she had on November 4.7
Defense counsel’s closing argument noted that the victim was “only eighty per cent sure” of his identification of the defendant in the photographic array. The prosecutor began his closing argument with the eighty per cent figure mentioned by defense counsel:
“I’d like to start off by agreeing with [defense counsel] on one point. If all we had in this case was [the victim] saying, ‘I’m about eighty per cent sure that this photograph is the person who robbed me,’ you probably wouldn’t be deliberating very long this morning. You’d go upstairs, you’d say eighty per cent [is] not high enough, not guilty, we’re done. But that’s not where this case stops.”
After arguing why the jury should discredit the testimony of Pacheco and Dias, and discussing the victim’s selection of the defendant’s photograph from the array, the prosecutor concluded by saying he would “like to make one other point which I think really really [sic] drives this case home.” He then argued:
*785“He’s shown six photos in the first array, six photos in the second array. Well, we also heard that there was a seventh option in each array. That was none of the above. An option [the victim] had, an option he didn’t take. So now you have two arrays with each seven options.[8] . . .
“Now, let’s think about this for a moment. Seven on the left, seven on the right. How many different combinations does that make? By my math, it’s forty-nine. . . . And who does he pick? Bosom buddies. Two men who’ve been seen hanging around together at least a dozen times by Officer Mello. He picks [the defendant] who is so close to Shawn Pacheco that he was physically carrying him to an ambulance in March 2006. [The defendant] who has a child with Shawn Pacheco’s sister. The odds of picking two men out of two arrays with forty-nine different combinations who are that type, one out of forty-nine. Two per cent. Two per cent. What are the odds that [the victim] would have picked two different people, some other combination? Ninety-eight per cent. Do you call that coincidence? I call that proof beyond a reasonable doubt.”
The defendant did not object to the prosecutor’s closing argument.
Discussion. The gist of the prosecutor’s argument to the jury was that there was a one in forty-nine chance that the victim would have identified as his assailants two persons who knew each other well and therefore a ninety-eight per cent probability that the victim had accurately identified the defendant as one of the assailants, which constituted proof beyond a reasonable doubt. The apparent simplicity of the mathematics belies the complexity of the conclusion regarding the probability of an accurate identification, and conceals the assumptions implicit in the conclusion.
It is true that, if the victim were shown two arrays of six photographs and given a seventh choice of “none of the above,” and if the victim closed his eyes and randomly selected one of the seven options in each array, there was a one in forty-nine chance that he would select any combination of two. But if the identification were truly random, it would have no evidentiary *786consequence. The victim’s eyewitness identification has potential evidentiary consequence only to the extent that it is not random, but reflects his recognition of two persons who had robbed him approximately ninety minutes before the identification procedure. For this reason alone, the prosecutor’s probability analysis is false and misleading. And without the implicit assumption of random selection, the probability analysis, at a minimum, becomes far more complex.
There are two other fundamental problems with the probability analysis. First, the victim testified that he saw the assailant in the hooded sweatshirt “a lot more,” and picked him from the array “[a]lmost right away.” Having identified Pacheco from the first array, the probability that the victim would randomly select the photograph of a person Pacheco knew well from the second array depended on how many people in the second array Pacheco knew well. But there was no evidence whether Pacheco knew the other persons depicted in the array, even though the prosecutor obtained an order of immunity and called Pacheco to testify. Nor was there any evidence as to the source of the photographs that comprised the array, apart from Officer Kimball’s testimony that he put the victim’s physical descriptions of the assailants into “our computer database” and “look[ed] up” people who had been arrested.9 If the computer database only contained photographs in the possession of the Fall River police department, and if the photographs were selected to reflect persons similar in age to Pacheco, it would not be surprising if Pacheco knew others whose photographs were in the second array apart from the defendant. Therefore, implicit in the prosecutor’s argument was the factual representation that the defendant was the only person in the second array whom Pacheco knew, a fact not in evidence. See People v. Collins, 68 Cal. 2d 319, 325, 327 (1968) (mathematician’s opinion that there was one chance in twelve million that another couple possessed distinctive characteristics of two defendants had inadequate evidentiary foundation).
Second, where the Commonwealth seeks to provide the jury *787with an analysis of mathematical probability, especially the mathematical probability that an eyewitness identification is accurate, the analysis must be offered through expert testimony, not the prosecutor’s closing argument. See Commonwealth v. Gomes, 403 Mass. 258, 275 (1988) (opinion based on statistical evidence admitted through expert testimony “where the statistical evidence is shown to be based on accepted scientific principles”), and cases cited; Commonwealth v. Drayton, 386 Mass. 39, 50 (1982) (opinion based on statistics “offered as evidence of identity, raises special problems”). If the prosecutor had offered this opinion to the jury through the testimony of an expert witness, the defendant could have challenged the admissibility of the expert opinion under Daubert v. Merrell Dow Pharms., Inc., 509 U.S. 579, 585-595 (1993); and Commonwealth v. Lanigan, 419 Mass. 15, 24-26 (1994). The judge, as gatekeeper, would then have had the opportunity to evaluate whether the Commonwealth had established the five foundational requirements necessary to admit expert testimony in a criminal case, including whether the expert’s opinion is based on facts or data of a type reasonably relied on by experts to form opinions in the relevant field, whether the theory underlying the opinion is reliable, and whether the theory is applied to the particular facts of the case in a reliable manner. See Commonwealth v. Bar-bosa, 457 Mass. 773, 783 (2010). We have no doubt that the opinion articulated by the prosecutor would not have met these requirements and therefore would not have been admitted in evidence. See People v. Collins, supra at 328 (mathematician’s opinion had “glaring defect” — “inadequate proof of the statistical independence” of factors claimed to be mutually independent). The prosecutor cannot use closing argument to provide the jury with an opinion based on probability analysis that would not be admissible through an expert.
The prosecutor also erred in equating proof beyond a reasonable doubt with a numerical percentage of the probability of guilt, in this case, ninety-eight per cent. “[T]o attempt to quantify proof beyond a reasonable doubt changes the nature of the legal concept of ‘beyond a reasonable doubt,’ which seeks ‘abiding conviction’ or ‘moral certainty’ rather than statistical probability.” Commonwealth v. Rosa, 422 Mass. 18, 28 (1996). “The idea of *788reasonable doubt is not susceptible to quantification; it is inherently qualitative.” Commonwealth v. Sullivan, 20 Mass. App. Ct. 802, 806 (1985). See Commonwealth v. Mack, 423 Mass. 288, 291 (1996) (“the concept of reasonable doubt is not a mathematical one”).
Where, as here, no objection was made at trial, we must determine whether the prosecutor’s closing argument error resulted in a substantial risk of a miscarriage of justice. See Commonwealth v. Freeman, 352 Mass. 556, 563-564 (1967). See also Commonwealth v. Kozec, 399 Mass. 514, 518 & n.8 (1987). The error here was particularly dangerous because it carried the aura of mathematical simplicity and precision — that the determination of guilt was as simple and inexorable as multiplying 1/7 by 1/7. See Commonwealth v. Ferreira, 11 Mass. App. Ct. 675, 685 n.6 (2010) (Milkey, J., dissenting) (“superficial plausibility of the prosecutor’s argument masked its profound flaws”). “[T]he very mystery that surrounds mathematical arguments — the relative obscurity that makes them at once impenetrable by the layman and impressive to him — creates a continuing risk that he will give such arguments a credence they may not deserve and a weight they cannot logically claim.” Tribe, Trial By Mathematics: Precision and Ritual in the Legal Process, 84 Harv. L. Rev. 1329, 1334 (1971).
We conclude that the prosecutor’s closing argument error created a substantial risk of a miscarriage of justice because of the danger that the jury gave undue weight to a mathematical probability analysis that supposedly demonstrated that the lone eyewitness identification on which the prosecutor’s case wholly rested constituted proof beyond a reasonable doubt, the victim’s admitted uncertainty as to the accuracy of the identification, and our recognition that “[e]yewitness identification of a person whom the witness had never seen before the crime or other incident presents a substantial risk of misidentification and increases the chance of a conviction of an innocent defendant.” Commonwealth v. Silva-Santiago, 453 Mass. 782, 796 (2009), quoting Commonwealth v. Jones, 423 Mass. 99, 109 (1996). Our conclusion is strengthened by the evidence of the defendant’s innocence: Pacheco’s testimony that Dias, not the defendant, committed the robbery with him; Dias’s testimony that he committed the robbery with Pacheco, and Bennett’s testimony *789that the defendant was baby-sitting her children on the evening of the robbery. While the jury apparently did not credit this evidence, it cannot be ignored in evaluating whether there was a substantial risk of a miscarriage of justice.10
Conclusion. Because the prosecutor’s closing argument error created a substantial risk of a miscarriage of justice, the judgment is reversed, the verdict and the habitual offender finding are set aside, and the case is remanded to the Superior Court for a new trial.
So ordered.