
810 A.2d 401 (2002)
Danny C. BELLAMY, Appellant
v.
UNITED STATES, Appellee
No. 99-CF-1173.
District of Columbia Court of Appeals.
Argued November 20, 2001.
Decided November 14, 2002.
*402 M. Elizabeth Kent, Washington, DC, appointed by the court, for appellant.
Stephanie J. Meltzer, Assistant United States Attorney, with whom Kenneth L. Wainstein, United States Attorney at the time the brief was filed, and John R. Fisher, Thomas J. Tourish, Jr., and Edward B. Parks, Assistant United States Attorneys, were on the brief, for appellee.
Before TERRY, FARRELL, and GLICKMAN, Associate Judges.
TERRY, Associate Judge:
After a jury trial, appellant Bellamy was convicted of distributing cocaine in a drugfree zone and possession of cocaine with intent to distribute it ("PWID") in a drug-free zone.[1] On appeal he contends (1) that he was prejudiced by a prosecution letter that allegedly misrepresented the testimony of an expert narcotics witness, (2) that the trial judge erred by neglecting to instruct the jury on the drug-free zone element of the charges against him, and (3) that he was entitled to a full evidentiary hearing on his claim of juror misconduct. We find no reversible error and affirm the convictions.

I
On April 29, 1998, the Metropolitan Police were involved in a drug investigation in the vicinity of 703 24th Street, Northeast, which is near Spingarn High School and within a drug-free zone.[2] Detective Gary Curtis, working undercover, engaged in a conversation to buy crack cocaine from an unidentified man.[3] This man then walked a short distance away to another man, later identified as appellant, made an exchange, and returned to Detective Curtis with a ziplock bag containing what appeared to be crack cocaine. Curtis paid for the cocaine with a $20 bill whose serial number had been pre-recorded. After the transaction, Detective Curtis went to his car and performed a field test on the contents of the ziplock bag. When the test yielded a positive result, Curtis broadcast *403 a lookout for the two men involved in the transaction.
Within minutes, police officers on the arrest team stopped appellant, who matched one of Curtis' descriptions, as he was coming out of a building at 703 24th Street. As Investigator Keith Tabron seized appellant and placed him up against a wall, he heard something hit the ground. Looking down, he saw a large ziplock bag containing twelve smaller bags of what appeared to be crack cocaine on the ground next to appellant. Moments later, Detective Curtis drove slowly past the scene and identified appellant as one of the men involved in the drug transaction. Appellant was then placed under arrest. A search of his person incident to his arrest yielded $202 in cash, including the $20 bill that Curtis had given him in payment for the cocaine.
Laboratory tests established that the ziplock bag purchased by Detective Curtis contained 85 milligrams of cocaine base at 82 percent strength. The twelve ziplock bags retrieved from the ground by Investigator Tabron contained 960 milligrams of cocaine base at 81 percent strength. Detective Charles Culver, an expert witness for the government, testified that the tests showed that cocaine in both samples came from the same batch. He also said that a person with crack cocaine packaged in twelve bags worth $20 each was more likely to have such drugs for distribution rather than for his own personal use.
Appellant denied selling any drugs and explained his possession of the $20 bill by presenting three witnesses. The substance of their testimony (appellant himself did not testify) was that a man known as "Hubcap" approached appellant and asked him for change for a $20 bill as appellant stood in front of 703 24th Street talking to a group of people. Although appellant "brushed Hubcap off," Hubcap followed him inside. One of the witnesses testified that, as he walked past the building, he saw appellant peeling off money from a roll of bills for Hubcap, who had a $20 bill in his hand. Appellant's defense as to the twelve ziplock bags found at his feet was that either the drugs belonged to someone else or the police planted the drugs on him.

II
On May 19, 1998, more than a year before trial, the prosecutor sent a letter to defense counsel containing a plea offer, a summary of the government's evidence, and certain discovery requests. The letter gave counsel notice that "should this matter proceed to trial, the government intends to call a narcotics expert," and that the expert "may testify," among other things,
(f) that possession of narcotics for use can be distinguished from possession of narcotics with intent to distribute, and that, for a variety of reasons, possession of individually packaged narcotics numbering more than ten (10), or possession of fewer packages but accompanied by other indicia of sale, is generally more consistent with an intent to distribute than with mere possession for personal use.
At trial, Detective Culver testified that possession of twelve $20 ziplock bags was inconsistent with possession for personal use, but he did not articulate the "ten bag" policy that was set forth in the letter. On cross-examination, defense counsel asked Detective Culver about the ten bag policy, but Culver said that no such policy currently existed.[4] Defense counsel then *404 moved to strike Culver's entire testimony because it "does not agree with what the government has told me their expert would be testifying to." The court denied the motion.
Appellant now argues that the prosecutor's letter affirmatively misled defense counsel about Detective Culver's testimony and that, because of this misrepresentation, the trial court abused its discretion in refusing to strike that testimony. He relies on a line of cases in which this court has held that certain actions by the government may be regarded as implied and binding promises. E.g., Wilson v. United States, 606 A.2d 1017, 1023 (D.C.1992) (reversing a conviction after the prosecutor failed to honor a representation that he would not use prior convictions for impeachment), overruled in part on other grounds by Lyons v. United States, 683 A.2d 1066, 1067 (D.C.1996) (en banc); Rosser v. United States, 381 A.2d 598, 607-609 (D.C.1977) (reversing a conviction after the prosecutor failed to disclose the defendant's complete grand jury testimony and then used the undisclosed portions to impeach him); see also McCall v. United States, 596 A.2d 948, 960 (D.C.1991) (Schwelb, J., dissenting). In Wilson we said that reversible error occurs when (1) the prosecutor makes a "pretrial representation of the government's intention [that] creates an obligation to inform the defense and the trial court of new information in a timely manner," (2) "the prosecutor fails to honor his or her pretrial obligation," and (3) "the defendant is prejudiced." Wilson, 606 A.2d at 1020.
In this case, even if we assume for the sake of argument that the government's letter constituted such an implied promise,[5] appellant has not shown any conceivable prejudice. He claims that he intended to argue that the twelve ziplock bags he possessed, only slightly more than the ten bags that show an intent to distribute according to the supposed policy, were not meant for distribution. Appellant's case, however, fits squarely within the policy about which he expected Detective Culver to testify. He possessed twelve bags of cocaine, two bags more than the maximum of ten bags which, under the policy, reflected the outer limits of simple possession. Even had the policy been still in effect at the time of appellant's trial, Culver stated during his testimony that at no point in his career would he have agreed that the twelve ziplock bags found on appellant were for personal use. Furthermore, the issue of whether the twelve bags of cocaine were for personal use or for distribution played no significant part in appellant's defense, because the defense theory was simply that the drugs did not even belong to appellant at all. Finally, regardless of the change in the government's policy, defense counsel had ample opportunity to cross-examine Detective Culver about the policy and the reasons *405 behind it. On this record we can discern no prejudice to appellant's case, and thus no reversible error.

III
At the close of all the evidence, the prosecutor and defense counsel met with the judge to go over the jury instructions. During their discussion, defense counsel requested that the jury be allowed to consider lesser included offenses. The court agreed to give instructions on possession of a controlled substance, possession with intent to distribute, and distribution. After a short recess, the following conversation took place:
MR. PARKS [the prosecutor]: I assume for both counts the jury could find that he wasn't in the drug-free zone, in which case ... he'd only have possession with intent to distribute.
MR. COOPER [defense counsel]: Your Honor, I think the top instruction has to read, for example, distribution in a drug-free zone, and then work its way down from there.
THE COURT: Give that to me again.
MR. COOPER: Given that the charge is, one of the charges is distribution in a drug-free zone
THE COURT: Is there an instruction on that?
MR. COOPER: I know there is.
* * * * * *
THE COURT: I thought I saw a drug-free zone [instruction]. I saw one the other day when I was looking. I have it in my book. I've flipped through my book and haven't seen it. I'm just not sure where it is.
MR. PARKS: I imagine you should find that this was more than a thousand feet from a school zone, if you find Spingarn High School was within a thousand feet of the crime in this case.
THE COURT: Okay. I'll see what I can put together.[6]
Because it was getting late, the judge decided to excuse the jury for the day and invited counsel back to his chambers to finish work on the jury instructions.
The next morning, at a meeting before the jury instructions were given, the prosecutor noted that the verdict form did not mention the drug-free zone charges. The judge said he would have his secretary correct the error. The judge then instructed the jury, but neglected to give any instructions on the drug-free zone element of any of the charges. At a bench conference immediately thereafter, the prosecutor again brought up the drug-free zone element:
MR. PARKS: I think it could be explained with the verdict form where exactly was the drug-free zone
THE COURT: I haven't said anything about it.
MR. PARKS:anything about a thousand feet from Spingarn High School.

*406 THE COURT: Haven't said anything about?
MR. PARKS: Spingarn High School.
THE COURT: Right. What about it?
MR. PARKS: I guess I mean we could do it on the verdict form.
Defense counsel did not mention anything about the drug-free zone element, and in fact said he was satisfied with the instructions as given.
Appellant now argues that the judge's failure to instruct the jury on the drug-free zone element of his charges violated his constitutional right to a jury trial under the Sixth Amendment. Because he did not object to the jury instructions below, however, we review only for plain error.[7]E.g., Johnson v. United States, 520 U.S. 461, 465-466, 117 S.Ct. 1544, 137 L.Ed.2d 718 (1997). To obtain reversal on the ground of plain error, appellant must show (1) that there is error; (2) that the error is plain; (3) that it affects substantial rights; and (4) that it "seriously affect[s] the fairness, integrity, or public reputation of judicial proceedings." Id. at 467, 117 S.Ct. 1544 (citing United States v. Olano, 507 U.S. 725, 732, 113 S.Ct. 1770, 123 L.Ed.2d 508 (1993)).
The trial judge here neglected to instruct the jury on the drug-free zone element of the charges even after the prosecutor repeatedly reminded him about it. This was certainly error. The question for us to decide, however, is whether it was plain error requiring reversal.
In a similar situation, we have said:
[A]t least where no objection is raised at trial to the instructions as given, any such error will not be a cause for reversal where the relevant facts are so closely related that no rational jury, shown by its verdict to have found the facts necessary to convict the defendant under the instructions as given, could have failed, if fully instructed on each element, to have found in addition the facts necessary to comprise the omitted element.
White v. United States, 613 A.2d 869, 879 (D.C.1992) (en banc). In White the defendant was charged with forgery of "a bank check having a value of $250 or more." At trial, the judge neglected to instruct the jury that to find her guilty of the crime charged, the jury also had to find that the value of the check was at least $250. We held that the failure to give such an instruction was not reversible error because "no rational jury" could have found that the writing was false and yet not have found that the check was for $250 or more. Id. at 879.
We followed White in Peterson v. United States, 657 A.2d 756 (D.C.1995), in which the appellant argued that the trial court committed reversible error when it failed to instruct the jury on operability as an element of possession of a prohibited weapon. This court concluded that the circumstantial evidence "was such that no *407 rational jury could have failed to find that the shotgun was operable if fully instructed on that element." Id. at 763 (citations omitted).
In the case at bar, the evidence was also "such that no rational jury could have failed to find" that the crimes occurred within a drug-free zone. First, although the judge did not orally instruct the jury on the drug-free zone element, the jury made that finding on the verdict form. That form, which the judge amended just before he began his jury instructions, asked the jury whether appellant was guilty of Count I, "Distribution of Cocaine Within 1000 Feet of Spingarn Senior High School." The form then went on to say, "If you are unable to reach a verdict on Count I, you may consider the lesser included offense of Distribution of Cocaine." The form was structured similarly for Count II, the PWID charge. In each instance, the jury foreman checked the line which indicated that the jury found appellant guilty of committing the crime "within 1000 feet of Spingarn Senior High School."
Second, even without the verdict form, the evidence was such that a jury could not have found appellant guilty of distributing and possessing drugs without finding that he did so within a drug-free zone. There was uncontroverted testimony that the building where appellant was found with twelve ziplock bags of crack cocaine, in front of which he was standing at the time of Detective Curtis' drug transaction, was 357 feet away from the school. Because the jury found that appellant had been guilty of distribution and PWID, there was no logical way in which it could have found, given the proper instructions, that those crimes did not take place within a drug-free zone.
Moreover, regardless of whether the facts of this case are so closely related under White, appellant cannot meet his burden under the more recent Supreme Court decisions in Neder v. United States, 527 U.S. 1, 119 S.Ct. 1827, 144 L.Ed.2d 35 (1999), and Johnson v. United States, 520 U.S. 461, 117 S.Ct. 1544, 137 L.Ed.2d 718 (1997).[8] In Neder the Court held that "an instruction that omits an element of the offense does not necessarily render a criminal trial fundamentally unfair or an unreliable vehicle for determining guilt or innocence" and is thus subject to harmless error review. Neder, 527 U.S. at 9, 119 S.Ct. 1827 (emphasis in original). In Johnson the Court applied the plain error test to a case in which the judge failed to instruct the jury on the element of materiality in a perjury charge. The Court concluded that the error did not seriously affect "the fairness, integrity or public reputation of judicial proceedings" because the evidence of materiality was "overwhelming." Johnson, 520 U.S. at 470, 117 S.Ct. 1544. Similarly, in this case, the government's evidence showed that the crimes took place within 1000 feet of the school, and appellant did not dispute it.
For these reasons, we conclude that the judge's failure to instruct the jury on the drug-free zone element did not amount to plain error.

IV
During voir dire, the judge asked, "Does anyone live or work or have any regular contact with the area in question, Spingarn High School, which is ... 703 24th Street, Northeast?" Juror number 566, who was *408 eventually selected for the jury, did not indicate that he had any familiarity with the area.
After the verdict, the judge invited the jurors to remain behind to talk with the lawyers if they wished. Juror 566 stayed and told defense counsel that he was aware that the area around Spingarn High School was a high drug area because he had previously taught at the school. Realizing that this juror had not revealed any knowledge of the area during voir dire, counsel prepared an affidavit in which he described his encounter with the juror and stated, "I have no doubt that juror No. 566 intended to mislead the Court and that such lies were not the product of accident or mistake." He filed the affidavit along with a motion for a new trial.
The trial judge decided to hold a limited hearing in order to question the juror. The hearing occurred in the jury room without the defendant present,[9] and with questions asked only by the judge. The hearing was conducted in this manner because, the judge said, he "didn't want to make it seem like it was an adversarial system hearing." Defense counsel, however, was allowed to submit suggested questions,[10] and was also given an opportunity to ask follow-up questions. At the hearing, the juror stated that he had never lived or worked in the area of Spingarn High School and that he had no special familiarity with that neighborhood. He did say, however, that he had worked at Coolidge High School and had administered a test at Ballou High School in the early 1990s. Furthermore, he said that when he saw the picture of Spingarn High School at the trial, he thought at first that it might be Ballou High School, but he now knew it was not Ballou.
After the juror was excused, defense counsel said to the judge, "I know what he told me. I don't think there's anything more that the court can ask him." The judge then heard arguments on the motion for new trial, during which defense counsel requested permission to conduct a full investigation about the juror, including the opportunity to subpoena records from Spingarn High School. The judge, however, denied the motion because he credited the juror's testimony that he had never worked at Spingarn, but had in fact confused it with Ballou. Appellant now argues that the judge erred in not allowing his counsel to conduct a full investigation, not allowing counsel to ask all of his proposed questions, and not holding a full evidentiary hearing with a broader opportunity for cross-examination.
This court has held that "the determination of juror bias or prejudice lies particularly within the discretion of the trial court, reversible only for a clear abuse of discretion ... and the findings of fact underlying that determination are entitled to `great deference.'" Washington v. Washington Hospital Center, 579 A.2d 177, 185 (D.C.1990) (citation omitted); accord, e.g., Leeper v. United States, 579 A.2d 695, 698 (D.C.1990). Furthermore, a trial judge also has considerable discretion in conducting an investigation into alleged juror misconduct. See id. at 699 ("the extent and type of the trial court's investigation into the improper contact are confided to the *409 court's discretion and reviewable only for abuse" (citations omitted)); see also Wilson v. United States, 663 A.2d 558, 562 (D.C.1995). Although we have not yet ruled on the proper scope of a hearing about alleged juror misconduct, several federal courts have held that the hearing does not need to be a full evidentiary hearing, so long as it is sufficient to uncover any prejudice. See United States v. Boney, 314 U.S.App. D.C. 287, 291, 68 F.3d 497, 501 (1995) ("although juror-bias hearings need not be conducted as full evidentiary proceedings, the inquiries put to the juror must be `sufficiently detailed to permit the judge to determine whether any prejudice is likely to [have occurred]'" (citation omitted)); United States v. Boylan, 898 F.2d 230, 258 (1st Cir.), cert. denied, 498 U.S. 849, 111 S.Ct. 139, 112 L.Ed.2d 106 (1990) ("The trial judge may, but need not, convene a full-blown evidentiary hearing.... So long as the district judge erects, and employs, a suitable framework for investigating the allegation and gauging its effects, and thereafter spells out his findings with adequate specificity to permit informed appellate review... his `determination that the jury has not been soured deserves great respect'" (citations omitted)); United States v. Calbas, 821 F.2d 887, 896 (2d Cir.1987), cert. denied, 485 U.S. 937, 108 S.Ct. 1114, 99 L.Ed.2d 275 (1988) (the limited hearing conducted by the trial judge was proper because it prevented the inquiry from becoming an adversarial evidentiary hearing).
Appellant, in arguing that the trial court did not adequately investigate the matter, relies principally on United States v. Boney, in which it was discovered that a juror had lied about his status as a convicted felon. The trial judge in Boney held a hearing on the matter and "asked the juror only two, overly general, questions regarding bias, and refused to ask several more probing inquiries submitted by Boney's counsel." Boney, 314 U.S.App. D.C. at 288, 68 F.3d at 498. The appellate court held that the hearing was not sufficient because the juror had "undisputedly lied," the lie raised an inference that the juror had an undue desire to participate in the case, and more extensive interrogation, including cross-examination, would not have affected the outcome of the case since the trial was already over. Id. at 292-293, 68 F.3d at 502-503.
This case is different. Here there was no clear evidence that the juror had purposely lied, as there was in Boney. The only evidence of misconduct was an affidavit by defense counsel which said that the juror had indicated a familiarity with the area of the crime which he had failed to disclose during voir dire. At the hearing, the judge directly asked specific questions about whether the juror had ever worked at Spingarn High School or had any special knowledge of the area. The juror denied that he had, but he did say that he had confused Spingarn with Ballou High School. Defense counsel was given an opportunity to propose further questions but declined to do so. The judge then found the juror's testimony credible and denied both the motion for new trial and the request for further investigation.
On this record, we are satisfied that the judge did not abuse his discretion in concluding that no broader investigation was required. Although a full-blown investigation into the matter might have allowed the judge to establish definitively whether the juror had lied about his employment, we have often warned trial judges to exercise great caution in allowing jurors to impeach their own verdicts. See, e.g., Posner v. Holmes, 739 A.2d 358, 364 (D.C. 1999); Bates v. United States, 619 A.2d *410 984, 986 (D.C.1993); Sellars v. United States, 401 A.2d 974, 981-982 (D.C.1979); see also Tanner v. United States, 483 U.S. 107, 120-121, 107 S.Ct. 2739, 97 L.Ed.2d 90 (1987) ("full and frank discussion in the jury room, jurors' willingness to return an unpopular verdict, and the community's trust in a system that relies on the decisions of lay people would all be undermined by a barrage of post-verdict scrutiny of juror conduct"). No doubt the judge could have taken more steps to determine whether the juror had lied, but, given the rule against allowing jurors to impeach their own verdicts, set forth in Sellars and many other cases, we cannot say that the judge's actions here amounted to an abuse of discretion.

V
We hold that appellant was not prejudiced by the government's failure to adhere to its ten bag drug policy, that the trial court did not commit plain error in failing to give a drug-free zone instruction, and that the court did not abuse its discretion in limiting the scope of the juror bias hearing. The judgment of conviction is
Affirmed.
NOTES
[1]  D.C.Code §§ 33-541 (distribution and PWID) and 33-547.1 (1998) (drug-free zones), recodified, as amended in part, as D.C.Code §§ 48-904.1 and 48-904.07a (2001), respectively.
[2]  "All areas within 1000 feet of a public or private ... school ... shall be declared a drug-free zone." D.C.Code § 33-547.1(a) (1998), recodified, as amended, as D.C.Code § 48-904.07a (2001).
[3]  This man was apparently never apprehended. The police did stop a suspect, but he was later released when Detective Curtis stated that he was not the man involved in the transaction.
[4]  Detective Culver explained that he had personally changed the policy two years before this case was tried because it was being misapplied in cases in which defendants relied on the packaging of large quantities of crack cocaine in just a few bags to claim that the drugs were for personal use rather than distribution.
[5]  The government disputes that it ever made a promise because the letter stated only that a narcotics expert "may testify" in the manner quoted above, not that the expert "will" so testify. Because we are satisfied that appellant was not prejudiced by Detective Culver's testimony, we need not decide whether the difference between "may" and "will" is legally significant. We note, however, that the prosecutor's letter explicitly declared that "should this matter proceed to trial, the government intends to call a narcotics expert at trial." The letter went on to say that the expert "may be Detective Charles Culver, [another named detective], or some other officer of the Metropolitan Police Department or other law enforcement agency."
[6]  The standard instruction for distribution of a controlled substance in a drug-free zone is found in bracketed language in CRIMINAL JURY INSTRUCTIONS FOR THE DISTRICT OF COLUMBIA, No. 4.30, Alternative B (4th ed. 1993). The instruction states in relevant part:

That the defendant distributed the controlled substance in, on or within one thousand feet of, the real property of a ... school.... The government is not required to prove that the defendant actually knew that s/he was within a thousand feet of a school when s/he committed the offense. The government must only prove that the defendant was in fact within one thousand feet of the school when s/he committed the offense.
The instruction for PWID in a drug-free zone is located in the notes to Instruction No. 4.29, and is nearly identical.
[7]  Appellant argues that the standard of review could be either plain error or constitutional harmless error because the discussion of jury instructions was completed off the record in the judge's chambers. According to appellant, "[f]or all we know, one or both parties may have requested [a drug-free zone] instruction and agreed, with the trial court's acquiescence, that all previous suggestions and objections would be preserved without further ado and without any need to repeat objections at the bench following instructions." The record does not even hint at any such agreement, however, and we cannot speculate that there was one. Moreover, "it is appellant's duty to present this court with a record sufficient to show affirmatively that error occurred." Cobb v. Standard Drug Co., 453 A.2d 110, 111 (D.C.1982).
[8]  The Supreme Court's decisions in Neder and Johnson may have some effect on our holding in White. However, because we conclude that appellant's Sixth Amendment right to a trial by jury was not compromised under either test, we need not decide in this case whether Neder and Johnson supersede White.
[9]  In excluding appellant from the hearing, the judge cited Super. Ct.Crim. R. 43(c)(3), which states that a defendant need not be present at a proceeding which "involves only a conference or hearing on a question of law." Appellant does not challenge this ruling on appeal.
[10]  Although defense counsel submitted ten questions, several of them were excluded as irrelevant.
