
534 F.3d 1290 (2008)
UNITED STATES of America, Plaintiff-Appellee,
v.
Thomas Guy CARAWAY, Defendant-Appellant.
No. 07-3229.
United States Court of Appeals, Tenth Circuit.
July 28, 2008.
*1292 Ronald E. Wurtz, Assistant Federal Public Defender, (David J. Phillips, Federal Public Defender, with him on the briefs), Topeka, KS, for Defendant-Appellant.
James A. Brown, Assistant United States Attorney, (Eric F. Melgren, United States Attorney, with him on the brief), Topeka, KS, for Plaintiff-Appellee.
Before HARTZ, Circuit Judge, McWILLIAMS, Senior Circuit Judge, and HOLMES, Circuit Judge.
HARTZ, Circuit Judge.
Thomas Guy Caraway was convicted by a jury in the United States District Court for the District of Kansas of causing an explosive device to be delivered by U.S. Mail and of possession of an explosive device during and in relation to a crime of violence. He challenges his convictions on the grounds of insufficient evidence to support the verdict and improper admission of evidence-namely, (1) testimony about a witness's prior statement, (2) a written copy of the prior statement, and (3) testimony that a bomb-construction manual had been found on his property. Exercising jurisdiction under 28 U.S.C. § 1291, we hold that there was no reversible error and affirm his conviction.

I. BACKGROUND
On January 29, 2004, a rural mail carrier delivered a package to the home of Daniel "Spud" Owens in Washington, Kansas. Because the package was too large for the mailbox, the carrier left it in Owens's pickup truck. The next day, Owens's son, Daniel, found the package in the truck and brought it into the living room. Although the package was addressed to his father, Daniel, who had turned 13 two days before, thought that it could be a birthday present for him. He and his father opened the cardboard packaging and removed what appeared to be a toolbox. They set the toolbox on the couch and opened its latches. The box exploded, propelling shotgun pellets in all directions. Daniel was hit in the shoulder, and Owens was hit with 14 or 15 pellets in his left hand.
Inspection later revealed that the device had the following components: First were two improvised shotgun barrels, each holding a 12-gauge shotgun shell, welded onto a metal plate on the bottom of the device. The rear of each barrel was covered by a threaded-on end cap with a hole large enough to accommodate the passage of a firing pin. The firing pin was cocked and held in place by a spring. When the spring was released, the firing pin would strike the shell and fire it. One of the firing pins was equipped with a washer, so that when the pin was released, the washer would hit a toggle switch, flipping the switch to complete an electrical circuit connected to four batteries. The batteries lit a model-rocket ignitor inside a bag of gun powder within a PVC pipe, which would explode upon ignition. As a final touch, a bottle of gasoline in the device was apparently intended to cause a fire after the explosion. Luckily, only one shotgun shell fired when the box was opened, and explosives experts safely dismantled the rest of the device.
From the postal sticker on the package in which the device was shipped, investigators determined that it had been mailed the day before from Wamego, Kansas. The postal clerk in Wamego who had accepted the package recalled that it had been wrapped in paper and tied with string and that the string had to be removed before it could be mailed. He also remembered that the "to" and "from" addresses had been reversed and had to be corrected. He described the sender as a man of average build in his early twenties with light brown hair. A computer record of the transaction showed that the postage had been paid with a $20 bill.
*1293 Mr. Caraway became a suspect in the investigation when investigators learned that his ex-wife Denise, who had left him about seven months before, had become involved in a relationship with Owens, Mr. Caraway's former best friend. On February 6, 2004, investigators conducted a search of Mr. Caraway's property in Delia, Kansas. They discovered about a dozen items that resembled items recovered from the explosive device.
The day after the search, postal inspectors interviewed Mr. Caraway's son, Shawn Caraway, who had been living with Mr. Caraway in Delia. Shawn, about 22 years old at the time, told the inspectors that he had seen the device sitting on the coffee table in their living room. He gave them a description of the device and also drew a diagram of it, but he denied having mailed it, a position he maintained for nearly the next three years. On October 26, 2006, however, Shawn told investigators that his father had given him a package and a $20 bill and asked him to mail the package at the Wamego post office. He said that his sister, Jessica Caraway, had given him a ride to the post office. Jessica had previously denied knowing who mailed the bomb, but a week after Shawn told the inspectors that he had mailed the package, she gave a written statement to postal inspectors that she had driven Shawn to the Wamego post office.
On November 8, 2006, Mr. Caraway was charged in a three-count indictment with causing an explosive device to be delivered by the U.S. Postal Service, with intent to kill or injure another, see 18 U.S.C. § 1716(a) and (j)(2); conspiracy to mail an explosive device, see id. §§ 371, 1716(a) and (j)(2); and possession of an explosive device during and in relation to a crime of violence, see id. § 924(c)(1)(A) and (B)(ii). The conspiracy charge was dismissed before trial. A jury found Mr. Caraway guilty of both remaining counts, and he was sentenced to 30 years in prison.

II. DISCUSSION

A. Sufficiency of the Evidence
Mr. Caraway argues that his conviction must be reversed because the evidence at trial was insufficient to support the jury's verdict. We review sufficiency-of-the-evidence challenges de novo, viewing all evidence in the light most favorable to the government. United States v. Lamy, 521 F.3d 1257, 1267 (10th Cir.2008). We will affirm the jury's verdict "unless no reasonable jury, when presented with the evidence introduced at trial together with the reasonable inferences therefrom, could find the defendant guilty beyond a reasonable doubt." Id. (internal quotation marks omitted). "While the evidence supporting the conviction must be substantial and do more than raise a mere suspicion of guilt, it need not conclusively exclude every other reasonable hypothesis and it need not negate all possibilities except guilt." United States v. Burkley, 513 F.3d 1183, 1188-89 (10th Cir.2008) (brackets and internal quotation marks omitted).
Several witnesses testified that Mr. Caraway had a difficult time with his wife's departure. His daughter Jessica testified that he was "mad" when Denise left and that he called her a "bitch." R. Vol. 5 at 415. Shawn said that his father was initially hurt and surprised and then was "very obsessed" for several months. Id. Supp. Vol. 2 at 112. Mark Hight, a friend of Mr. Caraway's, described him as "depressed" and "possibly suicidal" after Denise left. Id. Vol. 6 at 514. Denise testified that her ex-husband called her "all the time" after she left him and that he was "very insistent" in urging her to return. Id. Vol. 5 at 469.
Abundant evidence was presented that Mr. Caraway had made threats against *1294 Owens and Denise. Shawn testified that his father had said that Denise "should get what's coming to her," that she "should die for what she did," and that he was going to "take her out." Id. Supp. Vol. 2 at 16. Shawn also said that his father, referring to both Denise and Owens, had said that he was going to "blow them up." Id. at 19. Jessica testified that her father had made threats against Denise, such as that he would "put a cap in [her] ass," id. Vol. 5 at 417, and made "joking" threats against Owens, saying that he would "stomp a mud hole in his head" or "take Spud down," id. at 418. She said that the threats against her mother had continued until the time of the explosion. Denise testified that Mr. Caraway had once told her that "I ought to just put a cap in your ass," id. at 474, and once told one of their children that he could blow up her car. And Owens testified that two or three months before the device exploded at his house, Mr. Caraway had called him, quizzed him about Denise, and said something like "[you] better not do this." Id. at 401-02.
The prosecution also offered evidence that Mr. Caraway had the technical ability and experience to build the device. Several witnesses testified that Mr. Caraway was a very good mechanic, a "jack-of-all-trades," Id. Supp. Vol. 2 at 25, who knew how to weld and who was capable of fixing nearly anything on a vehicle. Shawn testified that his father had made firecrackers; Denise said that he had made pop-bottle bombs and once blew up a washer or a dryer. Both Shawn and Jessica recounted hearing loud bangs or explosions at the Delia property in 2003 and the beginning of 2004. Hight testified that he had found a book in Mr. Caraway's workshop that explained how to construct an explosive device using barrels for single-shot 12- and 20-gauge shotguns.
In addition, there was substantial physical evidence suggesting a connection between the explosive device and Mr. Caraway. When investigators searched the Caraway property, they found a number of items that resembled those found in the device, including duct tape, red tape, white PVC pipe of 1-inch diameter, shotgun pellets, shotgun shell casings, blue and black wires, springs, Triple Seven-brand gun powder, and toggle switches. Investigators also recovered next to an empty paper dispenser a scrap of greenish paper that was of about the same color and thickness as the greenish paper used to wrap the package. Shawn testified that he and his father had used the paper when they painted cars and that rolls of the paper had been present before the property was searched. He did not know why none had been found when the search took place, although he did testify that his father had warned him a day or two before the search that he should "get stuff cleaned up" because the police might be searching the property. Id. at 78. Welding equipment and shotgun-shell reloading tools were also found on the property.
Finally, Shawn's testimony provided a direct link between the explosive device and his father. Shawn testified as follows: Sometime before reading about the explosion in the paper, he entered the house and saw the device on the coffee table. His father was in the room and asked Shawn to get him a shotgun shell. His father then explained to him how the device worked. At the time, Shawn thought his father was just "goofing around." Id. at 47. On January 28 his father asked him to mail a package at the post office in Wamego, even though Delia had its own post office. He gave Shawn the return and destination addresses and a $20 bill for postage. Shawn asked Jessica to give him a ride to Wamego, which she did. At the post office Shawn gave the package to the clerk, paying with the $20 bill. When Shawn read about the explosion in the *1295 paper, he realized that the package he had mailed had contained the device he had seen on the table.
Along with Shawn's testimony, the government introduced the diagram of the device that Shawn had drawn for investigators on February 7, 2004, the day after the Delia property was searched. Mr. Caraway concedes that this diagram is sufficiently detailed to establish that Shawn had seen the device before it detonated, so we need not attach to this opinion a copy of Shawn's drawing and a photograph of the device after it detonated.
Despite evidence that he possessed the materials to assemble the device, the technical knowledge and skill to construct it, and a motive to send it, and despite testimony directly tying him to the device, Mr. Caraway argues that the evidence presented at trial is insufficient to support his conviction. The essence of his argument is that the government's case depended on Shawn's unbelievable testimony. He points to several reasons to disbelieve Shawn: Shawn was testifying under an agreement with the government, he had previously given inconsistent statements about his role in mailing the device, he had been a heavy methamphetamine user around the time the bomb was mailed, and his relationship with his father admittedly had been rocky at times. Although these matters could convince a jury to give less weight to Shawn's testimony, the determination of credibility is exclusively for the jury. See United States v. Bowen, 527 F.3d 1065, 1076 (10th Cir.2008). In general, we "simply determine whether the evidence, if believed, would establish each element of the crime." United States v. Hanrahan, 508 F.3d 962, 968 (10th Cir. 2007). It clearly would. Mr. Caraway is not entitled to a reversal of his conviction for insufficient evidence.

B. Evidence of Jessica's Prior Statement
To begin our discussion of Mr. Caraway's first two evidentiary challenges, we provide a brief overview of some legal principles regarding use at trial of a witness's prior statements. Ordinarily a prior statement by a testifying witness cannot be used at trial. If offered for the truth of the matter asserted, it is inadmissible hearsay unless, roughly speaking, the statement was sworn testimony inconsistent with the witness's trial testimony or, if consistent with the witness's trial testimony, is offered to rebut a charge that the witness's trial testimony is a recent fabrication or the result of an improper influence or motive. See Fed.R.Evid. 801(d)(1). Evidence of a prior inconsistent statement can be used to impeach the witness, see United States v. Bao, 189 F.3d 860, 866 (9th Cir.1999), but under the law of this circuit, a written statement offered for impeachment cannot itself be introduced as evidence if the witness admits making that statement, see United States v. Soundingsides, 820 F.2d 1232 (10th Cir.1987). Also, counsel may use a prior statement to refresh the recollection of a witness who cannot remember a past event, see 4 Jack B. Weinstein & Margaret A. Berger, Weinstein's Federal Evidence, § 612.03 (2d ed.2008), but doing so does not render the document thereby admissiblethe best practice being for the trial court to have the witness silently read the material. See N.M. Sav. & Loan Ass'n v. U.S. Fid. & Guar. Co., 454 F.2d 328, 337 (10th Cir. 1972); see also Fed.R.Evid. 612 (use by opposing party of document used to refresh witness's recollection). Finally, in recognition of the difficulty jurors can have in considering a prior inconsistent statement for impeachment purposes without also improperly using the statement as substantive evidence of what it asserts, the Carter doctrine in this circuit sets limits on *1296 prosecutorial use of prior inconsistent statements. See United States v. Carter, 973 F.2d 1509 (10th Cir.1992). Under Carter a prosecutor cannot ask a witness about a prior inconsistent statement by the witness if the primary purpose of calling the witness is to have the jury hear the prior statement and use it as substantive evidence of guilt,[1] even though it is ostensibly offered only for impeachment.
We now turn to the specifics of this case. On direct examination by the government, Jessica testified that in November 2006 she had given postal investigators a written statement that she had driven Shawn to the post office on January 28, 2004; in response to a later question, she asserted that she had not taken Shawn to the post office and that the written statement had been a lie. A copy of her written statement was admitted into evidence. (Although the record does not reveal whether it was ever given to the jury, we will assume that it was). In closing argument the prosecutor said that Jessica's written statement corroborated Shawn's testimony, although the district court later instructed the jury that the statement was to be considered only for impeachment. Mr. Caraway argues (1) that it was error to admit the oral testimony regarding her prior statement and for the prosecutor to use it as substantive evidence, and (2) that it was error to admit the written statement into evidence.

1. Presentation of Evidence at Trial
Leading up to the testimony in question, the prosecutor had asked Jessica about her father's reaction to her mother's leaving, his threats to harm Owens and her mother, whether she had heard explosions at her house in 2003, the type of mechanical work her father did, and whether he had experimented with explosives. On several occasions, when Jessica's answer was inconsistent with her grand-jury testimony, the prosecutor reminded her of that testimony, and she sometimes modified her answer. The following exchange then took place:
Q. Okay. And then also at the grand jury you deny knowing who mailed the bomb; is that correct?
A. Yes.
Q. Okay. And then later on, you gave a statement stating that you had mailed the bomb or notexcuse me, I'm sorry, I misspoke. You gave a statement that you drove Shawn Caraway to the post office and he mailed the bomb at the post office. Do you remember that?
A. Yes, I made that statement.
R. Vol. 5 at 422. Jessica explained that she had felt pressured by her mother and Owens to make the statement and that she had wanted investigators to leave her alone because she was seven-months pregnant. The prosecutor then gave her the written statement to review. He asked her to familiarize herself with it, to confirm that it was written in her own hand, and to confirm that it was prefaced with language stating that she had made the statement "of [her] own free will and accord, without coercion, threat, or promise of reward." Id. at 424 (internal quotation marks omitted). At that point, defense counsel objected that the prosecutor's questions were leading. The court overruled the objection. The prosecutor then asked, "Did you drive Shawn Caraway to the Wamego post office?" and Jessica answered, "No, I did not." Id. at 425. The prosecutor again asked if the statement said that it was made without coercion and if it was written in her own hand. Jessica again offered *1297 an explanation for giving the statement and said that she had lied in giving it. The prosecutor read the statement line by line, asking Jessica to confirm or refute each sentence in it. She said that it was all a lie.
After an interlude in which the prosecutor pursued an unrelated line of questioning, the prosecutor moved to admit Government Exhibit 12, Jessica's written statement to the postal inspectors on November 2, 2006. Defense counsel objected:
The statement, of course, is technically and clearly hearsay. She has admitted every line in it. To admit it and give it to a juryto the jury would be prejudicial. She's explained it, she's talked about it. And I think under Rule 403 it's far more prejudicial than it is probative, especially since she's gone through every line and explained every one already and the jury has heard it. To have this go into the jury room serves no purpose, other than to bring contrary testimony into the jury.
Id. at 432. The prosecutor responded, "As far as prejudice goes, any damage is already done because it's been read in open court." Id. at 433. He added, "And to feel better, we can give a limiting instruction stating that it can be offered for impeachment." Id. To this, defense counsel objected that it was not impeachment. The court then admitted the written statement.
After Jessica's testimony, the government called Postal Inspector Paul Mezzanotte, who had taken Jessica's statement. He testified that he had administered an oath to Jessica before taking her statement and that she had not appeared to be under the influence of any coercion, threat, or promise. He said that she had been a little upset but polite and that she had not hesitated in making the statement that she drove Shawn to the post office. Mezzanotte testified that after giving the statement, Jessica had met her mother and Owens outside and had given them both hugs.
In closing argument the prosecutor addressed the credibility of Shawn's testimony. He asked them to consider whether Shawn appeared to be lying on the stand and to take into account that Shawn's motive for initially lying about mailing the bomb was to protect himself and his sister. The prosecutor suggested that Shawn's story had always been consistent on its most important detailthat he had seen the device at his house. He also urged the jury to consider whether Shawn's story was corroborated, and he said that it was in several respects: Shawn's description and diagram of the bomb matched the actual bomb, the types of materials found in the house supported Shawn's assertion that the bomb was made in the house, and Jessica's testimony about the threats and hearing explosions matched Shawn's. He continued: "Also, before she denied it on the stand, Jessica Caraway corroborated his statement to the postal inspectors that he mailed it because, remember, Jessica Caraway had made a statement saying that she drove Shawn to the Wamego post office before she denied it on the stand." Id. Vol. 8 at 659-60. This was a substantive use of Jessica's prior inconsistent statement; it was not being used just to impugn her veracity when testifying that she did not drive Shawn to the post office.
At the close of trial the court gave the jury the following instruction:
You have heard evidence that before this trial certain witnesses made statements that may be different from his or her testimony in court. This is known as impeachment with a prior inconsistent statement. These earlier statements were brought to your attention only to help you decide how believable the testimony in this trial was. You *1298 cannot use the earlier statements as proof of anything else. You can only use them as one way of evaluating the testimony here in court. Exhibit 12 and Exhibit C-4 are examples of earlier statements introduced as prior inconsistent statements.
Id. Vol. 1, Doc. 59, Instr. No. 33. Exhibit 12 was Jessica's written statement.

2. Admissibility of Oral Testimony of Prior Statement
Mr. Caraway argues that it was error to admit Jessica's oral testimony about the prior statement and error for the prosecutor to use it as substantive evidence. Because Mr. Caraway made no objection to the admission of the oral testimony about the prior statement, we apply plain-error review. See United States v. Lamy, 521 F.3d 1257, 1265 (10th Cir.2008); Fed.R.Evid. 103(d). "Plain error occurs when there is (1) error, (2) that is plain, which (3) affects substantial rights, and which (4) seriously affects the fairness, integrity, or public reputation of judicial proceedings." United States v. Gonzalez-Huerta, 403 F.3d 727, 732 (10th Cir.2005) (en banc) (internal quotation marks omitted).
Mr. Caraway argues that the introduction of the testimony was a violation of the Carter rule. See Carter, 973 F.2d at 1512-14. Carter, however, does not apply to this fact situation. In Carter a witness for the government gave testimony unfavorable to the government. Id. at 1511. The prosecutor then impeached the witness with his prior statements to the contrary. Id. The defendant argued on appeal that the court erred in allowing the impeachment because the prosecutor knew how the witness was going to testify and called him for the purpose of placing into evidence the impeachment testimony. Id. at 1512. We disagreed with the defendant's assessment of the prosecutor's purpose but acknowledged that "[t]he government may not introduce evidence of prior statements under the guise of impeachment for the primary purpose of placing before the jury substantive evidence which is not otherwise admissible." Id. (emphasis and internal quotation marks omitted). Carter, because of the risk that the jury will misuse impeachment evidence, would exclude otherwise admissible evidence when the record "clearly and unequivocally" establishes that the party's primary purpose in calling the witness or in asking the question was to utilize a prior hearsay statement as substantive evidence. Id. at 1513. Because of the proof required, reversible Carter error is rare (there is no example in our circuit). Cf. United States v. Clifton, 406 F.3d 1173, 1185-86 (10th Cir.2005) (Hartz, J., concurring and dissenting in part) (discussing difficulties with the purpose inquiry).
The reason that the oral testimony concerning Jessica's prior statement does not raise a Carter question is that it was not offered for impeachment. When it was offered, she had yet to testify whether she had driven Shawn to the post office. The prosecutor asked Jessica about her prior statement before he asked for her current position. Although a different sequence of events could have presented facts suited to a Carter analysis, Carter does not provide the appropriate test here.
Nevertheless, it was improper for the government to elicit Jessica's prior statement before she gave inconsistent testimony and then to use that statement as substantive evidence in closing argument. As just noted, Jessica's statement to investigators was not impeachment evidence because she had yet not testified to the contrary. And it could not be used substantively because for that purpose it would be inadmissible hearsay: an out-of-court statement offered to prove the truth of the matter asserted, falling into no exception. *1299 See Fed.R.Evid. 801-804. Had the prior statement first been introduced at the proper timethat is, after she testified that she had not driven Shawnit would still have been error for the government to use it as substantive evidence in its closing argument. See Regan-Touhy v. Walgreen Co., 526 F.3d 641, 651 (10th Cir.2008).
In any event, even if we assume that there was Carter error, so that the prior statement was not admissible for any purpose, the error does not rise to plain error. We will not reverse a conviction for plain error unless all four prongs of the plain error test are satisfied. Here, the third prongprejudicehas not been satisfied.
Under the third prong, a defendant "must demonstrate a reasonable probability that, but for the error claimed, the result of the proceeding would have been different." United States v. Fields, 516 F.3d 923, 944 (10th Cir.2008) (internal quotation marks omitted). In other words, Mr. Caraway must demonstrate a reasonable probability that he would not have been found guilty if Jessica's prior statement had not been presented to the jury. Mr. Caraway argues that without Jessica's statement about giving her brother a ride to the post office, Shawn's testimony was "wholly uncorroborated" and was "significantly weaker." Aplt. Br. at 21. We disagree.
Shawn's claim that his sister drove him to the post office is a relatively minor detail, one that does not directly implicate Mr. Caraway. Contrary to Mr. Caraway's assertion, the core of Shawn's testimony was substantially corroborated, most critically by the diagram of the device he drew for investigators, which Mr. Caraway concedes is proof that Shawn had seen the device. In addition, both his sister and his mother corroborated his testimony about the threats his father made, his sister corroborated his testimony that there were explosions on the Delia property in 2003, and the postal clerk corroborated his testimony that he paid for postage with a $20 bill. Moreover, Shawn's testimony was merely one part of a strong case against Mr. Caraway. We cannot say that it was reasonably probable that Jessica's prior statement was a critical factor for the jury.

3. Admissibility of Written Statement
Mr. Caraway argues that it was error for the district court to admit into evidence the written copy of Jessica's statement to postal inspectors. We understand him to be making three arguments on appeal regarding the admission of this statement: (1) that it was hearsay, (2) that it violated the Carter rule, and (3) that it was improper to admit it because Jessica did not deny having made the statement. We address each in turn.
Mr. Caraway timely raised at trial a hearsay objection to the written statement. We have already explained why the content of the statement, if used substantively, would be inadmissible hearsay. If admitted for impeachment purposes, however, it is not hearsay. The district court gave a jury instruction that stated that prior inconsistent statements are admitted "only to help you decide how believable the testimony in this trial was," R. Vol. 1, Doc. 59, Instr. No. 33, and told the jury that it could not "use the earlier statements as proof of anything else," id. This instruction explicitly referred to Exhibit 12. Mr. Caraway has not argued that the instruction was inadequate in informing the jury that the statement was admissible only for impeachment. Thus, the prior statement was admitted only for the proper purpose of impeachment, and the jury was given a limiting instruction not to use it substantively. The court did not improperly admit hearsay.
*1300 As for the Carter issue, we have already observed that Mr. Caraway made no Carter objection at trial, and our review is therefore for plain error. Even if he could establish a Carter error here, Mr. Caraway cannot demonstrate plain error for the same reason discussed above: given the relative unimportance of the statement and the strong evidence of his guilt, he cannot show the required prejudice.
Mr. Caraway's last allegation of error regarding Jessica's written statement is that it was admitted "[n]otwithstanding her admissions on the stand." Aplt. Br. at 17. When the government moved to introduce the written statement, she had, as defense counsel protested, already "admitted every line in it." R. Vol. 5 at 432. In Soundingsides, 820 F.2d 1232, we held that the district court had erred in allowing the government to present extrinsic evidence of prior inconsistent statements by two witnesses, both of whom admitted making the statements. We explained: "Where the witness does not deny making a prior inconsistent statement, there is clearly no rationale for the introduction of a prior inconsistent statement." Id. at 1240 (internal quotation marks omitted). We did not address in Soundingsides whether those particular errors would in themselves require reversal; we held only that in light of the weak, circumstantial murder case against the defendant, the admission of the prior statements together with the improper admission of evidence of the defendant's beatings of a former girlfriend was not harmless error. Id. at 1243.
We hold that the improper admission of Jessica's written statement was harmless error. As previously stated, the evidence was hardly central to the trial. Moreover, the jury had already heard the entire statement, broken down line by line, and any additional emphasis on it that resulted from the admission of the written statement was minimal. "[A] non-constitutional error is harmless unless it had a substantial influence on the outcome or leaves one in grave doubt as to whether it had such effect." United States v. Griffin, 389 F.3d 1100, 1104 (10th Cir.2004) (internal quotation marks omitted). We do not believe that the admission of the statement had such an influence on the outcome of the case.

C. Evidence of Bomb Book
Mr. Caraway's friend Mark Hight testified at trial that he used to manufacture methamphetamine in the Caraways' garagea large, multiroom outbuilding also referred to as the "shop" or the "shed." Hight recalled that approximately six months to a year before the search of the property, he found a paperback book, an "ATF book" (although no connection was shown to the Bureau of Alcohol, Tobacco, and Firearms). It was in the middle of the shop, near some dressers and "some stuff somebody was storing there." R. Vol. 6 at 520. The book, Hight said, contained instructions and illustrations for making explosive devices, including how to make "bombs in barrels for single-shot 20-and 12-gauge shotguns." Id. at 522. Hight said that he took the book home with him and then loaned it to a friend, who did not return it. He never saw Mr. Caraway with the book, or any other book. On cross-examination, he said that the "stuff" near where the book was found belonged to a Calvin Mounkes.
Mr. Caraway argues that the district court erred in admitting Hight's testimony about the bomb book because (1) it was irrelevant, and (2) its minimal probative value was substantially outweighed by the danger of unfair prejudice. He raised this objection in a motion in limine and renewed it at trial. "We review challenges to admissibility of evidence solely for *1301 abuse of discretion." United States v. Chisum, 502 F.3d 1237, 1241 (10th Cir.2007) (internal quotation marks omitted).
Under Federal Rule of Evidence 401, evidence is relevant if it has "any tendency to make the existence of any fact that is of consequence to the determination of the action more probable or less probable than it would be without the evidence." Irrelevant evidence is inadmissible. Fed. R.Evid. 402. A matter of consequence in this case was whether Mr. Caraway had the knowledge to construct the explosive device. Evidence that there was at one time a book in Mr. Caraway's workshop that explained how to make an explosive device similar to the one sent to Owens makes his possession of that knowledge more probable.
Mr. Caraway argues, however, (1) that the evidence was too remote in time from the alleged construction of the device to be relevant, in that the book "had been removed from his place before the motive for revenge even arose," Aplt. Br. at 26, and (2) that the book was not adequately connected to Mr. Caraway (after all, it was found near someone else's belongings). We are not persuaded. First, we do not agree with this characterization of the timing. Denise testified that she left her husband in June 2003, which is approximately seven months before the device was mailed. Hight, although uncertain of the exact time frame, estimated that he saw the book 6 to 12 months before the mailing. Second, even if the book was removed from the shop before Denise left, and even though it may not have been among Mr. Caraway's belongings, his access to the book (it was on his property) sometime that year still supports an inference that he read the book before its removal and used the information he took from it when constructing the device. The striking similarity between the bomb and the portion of the book described by Hight could persuade a rational fact finder that the book's presence on the Caraway's property was no mere coincidence. The district court could properly determine that the evidence was relevant under Rule 401.
To be sure, relevant evidence is not necessarily admissible. Under Federal Rule of Evidence 403, relevant evidence "may be excluded if its probative value is substantially outweighed by the danger of unfair prejudice," which Mr. Caraway argues is the case here. Mr. Caraway characterizes the bomb-book testimony as "strong evidence," Aplt. Br. at 25, and aligns it with the evidence of his threats to Owens and his wife, his mechanical skills, his prior construction of bombs, and the items found on his property that matched bomb components. But probative evidence, even very probative evidence, is not to be confused with prejudicial evidence. "Evidence is not unfairly prejudicial simply because it is damaging to an opponent's case." United States v. Curtis, 344 F.3d 1057, 1067 (10th Cir.2003) (internal quotation marks omitted). To be unfairly prejudicial, the evidence must have "an undue tendency to suggest decision on an improper basis, commonly, though not necessarily, an emotional one." Fed.R.Evid. 403 advisory committee's note. Mr. Caraway points to no such tendency. If the jury inferred that Mr. Caraway had read the book, that inference was properly inculpatory. If the jury did not believe that he had read it, the jury was highly unlikely to hold against him that such a book was found on his property. Because Mr. Caraway has shown no danger of unfair prejudice, such danger could not have "substantially outweighed" the evidence's probative value, and the district court did not abuse its discretion in admitting it.

D. Cumulative Error
Mr. Caraway argues that the cumulative effect of any errors requires a new trial. *1302 We have identified both unpreserved errors (reviewed for plain error) and a preserved error (reviewed for harmless error). The unpreserved errors were the premature admission of evidence of Jessica's prior inconsistent statement and the prosecutor's improper substantive use of Jessica's prior statement in closing argument; we also assumed a Carter error, which was unpreserved. The preserved error was the admission of the written statement itself, as opposed to the oral testimony of its contents.
As this court has previously noted, there are inherent problems in cumulating unpreserved error (reviewed for plain error) with preserved error (reviewed for harmless error). See United States v. Fields, 516 F.3d 923, 951 (10th Cir.2008). For one, reversal on harmless-error review is mandatory when the error is sufficiently prejudicial; in contrast, reversal for unpreserved error on plain-error review is discretionary. See id. at n. 15. "Injecting issues rejected on plain-error grounds into cumulative harmless-error review would allow a party to argue for mandatory reversal based on matters that, having been forfeited at trial, should be left to the discretion of the appellate court guided by the four-part [plain-error] test." Id. For another, "the party who carries the burden also differs according to the review invoked," with the defendant bearing the burden of proving prejudice on plain-error review and the government bearing the burden of proving harmlessness on harmless-error review. Id. If we reviewed the cumulation of preserved and unpreserved error for harmless error, we would undermine plain-error review. For example, under that procedure if there was one trivial preserved error that could not possibly have influenced the jury, the government would need to prove the cumulative harmlessness of the unpreserved errors; whereas without the trivial preserved error, the defendant would have to prove prejudice (as well as the fourth prong of the plain-error test). As a result, a defendant who had identified unpreserved error but could not overcome the third and fourth prongs of the plain-error test need only find an inconsequential preserved error to reverse the burden of persuasion on the third prong (that is, require the government to show no prejudice) and eliminate the fourth prong altogether.
Therefore, when there are both preserved and unpreserved errors, cumulative-error analysis should proceed as follows: First, the preserved errors should be considered as a group under harmless-error review. If, cumulatively, they are not harmless, reversal is required. If, however, they are cumulatively harmless, the court should consider whether those preserved errors, when considered in conjunction with the unpreserved errors, are sufficient to overcome the hurdles necessary to establish plain error. In other words, the prejudice from the unpreserved error is examined in light of any preserved error that may have occurred. For example, the defendant may not be able to establish prejudice from the cumulation of all the unpreserved errors, but factoring in the preserved errors may be enough for the defendant to satisfy his burden of showing prejudice. If so, the fourth prong of plain-error review must then be examined.
Applying this approach, we first note that there is only one preserved error, so there is no cause for a cumulative harmless-error analysis. We thus ask whether Mr. Caraway, grouping the preserved error and the unpreserved errors together, can establish prejudice. We conclude that he cannot. The question whether the introduction of testimony regarding Jessica's prior statement affected Mr. Caraway's substantial rights was not a close one. Any additional harm resulting from the *1303 preserved error  the admission of the written statement (as opposed to the oral testimony describing the statement)  was far too slight to push the prejudice over the line. Mr. Caraway has not met his burden of showing that absent the errors that we have identified or (with respect to the Carter error) assumed, the outcome of the trial would have been different.

III. CONCLUSION
We AFFIRM Mr. Caraway's conviction.
NOTES
[1]  The test stated in Carter can be read as focusing on the primary purpose in introducing the evidence, id. at 1512, or as focusing on the primary purpose in calling the witness, id. at 1512-13. We need not decide here which version controls.
