Joanne, a friend of the defendant’s former girl friend, and her twelve-year-old daughter, Alyssa, were found stabbed to death in their apartment on January 7, 2004.2 The defendant was convicted by a jury in the Superior Court of murder in the first degree of both victims. As to Joanne, he was found guilty on the theories of extreme atrocity or cruelty and felony murder. The jury found the defendant guilty of Alyssa’s murder on both of those theories and on a theory of deliberate premeditation as well. He was also convicted of one count of aggravated rape of Joanne. The defense at trial was that a third party, most likely the victims’ landlord, committed the murders, and that the police investigation was inadequate.
*402On appeal, the defendant asserts that errors occurred at trial involving the introduction of scientific evidence; the admission of expert testimony by one expert on the results of a test performed by another; and the exclusion of certain evidence related to the victims’ landlord that the defendant contends would have assisted him in presenting both an inadequate investigation defense and a third-party culprit defense. The defendant requests also that we exercise our power under G. L. c. 278, § 33E, to reduce the convictions to a lesser degree of guilt or, in the alternative, to grant him a new trial.
We reject the defendant’s claims, affirm his convictions of murder in the first degree, and, after review of the entire record pursuant to G. L. c. 278, § 33E, decline to exercise our power to grant extraordinary relief. Because the defendant was convicted of the rape victim’s murder on theories of both felony-murder and extreme atrocity and cruelty, the judgment on the indictment charging aggravated rape is not duplicative, and that judgment is affirmed. See Commonwealth v. Felder, 455 Mass. 359, 370-371 (2009), citing Commonwealth v. Brown, 441 Mass. 199, 200 n.l (2004).
1. Facts and procedural background. We recite the facts that the jury could have found, reserving some details for later discussion. During the summer and fall of 2002, the defendant lived with Bobbie Jo Miller, the mother of their infant daughter, in Woburn. In October, 2002, their romantic relationship ended and the defendant moved to a rooming house in Lowell. However, Miller and the defendant continued to have a cordial relationship and he frequently visited their daughter. In the spring of 2003, Miller became friends with Joanne, who lived with her two children on the same street in Woburn. Miller introduced the defendant to Joanne and the three socialized together on occasion.
In November, 2003, Joanne and her children moved into a two-family duplex across the street from where they had been living. The owner of the house, Harinder Singh, his wife, and their children occupied the other unit in the building. The defendant came to the victims’ apartment on a few occasions to assist *403Joanne with household tasks such as fixing cabinets and connecting the cable television in her new apartment.
On the evening of January 4, 2004, the defendant called Miller to arrange a visit with his daughter; Miller refused him permission to visit because she suspected he was “under the influence.” Joanne came to Miller’s house around 7:30 p.m.; she seemed upset and appeared to have been crying. She told Miller that she “needed to get out of the house,” but left after a few minutes. Joanne was not seen alive after that point. Computer records indicated that she used the Internet until approximately 9:30 p.m. She spoke by telephone with Singh and his wife at approximately 10 p.m. Although they did not talk in person, the defendant telephoned Miller several times that evening from his home, and she telephoned him at approximately 11:30 p.m. The defendant’s home telephone was in use until approximately 11:45 p.m.
On January 5, 2004, Alyssa was absent from school and the defendant did not go to work. He told his supervisor and Miller’s mother that he had been up all night fighting with Miller and that he overslept.3 He told Miller that he missed work because he was “tired and hung over.” The defendant reported to work as usual on January 6, 2004. Between January 5 and January 7, 2004, Joanne’s car did not move from her driveway and the shades on her house remained drawn. Telephone calls to Alyssa and Joanne were unanswered.
On January 7, when Joanne did not respond to Miller’s persistent knocking, Miller contacted Joanne’s parents. They entered Joanne’s apartment through the kitchen door, which was open but showed no signs of forced entry. They discovered her partially clad body face down on the living room sofa; she was lying in a pool of blood, dressed only in a black shirt and white sweater that had been sliced open.4 Her wrists and ankles bore ligature marks. A rope with duct tape attached was tied around one wrist and there was duct tape on her sweater. Her underwear, *404a rolled towel containing her saliva, a comforter, and a pillow were nearby. The medical examiner determined that Joanne had been gagged with the towel, bound, and sexually assaulted. The cause of death was seven stab wounds to the neck, one of which nearly severed a carotid artery, and a “contributing factor” of blunt force trauma to the head.
Alyssa’s body was found on the floor in an upstairs bedroom, also in a pool of blood. The room showed signs of a severe struggle. Blood spatter and handprints on the wall, defensive wounds on Alyssa’s hands and arms, and blood under her fingernails indicated that she had fought strenuously with her attacker, attempted to escape, and been stabbed at several different locations in the room. Trauma to her face, and damage to the inside of her lip from her braces, showed that she had been punched repeatedly in the face. The medical examiner determined that Alyssa died of stab wounds to the neck that severed both carotid arteries.
Joanne’s two year old son was found in his crib in his bedroom. He was dehydrated and wearing a heavily soiled diaper, but was otherwise unharmed.
Police investigation focused initially on Singh.5 However, Singh’s deoxyribonucleic acid (DNA) profile was excluded as a match to the DNA found in semen on Joanne’s body and clothing. Several days after the discovery of the bodies, police found and developed several rolls of film from Joanne’s apartment. They questioned her friends, particularly Miller, about the individuals in the photographs, a number of which included the defendant. Based on these photographs and interviews, police sought to question the defendant. He agreed to speak with police at his home. He gave conflicting statements about his relationship with Joanne, whom he claimed not to know well; denied having had a sexual relationship with her; and stated that he did not own a vehicle. He had cuts on the knuckles of both hands and was unable to explain their origin.
After a hearing, a Superior Court judge ordered the defendant to produce a DNA sample.6 The sample resulted in a confirma*405tory match to DNA in semen found on Joanne and on her clothing. The defendant was arrested following the results of the confirmatory match. Later, testing of blood beneath Alyssa’s fingernails could not exclude the defendant as a contributor to that blood.
2. Admission of DNA evidence. The defendant asserts that DNA profile results obtained from evidence found on both victims should not have been admitted. Citing Daubert v. Merrell Dow Pharms., Inc., 509 U.S. 579 (1993), and Commonwealth v. Lanigan, 419 Mass. 15, 24-26 (1994), he claims that the statistical methods employed by the Commonwealth in interpreting the DNA evidence are unreliable, prejudicial, and irrelevant. Alternatively, the defendant contends that the judge erred by declining to conduct an evidentiary hearing on the admissibility of the DNA evidence.
During the course of the investigation, DNA from semen found on the adult victim’s body was compared against DNA profiles of convicted offenders stored in the Combined DNA Indexing System (CODIS) database. A DNA match resulting from such a database comparison is known as a “cold hit.” The CODIS database revealed a possible match between the defendant’s DNA profile and the DNA found on the adult victim. Comparison of DNA in a blood sample taken from the defendant after a hearing, and the DNA from the semen on the adult victim’s body, resulted in what is known as a “confirmatory match.”7 The DNA expert testified that the likelihood of a random individual having the same profile was one in 101.5 quadrillion in the Caucasian population,8 one in 1.503 quintillion in *406the African American population, and one in 329.8 quadrillion in the Hispanic population. The DNA expert stated that a quadrillion is one million times the population of the earth. A partial match to the defendant’s DNA was also found in blood under the child victim’s fingernails. That blood sample was small and degraded, yielding a partial Y-STR profile match to the defendant’s DNA at seven of twelve loci.9 While the defendant could not be excluded from that partial DNA profile, 99.6 per cent of the general Caucasian population could be excluded as contributors to the sample.10
Prior to trial, the defendant filed a motion in limine to exclude all of the DNA evidence. That motion was denied after a nonevidentiary hearing. The defendant then filed a motion in limine to exclude the Y-STR DNA evidence found under the child victim’s fingernails. A second motion judge denied that motion on the ground that the defendant had not shown that there was disagreement in the scientific community on the validity and reliability of the Y-STR testing used on that sample. At trial, the defendant filed a motion in limine to exclude any results from the CODIS database testing, see note 6, supra; that motion was allowed on the Commonwealth’s representation that it did not intend to introduce the results of the CODIS match.
The defendant claims that, even though the jury heard no evidence of the CODIS search and its results,11 the admission of the confirmatory match results was error. The defendant contends that there is a significant debate in the scientific community concerning the preferable statistical method for describing the likelihood of a confirmatory match after a cold hit database match. He argues that random match probability (RMP), the statistical method used to determine the probability of a randomly selected individual in the general population having contributed to the DNA profile in the sample, while generally accepted in the scientific community for a “probable cause” match (a match *407where a single suspect’s DNA is tested and found to match DNA from a crime scene), is unreliable and more prejudicial than probative when it is used after an earlier cold hit.12
There was no error in the first motion judge’s decision that a Lanigan hearing was not necessary and that the DNA evidence was admissible. As that judge stated in denying the defendant’s motion, the use of the DNA analysis involved here is generally accepted within the scientific community. See Commonwealth v. Rosier, 425 Mass. 807, 813, 815-817 (1997). There is no scientific debate over the methodology by which such DNA matches are obtained or their validity. See Commonwealth v. Gaynor, 443 Mass. 245, 268 (2005). See also Commonwealth v. Dixon, 458 Mass. 446, 453-454 (2010).
The defendant’s claim that there is a dispute concerning which statistical measure to use in analyzing the significance of a confirmatory match following a cold hit confuses the purpose of the statistical measures, which are not in conflict.13 The dif*408feting statistical methods that the defendant discusses, including the RMP (or rarity calculation), are each used (separately or in combination) to describe the significance of such a hit. There is no debate over the calculation involved in each method, what it measures, or its reliability. The scientific debate concerns only which method or combination of methods is most relevant in describing the statistical significance of this type of match. See United States v. Jenkins, 887 A.2d 1013, 1022-1025 (D.C. 2005). Statisticians agree that the question of “ascertainment bias”14 created by a cold hit match is a significant issue, and that presentation of the significance of the cold hit match must account for ascertainment bias in some manner. See id. As the first motion judge concluded, however, questions of relevancy do not involve the validity or reliability of the testing method. The judge observed that the defendant would be “free to challenge the relevance of rarity statistics” at trial, and the defendant did so on cross-examination of both DNA experts.
Moreover, the defendant’s argument would eviscerate the purpose for which the Legislature created the CODIS database. The defendant claims, in essence, that the statistical significance of the probability of a given DNA profile in the general population, widely accepted in the scientific community and courts throughout the United States (and bearing no relationship to whether an individual’s DNA profile is in a CODIS database), somehow becomes irrelevant and unreliable when the individual involved was first identified through a CODIS database. Were we to accept the defendant’s argument, DNA evidence from convicted *409offenders whose DNA is stored in a CODIS database could never be used at trial unless it was obtained without using the CODIS database. Exclusion of DNA evidence from convicted offenders was surely not the result intended by the Legislature in enacting G. L. c. 22E, § 3, to create a database designed to aid law enforcement personnel in investigating serious crimes.
Having concluded that there was no error in the motion judges’ decisions prior to trial to permit introduction of the DNA evidence, we turn to the admission of the DNA evidence presented at trial. See Commonwealth v. Mathews, 450 Mass. 858, 872 n.15 (2008) (judge accorded substantial discretion in deciding whether evidence is relevant, and, if so, whether it should nevertheless be excluded as less probative than prejudicial). Pursuant to the defendant’s third motion to exclude DNA evidence (his only motion with regard to the DNA evidence that was made at trial), all mention of the CODIS database was excluded15 and the jury heard only testimony concerning the DNA evidence obtained from the defendant by court order. See Commonwealth v. Guy, 454 Mass. 440, 447 (2009) (“Although [the CODIS database] ‘hit’ focused attention on the defendant and led to the Commonwealth’s obtaining a new sample of his DNA which then was sent to the . . . laboratory, it was the comparison by the . . . laboratory of his known DNA sample with the male DNA samples from the crime scene that led to the critical match used at trial”).
We have adhered to the rule that test results showing a DNA match should not be admitted without statistical evidence of the likelihood of that match occurring. See Commonwealth v. Curnin, 409 Mass. 218, 222 n.7 (1991). In Commonwealth v. Mattei, 455 Mass. 840, 851-855 (2010), we held that DNA evidence that a particular individual could not be excluded as a potential contributor of the DNA at issue should not be admitted without *410accompanying statistical evidence of the likelihood that the test could not exclude other individuals in a given population.
Here, each of the two DNA experts who extracted the DNA profiles (one from the sperm and one from the blood samples) testified at trial. The former stated that the defendant’s DNA matched DNA samples taken from the adult victim, and the latter testified that the defendant could not be excluded from DNA found on the twelve year old victim. As discussed supra, each described in detail the statistical likelihood that a randomly selected individual in the general population could be excluded as a contributor to these samples.
3. Chemist’s report. The defendant contends that the admission of testimony by a State police chemist, Eugene Hagan, who examined evidence at the crime scene but was not the expert who conducted the tests at issue, violated his rights under the Sixth Amendment to the United States Constitution to confront and cross-examine witnesses. See Crawford v. Washington, 541 U.S. 36, 40 (2004). In addition to describing his own investigation, Hagan testified concerning the results of a report prepared by another State police chemist, Thomas Sendlenski, who performed tests on a number of objects found near Joanne’s body. These tests included two methods of screening for the possible presence of semen. Sendlenski found no evidence of semen on the sofa on which Joanne was lying, the pillows near her, or the comforter that an initial responder had used to partially cover her body. The absence of semen on the comforter, in particular, was contrary to the defense theory that the defendant and Joanne had engaged in consensual sexual relations several days before the crime, and that semen from the comforter had been transferred to her shirt when rescue personnel picked it up to cover her.
The defendant did not object to the admission of this testimony; therefore, we review the defendant’s claim to determine whether there was error and, if so, whether the error created a substantial likelihood of a miscarriage of justice. See Commonwealth v. Avila, 454 Mass. 744, 763 (2009).
Hagan’s testimony concerning the presence of semen at the crime scene should not have been admitted. We have held that expert opinion testimony based on hearsay does not violate a *411defendant’s right to confront witnesses against him, and is admissible, so long as the hearsay is independently admissible and is of a kind customarily relied on by experts in that field as a basis for opinion testimony, and the expert does not testify to the details of the hearsay on direct examination. See Commonwealth v. Banville, 457 Mass. 530, 541-542 (2010). The expert giving the opinion may testify to his or her own actions and observations, the identification of the hearsay material on which he or she relied, the reliance by professionals within his or her area of expertise on such material, and his or her own opinion. Id. The expert is subject to cross-examination on these matters. See Commonwealth v. Nardi, 452 Mass. 379, 390 (2008) . See also Commonwealth v. Avila, 454 Mass. 744, 761-762 (2009); Commonwealth v. Hensley, 454 Mass. 721, 732 (2009) .
Here, Hagan testified that he had reviewed Sendlenski’s report, and that it was the type of report on which chemists are accustomed to rely. Had Sendlenski testified, his findings would have been independently admissible. However, Hagan did not rely on Sendlenski’s findings in presenting his own opinion. Rather, Hagan recited the results of Sendlenski’s tests and presented to the jury Sendlenski’s underlying factual findings. Hagan stated that, although he worked overnight at the crime scene, he left the scene while the tests were being conducted. He described in detail the tests used to detect the presence of semen, and, relying on the report, presented the locations at which semen was or was not found. This testimony was not permissible opinion testimony based on the results of tests performed by another expert. The testimony introduced factual findings through the hearsay notes of a nontestifying expert, and should not have been admitted. See Commonwealth v. Avila, supra at 762. In addition, the prosecutor emphasized this testimonial hearsay as fact during her closing argument, when she referred to the absence of semen on the comforter to refute the defendant’s argument that he had had consensual sex with Joanne several days before the crime.
Nonetheless, the admission of this evidence could have had little or no effect on the verdicts, and there was no substantial likelihood of a miscarriage of justice. See Commonwealth v. Linton, 456 Mass. 534, 560 (2010). The properly admitted *412evidence against the defendant was overwhelming. DNA matching the defendant’s DNA profile was found in semen at two locations in and on the adult victim’s body, and one location on her clothes. The semen on her clothes was on a shirt that that victim had been wearing on the last day she was seen alive, and that had been ripped open in front. The semen had been deposited no more than five to seven days before the sample was taken; a DNA expert testified that it would have been eliminated by physical activity or bathing. The semen had a very rare DNA profile; as stated, see part 2, supra, the probability of a randomly selected, unrelated individual matching it was one in 101.5 quadrillion of the Caucasian population. Similarly, whereas 99.6 per cent of the Caucasian population could be excluded, the defendant could not be excluded from the partial DNA profile from the blood under the child victim’s fingernails. See part 2, supra.
In addition to the DNA evidence, other physical evidence connected the defendant to the crime. Rope consistent with the type of rope found on the adult victim’s arm was discovered in the truck, customarily driven by the defendant, that was parked outside the defendant’s rooming house. Police recovered bloody jeans from the defendant’s room; the blood matched the defendant’s.16
There was also substantial consciousness of guilt evidence. Police found a letter in the defendant’s room addressed to his daughter, dated January 5, 2004, stating that he had done “some bad things” and was taking “the easy way out,” but would be watching her from a star. On the night of January 7, 2004, after the victims’ bodies were discovered, the defendant went to visit Miller, stating that he wanted to console her. He had hives on his forehead which appeared when he was angry or nervous. He also had cuts on his hands, and, in Miller’s opinion, he looked “stressed out.” When Miller asked him what was wrong, the defendant said that he had “a lot on his mind.”
Moreover, the erroneously admitted testimony (no semen present on the sofa, the comforter, and the pillows) was not incriminating because it, by itself, did not tie the defendant to the crime. The defendant contends, however, that he would have benefited from the presence of semen on the comforter *413and therefore that the evidence of the lack of semen on the comforter impaired his defense.
He reasons in that regard that Joanne customarily slept on the couch and used the comforter and pillows found near her body as bedding. The defendant’s counsel suggested in his closing (with no basis in the evidence) that the defendant had consensual sex with Joanne on January 2, 2004, and that semen from that hypothesized sexual encounter remained on the comforter for days afterward. Counsel contended that the semen was transferred to Joanne’s shirt when the comforter was placed over her by initial responders, and that this would explain the presence of the defendant’s DNA on her shirt. But absent the presence of semen on the bedding, this theory became untenable, and the jury were left with the logical alternative that the defendant’s semen was deposited on Joanne’s shirt at the time she was killed. The entire argument fails for lack of any evidentiary support for the theory, there being no evidence whatever that Joanne and the defendant had a consensual sexual encounter on January 2, 2004, or at any other time.
4. Evidence of the adequacy of the police investigation. The defendant contends that he was deprived of his constitutional right to challenge the police investigation, see Commonwealth v. Avila, supra at 753, when the judge excluded certain evidence, detailed infra, regarding Joanne’s landlord, Singh. As stated, Singh was the initial target of the police investigation. The defendant claims that, although police interviewed Singh, searched his house cursorily, and obtained a DNA sample from him, once they learned of the defendant’s DNA profile match they dropped their investigation of Singh and focused only on the defendant. See part 5, infra. The defendant argues that the excluded evidence would have demonstrated to the jury that the police should have investigated Singh more extensively and would have supported the defendant’s claim that the police investigation was inadequate.
A defense of inadequate police investigation suggests to the jury “that the evidence at trial may be inadequate or unreliable because the police failed to conduct the scientific tests or to pursue leads that a reasonable police investigation would have conducted or investigated,” with the result that the police may have missed “significant evidence of the defendant’s guilt or *414innocence.” Commonwealth v. Silva-Santiago, 453 Mass. 782, 801 (2009). “Under the so-called Bowden defense, a defendant [may] challenge the adequacy of a police investigation and may use information concerning third-party culprits to question whether the police took reasonable steps to investigate the crime.” Commonwealth v. Ridge, 455 Mass. 307, 316 (2009), citing Commonwealth v. Bowden, 379 Mass. 472, 486 (1980). Because any statements introduced as part of such a defense are offered not for their truth, but to prove that the police did not take “reasonable steps to investigate,” those statements are not hearsay. Commonwealth v. Ridge, supra, citing Commonwealth v. Silva-Santiago, supra at 802. Evidence is admissible to show inadequate police investigation, however, only if police learned of it during the course of their investigation. Commonwealth v. Silva-Santiago, supra at 803. In deciding whether to admit such evidence, the judge must determine “whether the probative weight of the Bowden evidence exceeds the risk of unfair prejudice to the Commonwealth from diverting the jury’s attention to collateral matters.” Id.
The defendant objects first to the judge’s exclusion of statements made by Joanne to her friends and neighbors, which statements were then reported by those individuals to police and to the defendant’s investigator, regarding the relationship between Joanne and Singh. Several individuals told police that Joanne had started a new, “hush-hush” romantic relationship with an unidentified male. One friend told police that Joanne had said the man involved was Singh, and that friend suggested that secrecy was required because Singh was married. Another friend said that Singh had promised to build Joanne a deck, had offered to buy her a bed, and that he was “too nice” to her. After another individual informed police that Joanne said Singh entertained her on a futon in his basement, where they drank scotch, police obtained Singh’s consent for a search of the basement and found a futon, scotch, and glasses.17
At trial, the Commonwealth moved to exclude all “rape shield evidence,” see G. L. c. 233, § 21B (the rape shield statute),18 *415and sought to exclude all statements that Joanne was having “an affair” with Singh on the grounds that such statements were hearsay and “potentially” fell within the rape shield statute.19 The judge granted the defendant’s motion in limine to “allow evidence of inadequate investigation” but excluded “the hearsay statement that the victim was having an affair with the landlord” unless the defendant filed a motion pursuant to the rape shield statute and that motion were allowed.20 In addition, the judge denied a motion in limine by the defendant seeking to cross-examine a friend of Joanne regarding statements the friend made to the police about Joanne’s relationship with Singh.
We need not determine whether the judge was correct in finding the rape shield statute applicable in these circumstances. The judge acted within her discretion in excluding the statements because they did not meet the balancing requirement set forth in Commonwealth v. Silva-Santiago, supra at 801-804.21 The statements suggesting the existence of an alleged romantic relationship between Joanne and an unidentified male were of extremely dubious probative value and could have distracted the jury, to the Commonwealth’s prejudice. As the judge said, alleged statements that Joanne was having an “affair” did not constitute tips received by police that someone else committed the offense and provided no more than a “feeble” indication, see id. at 801, that police might have had reason to investigate *416such an individual. The innuendo that the unidentified man was Singh was similarly not probative. There was no evidence of any hostility on the part of Singh or motive that he might have had to harm the victims. See Commonwealth v. Reynolds, 429 Mass. 388, 391-392 (1999).
The defendant objects also to the exclusion of evidence in a police report involving an incident between Singh and a former tenant, five months earlier, in which the tenant called police after Singh went to her apartment uninvited. The report stated that, early in the evening on August 16, 2003, Singh telephoned the former tenant and told her that he wanted her out of the apartment. Later that evening, the tenant heard the door open and saw Singh at the bottom of the stairs, yelling at her; he had used his key to enter the apartment. She told him to leave and then called police. Singh left before police arrived, but, when police located him in his back yard, he was so intoxicated that they took him into protective custody for his own safety. Police were familiar with the tenant and Singh because, on numerous previous occasions, each had made complaints against the other (although police did not bring charges in response to any of those complaints). The defendant argued unsuccessfully at trial that the statements should have been admitted, “not for the truth of the matter, but to show whether the investigating officer had any intensity”22 in pursuing the investigation.
Because this information was not relevant to the crimes for which the defendant was on trial, its exclusion was not an abuse of discretion. During a sidebar conference on the question of admission of the information in the police report, the prosecutor stated that Singh went to the tenant’s part of the house to confront the unruly tenant who was engaged in a heated dispute with her boy friend. She offered to introduce evidence to show that the former tenant had a reputation of being difficult, and that the defendant’s description of the incident was not supported by the evidence. These collateral circumstances had no relationship to the offense at issue, and the judge ruled properly that the police report could not be admitted as evidence that police should have devoted more attention to Singh in their investigation.
*417The exclusion of the friends’ statements and the police report did not prevent the defendant from challenging the adequacy of the police investigation. The judge stated that the defendant would have “free rein” to pursue his theory of alleged deficiencies in the police investigation, and the defendant pursued this theory vigorously in cross-examination and in closing. He inquired of many of the police officers concerning tests that were not conducted and questions that they did not ask witnesses. He attacked the manner in which DNA and other scientific tests were performed and whether the samples had been contaminated, and brought out on cross-examination of the DNA expert that one in 204 people could have matched the partial DNA profile in the blood under Alyssa’s fingernails.
Despite the exclusions of which the defendant complains, he was permitted to produce other evidence at trial supporting his claim that police should have investigated Singh more carefully. See Commonwealth v. Ridge, 455 Mass. 307, 316 (2009) (“The defendant fails to address the fact that almost all of what he contends he wanted to raise was, in fact, asked”). The defendant cross-examined both Singh and his wife on Singh’s relationship and interactions with Joanne. In closing, the defendant emphasized the asserted failings in the police investigation, taking note not only of the investigative tests that the police did not undertake, but also of the evidence found in Singh’s house and the questions that police did not ask during their investigation of Singh. He argued vigorously that police did not do enough to pursue their investigation of Singh after they obtained the defendant’s DNA test results.
5. Third-party culprit evidence. The defendant claims also that the judge’s decision to exclude statements by Joanne and her friends and neighbors concerning her alleged relationship with Singh and the police report detailing Singh’s altercation with the former tenant, as described in part 4, supra, denied him the ability to present adequately a third-party culprit defense that Singh was the killer.
While a third-party culprit defense is offered frequently in conjunction with a defense of inadequate police investigation and both defenses may rely on the same evidence, the purposes to be served by introducing the evidence differ. Rather than seeking to undermine the strength of the investigation, “the *418inference to be drawn from third-party culprit evidence is simply that someone else committed the crime.” Commonwealth v. Silva-Santiago, 453 Mass. 782, 800-801 (2009). Although we have accorded wide latitude to judges in determining whether to admit evidence that a person other than the defendant may have committed the crime, see Commonwealth v. Mosher, 455 Mass. 811, 824 (2010), that discretion is limited in two respects. First, because the evidence is introduced to prove its truth (i.e., that a third party committed the crime), it may constitute hearsay. If such evidence does not fall within a hearsay exception, we have permitted its introduction “only if, in the judge’s discretion, ‘the evidence is otherwise relevant, will not tend to prejudice or confuse the jury, and there are other “substantial connecting links” to the crime.’ ” Commonwealth v. Silva-Santiago, supra at 801, quoting Commonwealth v. Rice, 441 Mass. 291, 305 (2004). Second, the evidence “must have a rational tendency to prove the issue the defense raises, and the evidence cannot be too remote or speculative.” Commonwealth v. Silva-Santiago, supra at 801, quoting Commonwealth v. Rosa, 422 Mass. 18, 22 (1996).23
The judge did not abuse her discretion in excluding, in the context of a third-party culprit defense, statements regarding Joanne’s alleged romantic relationship with Singh and the police report describing the incident with Singh’s former tenant. Offered for this purpose, those statements were hearsay. As the judge correctly concluded, and as noted in part 4, supra, the defendant’s offer of proof failed to show any “substantial connecting link” between Singh and the murders. The evidence was of a highly speculative nature and did not “have a rational tendency to prove” that Singh committed the crime. The judge characterized the affidavit submitted by the defendant’s investigator in support of the introduction of the evidence as “a lot of hearsay,” “useless,” and “not even close” to showing that *419Singh was a third-party culprit, and stated that the defendant was not entitled to present “a defense based on rumor.” We agree. The statements lacked probative value and could easily have led to confusion of the jury and unfair prejudice to the Commonwealth. See Commonwealth v. Silva-Santiago, supra at 801 (“feeble third-party culprit evidence poses a risk of unfair prejudice to the Commonwealth”); Commonwealth v. O’Brien, 432 Mass. 578, 588-589 (2000) (murder victim’s statements to friend about her fear of third party were hearsay and would be “confusing” to jury as “no more than an opinion of [his] involvement”).
Nor did the judge’s exclusion of these statements deprive the defendant of the ability to present a defense suggesting that Singh was the killer. Police officers testified that Singh was interviewed at the police station, that he voluntarily provided DNA samples on police request, and that he allowed police to search his house. An investigating officer described the search and the objects that police took from Singh’s house for testing, including footwear that might have matched shoe imprints found in Alyssa’s room and beverage glasses. The defendant was able to point to the glasses, alcohol, and other evidence found in Singh’s basement and to indicate that police sought this evidence “on information received.” Evidence concerning the incident with the former tenant, the promise to build a deck, and the offer to buy a bed was introduced on cross-examination of Singh and his wife.24 The defendant argued forcefully in closing that the investigation was inadequate and suggested that Singh was the culprit.
6. Review pursuant to G. L. c. 278, § 33E. Although the defendant does not raise the issues on appeal, our review of the record indicates that two aspects of the prosecutor’s closing argument were improper. Because there was no objection at trial to the first line of improper argument, we review to determine whether a substantial likelihood of a miscarriage of justice *420occurred. At the end of the prosecutor’s argument, the defendant objected to the second of the problematic arguments, and the judge gave a curative instruction immediately thereafter. The defendant did not object to the curative instruction, nor has he raised the point on appeal. Therefore, we again review only to determine whether a likelihood of a miscarriage of justice was created.
First, the prosecutor speculated, repeatedly and at length, about what Joanne and her daughter might have been thinking and what actions they might have taken during the commission of the crime. For instance, the prosecutor hypothesized that Alyssa might have come downstairs and seen the defendant raping her mother, then run back upstairs. She debated whether Joanne might have let the defendant into the house, and described in detail the possible thoughts that might have been running through her head, in particular concerning her children, during the rape. Prosecutors must limit the scope of their arguments to facts in evidence and inferences that may reasonably be drawn from the evidence. See Commonwealth v. Beaudry, 445 Mass. 577, 580 (2005); Commonwealth v. Good, 409 Mass. 612, 623 (1991). These statements were not supported by any evidence and should not have been permitted.
Second, the prosecutor urged the jury to place themselves in Alyssa’s shoes. The jury should not be asked to put themselves “in the shoes” of the victim, or otherwise be asked to identify with the victim. See Commonwealth v. Thomas, 400 Mass. 676, 684 (1987). See also Commonwealth v. Grinkley, 75 Mass. App. Ct. 798, 808-809 (2009); Commonwealth v. Saunders, 75 Mass. App. Ct. 505, 511-512 (2009); Commonwealth v. Jordan, 49 Mass. App. Ct. 802, 806 (2000); Commonwealth v. Harris, 11 Mass. App. Ct. 165, 176 & n. 9 (1981). Generally, the victim’s state of mind is not at issue in a homicide case. Such cases revolve around questions such as the identity of the killer, the killer’s intent, the means employed by the killer, and the like. Asking that the jury focus on the victim distracts attention from the actual issues, and invites the jury to decide guilt or innocence on the basis of sympathy. Nor can this type of argument be justified on the ground that it serves the Commonwealth’s effort to prove that the killing was committed with extreme atrocity or cruelty, a showing that turns largely on what the *421defendant has done. See Commonwealth v. Cunneen, 389 Mass. 216, 227 (1983) (factors which jury may consider in deciding whether crime committed with extreme atrocity or cruelty include [1] indifference to or taking pleasure in the victim’s suffering, [2] consciousness and degree of suffering of the victim, [3] extent of physical injuries, [4] number of blows inflicted, [5] manner and force with which blows were delivered, [6] nature of instrument or weapon used, and [7] disproportion between means needed to cause death and those employed).
We conclude that the improprieties in the closing argument, while serious, do not require a new trial. The evidence against the defendant, specifically the DNA evidence, was overwhelming. See Commonwealth v. Kent K., 427 Mass. 754, 761 (1998), quoting Commonwealth v. Santiago, 425 Mass. 491, 501 (1997) (“although troubling,” improper appeals to sympathy less crucial where “guilt is clear”). Additionally, the judge’s curative instruction on the issue of juror sympathy was forceful and was given immediately following the improper argument, separately from the remainder of the judge’s charge.
We have reviewed the entire record and discern no reason to order a new trial or reduce the convictions of murder in the first degree to a lesser degree of guilt.
7. Conclusion. The convictions of murder in the first degree are affirmed. Because the defendant was convicted of the rape victim’s murder on theories of both felony-murder and extreme atrocity and cruelty, the judgment on the indictment charging aggravated rape is not duplicative, and that judgment is affirmed. See Commonwealth v. Felder, 455 Mass. 359, 370-371 (2009), citing Commonwealth v. Brum, 441 Mass. 199, 200 n.l (2004).
So ordered.
Because the victims share a last name, and to protect their privacy given the nature of the crime, we refer to the victims by their first names.
Telephone records showed no contact between the defendant’s telephone number and Miller’s number after approximately 11:30 p.m., and no use of the defendant’s telephone from approximately 11:45 p.m. on January 4, 2004, until 9:50 a.m. on January 5, 2004, when he spoke with his supervisor.
These were the same clothes that she had been seen wearing on January 4, 2004.
Police also investigated the fathers of Joanne’s children and residents of a local half-way house.
On January 16, 2004, several days after their first interview of the defend*405ant, police initiated a Combined DNA Indexing System (CODIS) database search for a deoxyribonucleic acid (DNA) profile matching the foreign DNA found on the victims. The CODIS database is comprised of DNA samples from individuals convicted of offenses in Massachusetts punishable by imprisonment in a State prison. See G. L. c. 22E, § 3. This search produced a match to DNA taken previously from the defendant. The grand jury then sought a court order to obtain a new sample from the defendant. The defendant filed a motion in limine to exclude the CODIS results. To avoid prejudice to the defendant due to his prior conviction and his status as a level 3 sex offender, the parties stipulated that the police collect DNA from many people, and any mention of the CODIS database was excluded.
The defendant’s DNA profile matched the samples taken from the adult victim’s shirt and from her genitals, and was the major contributor to the sample taken from within her vagina.
The defendant is Caucasian.
Y-STR testing involves testing only the twelve male loci from a given sample. It is commonly used in situations such as that here, where there is a large amount of female DNA and potentially only a small amount of male DNA. See Commonwealth v. Linton, 456 Mass. 534, 543 & n. 8 (2010).
This random match probability statistic was calculated using a confidence interval to extrapolate from a database of 4,004 samples.
See note 6, supra.
The Commonwealth asserts that the defendant did not preserve this issue for review. The Commonwealth contends that, at trial, the defendant objected only to the evidence from the CODIS database, and, by failing to object at trial on the grounds raised in his pretrial motions, waived any objection to admission of the other DNA evidence. See Commonwealth v. Whelton, 428 Mass. 24, 26 (1998). The defendant argues, however, that the CODIS hit prompted police to obtain the confirmatory match used at trial, and that the statistical methodology required to present the confirmatory match to the jury is interwoven with the number and types of samples in the CODIS database. Moreover, in denying the defendant’s pretrial motion to exclude all the DNA evidence, the motion judge (who was not the trial judge) stated explicitly that the defendant’s objection was noted. See Commonwealth v. Kee, 449 Mass. 550, 553 n.5 (2007) (objection to denial of motion in limine before trial reviewed under harmless error standard where judge stated in denying motion that defendant’s objection was noted and his rights were “saved”). See also Commonwealth v. Galicia, 447 Mass. 737, 746-747 (2006) (objection to denial of motion in limine before trial, while not generally sufficient to preserve appellate rights, may suffice in particular circumstances). Accordingly, we consider the issue preserved.
The defendant points to a number of statistical methods that he alleges are topics of debate within the scientific community. As the District of Columbia Court of Appeals makes clear in a case discussing the same scientific issues, see United States v. Jenkins, 887 A.2d 1013, 1022-1025 (D.C. 2005), these statistical calculations do not conflict; they simply measure different things. The random match probability (RMP or rarity) calculation measures how rare a given DNA sample is among the general population. The database match probability calculation states the probability of obtaining a match to a known sample from a particular database. The calculation discussed by Professors *408David Balding and Peter Donnelly, on which the defendant relies, describes the probability, given that some number of other suspects have been eliminated through the database search, that a suspect identified by a cold hit is the source of DNA in a particular sample. See United States v. Jenkins, supra.
Ascertainment bias describes “the bias that exists when one searches for something rare in a set database.” United States v. Jenkins, supra at 1018-1019. For example, if the RMP of a defendant’s DNA profile is one in 100,000 in the general population, a search of a database of 100,000 unrelated DNA profiles will, statistically, produce one match. If the database searched instead contains 200,000 profiles, then two matches will be produced. The larger the database the more matches, and thus the more potential suspects, will be produced. As a result, “the more populated the database is, the less impressive a match becomes.” United States v. Jenkins, supra at 1019 n.10. The second and third calculations discussed in note 13, supra, are designed to address the issue of ascertainment bias.
We make no comment regarding the necessity of such a ruling. At this time, most courts in other jurisdictions that have considered the issue have held admissible evidence that a defendant was first identified through a match in an offender database. See, e.g., United States v. Davis, 602 F. Supp. 2d 658, 673-677 & nn.21, 23 (2009); People v. Nelson, 43 Cal. 4th 1242, 1247-1249, 1257-1267 (2008); United States v. Jenkins, supra at 1025 & n.20; State v. Bartylla, 755 N.W.2d 8, 11-14, 19-20, & nn.2-4 (Minn. 2008); State v. McMilian, 295 S.W.3d 537, 540-541 & n.4 (Mo. App. 2009).
The Commonwealth’s theory was that the defendant’s hands were cut while he was fighting with Alyssa.
There were no fingerprints on the glasses, no semen on the futon, and no evidence that Joanne had ever been in the basement with Singh.
General Laws c. 233, § 2IB (the rape shield statute), prohibits introduction of “[e]vidence of the reputation of a victim’s sexual conduct” where a defend*415ant is charged with any one of enumerated sex offenses, including rape. The statute allows introduction of specific instances of a victim’s sexual conduct with the defendant or “recent conduct of the victim alleged to be the cause of any physical feature, characteristic, or condition of the victim,” only on written motion and after an in camera hearing.
Before us, the Commonwealth asserts that evidence of Joanne’s alleged relationships with other men was not admissible under the rape shield statute because none of the proposed testimony would have served to explain anything about her physical condition at the time of the crime.
See note 18, supra. The defendant did not file a motion pursuant to the rape shield statute.
In Commonwealth v. Silva-Santiago, 453 Mass. 782, 803 (2009), we concluded that in circumstances such as these a trial judge should conduct a voir dire hearing. This case was tried prior to our decision in Commonwealth v. Silva-Santiago, and no such hearing was conducted in this case. In the absence of a voir dire, we review to determine whether the judge abused her discretion in excluding the evidence at issue. See id. at 804. See also Commonwealth v. Ridge, 455 Mass. 307, 317 & n.10 (2009) (reviewing for abuse of discretion where case was tried prior to Silva-Santiago decision and no voir dire conducted).
We take this phrase to mean whether the officers pursued their investigation diligently and thoroughly.
Although the latter restriction is applicable equally when evidence of a third party’s involvement is introduced as part of an inadequate police investigation defense, the former is not. For purposes of an inadequate police investigation defense, the evidence is not hearsay, and it need not fit within a hearsay exception or meet the “formidabl[e]” requirement of a substantial connecting link. See Commonwealth v. Ridge, supra at 317 & n. 11, quoting Commonwealth v. Silva-Santiago, supra at 802-803.
Singh’s wife confirmed that police were called, but testified that the former tenant had been difficult and that Singh had gone to the apartment because the tenant was making noise late at night. Singh stated on cross-examination that he offered to purchase a bed for Joanne after one of the couple’s children broke her bed while the Singh family was helping her move into the duplex.