
687 A.2d 1356 (1996)
Darius SMITH, Appellant,
v.
UNITED STATES, Appellee.
No. 94-CF-569.
District of Columbia Court of Appeals.
Argued October 18, 1995.
Decided December 30, 1996.
*1358 Richard Greenlee, Public Defender Service, with whom James Klein and David Reiser, Public Defender Service, were on the brief, for appellant.
Leanne Shaltis, Assistant United States Attorney, with whom Eric H. Holder, Jr., United States Attorney, and John R. Fisher, Roy W. McLeese, III, and Ronald L. Walutes, Assistant United States Attorneys, were on the brief, for appellee.
Before WAGNER, C.J., and SCHWELB and FARRELL, Associate Judges.
WAGNER, Chief Judge:
Appellant, Darius Smith, was indicted for premeditated first degree murder while armed (D.C.Code §§ 22-2401 and -3202 (1996)), possession of a pistol during a crime of violence or dangerous offense (D.C.Code § 22-3204(b) (1996)), and carrying a pistol without a license (D.C.Code § 22-3204(a) (1996)). Following a jury trial, Smith was acquitted of the first degree murder charge, but found guilty of the lesser offense of second degree murder while armed and both weapons offenses. He argues that the trial court erred in instructing the jury in a manner which shifted the burden of proof and provided no meaningful guidance on the application of the standard of proof beyond a reasonable doubt. He also contends that the trial court erred in permitting the government to introduce into evidence the plea agreements of two of its witnesses because the documents had the effect of allowing the government to vouch for and bolster the credibility of these witnesses. We hold that the court's instruction, taken as a whole, correctly conveyed the government's burden of proof under the reasonable doubt standard and did not shift or lessen that burden of proof. We also conclude that the admission of the plea agreement, including requirements for the witness to testify truthfully, was not error.

I. The Crime

Quillen Long testified that he was a close friend of the victim, Dontilous Hardy, who was shot and killed on November 21, 1992, by the passenger in a burgundy Maxima while Hardy was a passenger in the Honda which Long was driving. According to Long, he and Hardy had been visiting Hardy's girlfriend, Valawnda Ray, at her home before the shooting. He testified that as they drove away from Ms. Ray's house and stopped for a stop sign at the end of the street, he noticed two black men, one standing outside of a burgundy Maxima who stared "real hard" into Long's vehicle, and the other seated in the driver's seat of the car. Shortly thereafter, Long noticed that same car driving behind his vehicle. After driving two more blocks, the Maxima pulled into the right lane next to Long's car, and the passenger in the Maxima fired gun shots into Long's vehicle. *1359 Long ducked to avoid being shot, but Hardy was hit. At Hardy's request, Long drove him home. Long viewed two different lineups, but he was unable to identify either of the men who were in the Maxima that night.
Leo Spriggs testified that he and Smith had been "best friends." According to Spriggs, he and Smith were outside Smith's house, which was in the same neighborhood as Ms. Ray's home, when he saw the passenger in a white car, whom he recognized as Ms. Ray's boyfriend, give them a "staring glare." Smith said, "Do you want to get him?" Spriggs followed the victim, driving his Maxima with Smith in the passenger seat. According to Spriggs, as he pulled up next to the white car, Smith got into the back seat and fired two shots from the back window while Spriggs was attempting to pull past the white car. They returned to Smith's house where Smith resumed washing his own car.
Michael Allison, who had known Smith for a few years, testified that he knew Spriggs and Smith to be friends and had "hung out" with them before. Allison testified that on the day of the shooting, he saw Spriggs driving by in Spriggs' burgundy Maxima. Smith, who was in the passenger seat of the Maxima, nodded to Allison and the others in the alley as the car sped by. A few seconds or a few minutes later, Allison heard gunshots. He testified that he was sure he had seen Spriggs and Smith in the car from which the shots were fired.
Smith testified on his own behalf and denied involvement in the shooting. According to Smith, he was washing his car when Leo Spriggs and Lamont Spriggs, Leo's brother, drove up in the burgundy Maxima and stopped to talk with him. According to Smith, when the white Honda drove by, the Spriggs brothers jumped into the Maxima and drove off in the same direction. Smith stated that he saw Lamont Spriggs driving the Maxima "at a high speed" about five or ten minutes later, but Leo was not in the car. However, Leo came over to visit him again later. The Spriggs' father testified that both his sons had been with him that afternoon. He said that Lamont had remained at home with him watching a football game, while Leo went somewhere with Smith.
Valawnda Ray, the victim's girlfriend, testified that Hardy had visited her on the afternoon before his death. She testified that Leo Spriggs had telephoned her in the past and that she had the impression that he was romantically interested in her.

II. Claim of Instructional Error


A. The Trial Court's Instruction

Smith argues for reversal on the ground that, over his objection, the trial court gave a reasonable doubt instruction which diminished the government's burden of proof and shifted to the defense the burden of convincing the jury of a "real possibility" of his innocence. He contends that there is a reasonable likelihood that the jury understood the instructions to mean that conviction could be based on less than proof beyond a reasonable doubt. The government contends that the trial court's instruction adequately defined reasonable doubt and did not shift the burden of proof.
At trial, Smith expressed his preference for the standard Redbook instruction on reasonable doubt. Over Smith's objection, the trial court instructed the jury on reasonable doubt using portions of the FEDERAL JUDICIAL CENTER, PATTERN CRIMINAL JURY INSTRUCTIONS, (1988 ed.) (Federal pattern instruction).[1] The court's instruction *1360 on reasonable doubt is printed in the margin of this opinion.[2]
As Smith points out, omitted from the court's instruction to the jury, are the following familiar concepts which appear in the standard Redbook instruction defining reasonable doubt: (1) a doubt based on reason and for which you can give a reason; (2) a doubt which, after careful and candid consideration of all the evidence, leaves you so undecided that you are not firmly convinced of the defendant's guilt; (3) "a doubt as would cause a reasonable person to hesitate or pause in the graver or more important transactions of life"; (4) not a whimsical doubt or one based upon conjecture; and (5) not one which reaches "mathematical or scientific certainty."[3] In addition to these omissions, Smith contends that the instruction *1361 failed to distinguish proof beyond a reasonable doubt from the lower standard of "clear and convincing" evidence and that the "real possibility of innocence" language improperly shifted the burden of proof to the defendant.

B. Instructional Requirements

In a criminal trial, proof beyond a reasonable doubt is a requirement of due process. Victor v. Nebraska, 511 U.S. 1, 5, 114 S.Ct. 1239, 1243, 127 L.Ed.2d 583 (1994); Cage v. Louisiana, 498 U.S. 39, 111 S.Ct. 328, 112 L.Ed.2d 339 (1990) (citing In re Winship, 397 U.S. 358, 364, 90 S.Ct. 1068, 1072-73, 25 L.Ed.2d 368 (1970)) (other citation omitted). While most constitutional errors are subject to a harmless error analysis, the Supreme Court has made clear that an instruction which is constitutionally deficient in describing the government's burden of proof is not amenable to harmless error analysis and will require reversal. Sullivan v. Louisiana, 508 U.S. 275, 278-81, 113 S.Ct. 2078, 2081-82, 124 L.Ed.2d 182 (1993) (citation omitted);[4]see also Butler v. United States, 646 A.2d 331, 333-34 (D.C.1994). As the Supreme Court explained in Sullivan, the premise underlying the harmless error review under Chapman, supra note 4, 386 U.S. at 24, 87 S.Ct. at 828 is not present where the jury has not reached a verdict based on proof beyond a reasonable doubt. Sullivan, 508 U.S. at 279-81, 113 S.Ct. at 2082. The focus under the Chapman analysis is "whether the guilty verdict actually rendered in [the] trial was surely unattributable to the error." Sullivan, 508 U.S. at 279, 113 S.Ct. at 2081. Where the jury has reached a guilty verdict not based upon the reasonable doubt standard, "there has been no jury verdict within the meaning of the Sixth Amendment" to which to apply the harmless error analysis. Id.
This court and the United States Court of Appeals for the District of Columbia Circuit have approved the instruction on reasonable doubt contained in the Redbook which provides a definition for the term. Butler, supra, 646 A.2d at 334; Foreman v. United States, 633 A.2d 792, 793-94 (D.C. 1993); Moore v. United States, 120 U.S.App. D.C. 203, 204, 345 F.2d 97, 98 (1965). Attempts to improve upon the definition all too often fail to achieve the goal of clarity and result in constitutional challenges.[5]See Butler, *1362 646 A.2d at 336 (citing Holland v. United States, supra note 5, 75 S.Ct. at 137-38, 348 U.S. at 140; Taylor, supra note 5, 302 U.S.App.D.C. at 356, 997 F.2d at 1558) (citation omitted); see also United States v. Gibson, 726 F.2d 869, 874 (1st Cir.) cert. denied, 466 U.S. 960, 104 S.Ct. 2174, 80 L.Ed.2d 557 (1984). Given the risk of impermissibly lessening the burden of proof or of confusing the jury on an issue central to the determination of guilt or innocence, this court has urged, in the strongest way, that the trial court not deviate from the standard Redbook instruction. Butler, 646 A.2d at 337.[6] In Butler, this court determined that
for purposes of analysis in the future we shall consider trial court deviations from [the Redbook] instruction, over defense objection, to be improper  indeed, presumptively erroneous....
Id. Therefore, we begin our analysis in this case with that premise in mind. In determining whether the trial court's deviation from the standard instruction is sufficiently prejudicial to require reversal, we consider whether the instruction, viewed in context, creates "`a reasonable likelihood that the jury has applied the challenged instruction in a way' that violates the Constitution." Estelle v. McGuire, 502 U.S. 62, 72, 112 S.Ct. 475, 482, 116 L.Ed.2d 385 (1991) (quoting Boyde v. California, 494 U.S. 370, 380, 110 S.Ct. 1190, 1198, 108 L.Ed.2d 316 (1990)) (footnote omitted). In light of these standards, we examine the claimed deficiencies in the instruction asserted by Smith.

C. Claimed Instructional Deficiencies

Smith contends that the instructions in this case omitted improperly the "hesitate or pause" language which appears in the Redbook instruction and included a "real possibility" formulation which overstates the required level of doubt for acquittal and improperly shifts the burden of proof. With respect to the "hesitate or pause" language, which appears in the Redbook instruction, we have held that its omission from the reasonable doubt instruction is not constitutionally deficient where the instruction otherwise adequately conveys the concept of reasonable doubt to the jury. See Butler, supra, 646 A.2d at 335, 337; see also United States v. Velasquez, 980 F.2d 1275, 1278 (9th Cir.1992) (Omission of the "hesitate to act" language is not necessarily reversible error.); United States v. Barrera-Gonzales, 952 F.2d 1269, 1272 (10th Cir.1992) (Instructions which have "strayed from the `hesitate to act' formula have been upheld."). So long as the instruction, read in context, is not reasonably likely to be understood to allow conviction on less than proof beyond a reasonable doubt, merely omitting a phrase from the approved Redbook instruction will not require reversal. See Butler, 646 A.2d at 337. But see Proctor, supra note 6, (reversing where deletion of this language was combined with substitution of "belief" for "convinced" or "conviction"). Therefore, we construe the instruction as a whole for sufficiency in reviewing the omission.
In this case the trial court instructed the jury that "[p]roof beyond a reasonable doubt is proof that leaves you firmly convinced of the defendant's guilt." The D.C. Circuit, agreeing with the Ninth Circuit, concluded that "`[t]he phrases "firmly convinced" and "hesitate to act" are essentially *1363 two ways of conveying the same definition of reasonable doubt.'" Taylor, supra note 5, 302 U.S.App.D.C. at 355, 997 F.2d at 1557 (quoting Velasquez, supra, 980 F.2d at 1278); see also Barrera-Gonzales, supra, 952 F.2d at 1271 (instruction with the "real possibility" language is similar to approved language of "real doubt" of guilt). This court has encouraged the use of the "firmly convinced" language as used in the Federal pattern jury instruction. Foreman, supra, 633 A.2d at 795.[7] The use of the "firmly convinced" language apprises the jury of the high level of proof required for conviction in a criminal case and does not in itself lessen the standard of proof. United States v. Williams, 20 F.3d 125, 131 (5th Cir.1994); Velasquez, 980 F.2d at 1278. Given the use of the "firmly convinced" formulation here, absent some other deficiency in the instruction, reversal is not warranted.
However, Smith argues that by equating reasonable doubt with "a real possibility" of innocence, the trial court shifted the government's burden of proof to the defense. He contends that, at the very least, the language creates an ambiguity which appears to alter the burden of proof by focusing upon whether the defendant has shown a real possibility of innocence, rather than whether the government has met its burden of proving guilt beyond a reasonable doubt. Although not approved by this court, the language challenged here has been approved by several of the Federal circuit courts.
In Taylor, supra note 5, 302 U.S.App.D.C. at 356, 997 F.2d at 1558, the D.C. Circuit Court held that using a reasonable doubt instruction modeled on the Federal pattern jury instruction definition, like the one used in this case, is not reversible error where "there was no reasonable likelihood that the jury applied the challenged instruction in a way that violates the Constitution." Id. On appeal from conviction of various drug distribution and possession charges, the appellants in Taylor contended that the trial court erred in instructing the jury on reasonable doubt in a way that lowered the government's burden of proof and unconstitutionally shifted that burden to the defense. 302 U.S.App.D.C. at 355, 997 F.2d at 1557. In addition to challenging the "firmly convinced" language, the appellants contended that the "real possibility" language in the Federal pattern instructions impermissibly shifted the burden of proof to the defense. 302 U.S.App.D.C. at 354, 997 F.2d at 1556. The circuit court concluded that the instruction, taken as a whole, was not erroneous. 302 U.S.App.D.C. at 355, 997 F.2d at 1557. In rejecting the argument that the "real possibility" language shifted the burden of proof, the court observed that (1) the instruction did not indicate which side should be the source of showing a real possibility of innocence; and (2) the remainder of the instruction eliminated any concern that the jury would perceive that defendant was required to show a possibility of his innocence. Id. In Taylor, the District Court indicated, no less than thirteen times, that it was the government which was required to prove the defendant's guilt beyond a reasonable doubt. Id. Further, the court gave no other standard than proof beyond a reasonable doubt. The court also noted that although some circuit courts had criticized the Federal pattern jury instruction, "none had found its use reversible error." Id. (citations omitted).
In Barrera-Gonzales, supra, the Tenth Circuit rejected the argument that the real possibility language misdescribed the government's burden of proof where the "firmly convinced" formulation was also included in the instruction. 952 F.2d at 1271. Likewise, in Williams, supra, the Fifth Circuit rejected the argument that the use of language similar to the Federal pattern jury instruction on reasonable doubt was reversible error. 20 F.3d at 131. In Williams, the District Court explained to the jury that proof beyond a reasonable doubt was such as "leaves you firmly convinced of a defendant's guilt." Id. It further instructed

*1364 [i]f, based on your consideration of all the evidence, you are firmly convinced that a defendant is guilty of the crime charged, you must find him guilty. If, however, you think there is a real possibility that he is not guilty, you must give him the benefit of the doubt and find him not guilty.
Williams, supra, 20 F.3d at 127. The court concluded on appeal that the charge as a whole adequately instructed the jury on the required level of proof. Id. As to the "real possibility" formulation, the court stated:
We find no infirmity in this portion of the charge either. When read in the context of the charge as a whole, the instruction's "real possibility" formulation explains that the beyond a reasonable doubt standard does not require "proof that overcomes every possible doubt." In other words, the modifier "real" merely indicates that the jury is not to acquit a defendant if it can conceive of any possibility that the defendant is not guilty. This is because "absolute certainty is unattainable in matters relating to human affairs."
Id. at 131.
In analyzing jury instructions similar to the contested instructions here, the Fourth Circuit, which has urged trial courts not to attempt a definition of reasonable doubt absent a request from the jury because of the risk of confusion or burden shifting, nevertheless has upheld an instruction similar to the one in this case. See United States v. Porter, 821 F.2d 968, 973 (4th Cir.1987) (citing United States v. Love, 767 F.2d 1052, 1060 (4th Cir.1985)). The Court stated that:
Contrary to the appellant's protests, the instruction did not shift the burden of proof on the question of real possibility. Instead it failed to allocate the burden. Other instructions, however, compensated for this omission.... The instructions taken as a whole properly described the prosecution's burden and the protection the law affords the accused. Therefore, the error, which introduced the unnecessary concepts of being "firmly convinced" of guilt and a "real possibility" of innocence, did not affect the substantial rights of the accused. It should be disregarded.
Id. (citation omitted).
Similarly, the First Circuit has held that use of instructions like the one given in this case, while not preferable, does not constitute reversible error. See Gibson, supra, 726 F.2d at 874; accord, United States v. McBride, 786 F.2d 45, 52 (2d Cir.1986); United States v. Hunt, 794 F.2d 1095 (5th Cir.1986).
Each of these courts was persuaded that the charge, read as a whole, did not improperly shift the burden of proof or confuse the jury. While the "real possibility" formulation has been criticized by some federal courts, the language has not resulted in reversal where the instructions taken as a whole adequately apprised the jury that the government bears the burden of proof of guilt beyond a reasonable doubt. See, e.g., Porter, supra, 821 F.2d at 973; McBride, supra, 786 F.2d at 51-52.
In reviewing this type of claim of instructional error, the question is whether the court's instructions, "`as a whole ... correctly conve[y] the concept of reasonable doubt to the jury.'" Victor, supra, 511 U.S. at 5, 114 S.Ct. at 1243 (quoting Holland v. United States, supra, 348 U.S. at 140, 75 S.Ct. at 138). The proper focus is whether there is a reasonable likelihood that the jury applied the instruction in an unconstitutional manner. Victor, 511 U.S. at 6, 114 S.Ct. at 1243. In making that assessment, among the factors for consideration are: (1) whether the court misstated the law; (2) whether the required standard of proof was adequately explained; and (3) whether the instruction was confusing. See Butler, supra, 646 A.2d at 337-38.
Here, the trial court did not misstate the law and it adequately conveyed the concept *1365 of reasonable doubt to the jury.[8] The trial court instructed the jury at least five times that the burden was on the government to prove the defendant guilty beyond a reasonable doubt, as well as that the defendant was not required to prove his innocence or produce any evidence. Therefore, we perceive no possibility that the jury misunderstood where the burden of proof was lodged. The court also provided a workable definition of what proof beyond a reasonable doubt means, employing the previously approved "firmly convinced" language. We agree with those federal circuit courts which have held that the inclusion of the "real possibility" formulation, at least in context with the remainder of the instruction given here and in those cases, did not indicate to the jury that the government had a lesser burden. See, e.g., Porter, supra, 821 F.2d at 973; Gibson, supra, 726 F.2d at 874; McBride, supra, 786 F.2d at 51. However, we caution the trial court that use of the "real possibility" language might pose a problem if the remainder of the instruction is inadequate to assure that no confusion or burden shifting results. In light of the serious risk of misdescribing the burden of proof, we again admonish the trial court to adhere to the approved language in this area. See Butler, 646 A.2d at 337-38. In this case, we conclude that the instructions as a whole did not misdescribe the burden of proof or create the reasonable likelihood that the jury misunderstood the government's burden of proof.[9] Therefore, we find no reversible error.

III. Admission of Witness' Plea Agreements

Smith argues that the trial court erred in permitting the government to introduce, over objection, plea agreements with two of its witnesses. He contends that the agreements improperly vouched for and bolstered the credibility of the government's key witnesses and contained information having a prejudicial effect outweighing any probative value. The government argues that, where a defendant brings out that the witness is testifying pursuant to a plea agreement with the government, the prosecutor is permitted to elicit the contents of the agreement during its direct examination of the witness. It also contends that provisions of the agreement requiring truthful testimony from the witness have not been held not to amount to vouching for the witness' credibility when placed before the jury. The government also challenges Smith's claim of undue prejudice.
This court has not addressed specifically whether the government may introduce evidence on direct examination concerning its witness' plea agreement with the government or whether the agreement may be introduced in its entirety. The majority of the circuits, including the D.C. Circuit, have held that the prosecutor may introduce on direct examination its witness' plea agreement in its entirety. United States v. Spriggs, 996 F.2d 320, 324 (D.C.Cir.1993) (citations omitted); United States v. Lord, 907 F.2d 1028, 1029 (10th Cir.1990). The argument against introduction, during the direct examination of the witness, of the entire agreement which requires the witness to provide truthful testimony is that it allows the government to bolster the witness' testimony before it is attacked. United States v. Borello, 766 F.2d 46, 56 (2d Cir.1985); United States v. Edwards, 631 F.2d 1049, 1052 (2d Cir.1980). The case law in the Second Circuit precludes the introduction of such evidence prior to a challenge to the witness' credibility. Borello, 766 F.2d at 56. Recognizing that such agreements have the effect of both bolstering and discrediting, the Second Circuit created an exception which allows the government to introduce circumstances concerning the witness' motivation to testify for the government during its direct examination of the witness in order to keep the jury from inferring that the government is attempting to hide evidence of the witness' bias which is later brought out by the defense. Id. at 57. Although at least two circuits prohibit introduction *1366 of those portions of the agreement which require truthfulness until the witness' credibility is challenged, the difficulties attendent to making such redactions during trial have caused other courts to opt in favor of introduction of the entire agreement. See Lord, 907 F.2d at 1030 (citing United States v. Cosentino, 844 F.2d 30, 32-35 (2d Cir. 1988), cert. denied, 488 U.S. 923, 109 S.Ct. 303, 102 L.Ed.2d 322 (1988) and United States v. Cruz, 805 F.2d 1464, 1479-80 (11th Cir.1986), cert. denied, 481 U.S. 1006, 107 S.Ct. 1631, 95 L.Ed.2d 204 (1987)).
The D.C. Circuit has adopted the position of the majority of the circuits which hold that the agreement may be admitted in its entirety during the direct examination of the witness by the government. Spriggs, supra, 996 F.2d at 324. The court reasoned that the provision of the agreement providing that the witness can be prosecuted for perjury for false testimony referenced no new law or incentive which would suggest to the jury that special credence should be attached to the witness' testimony. Id. Moreover, the court observed, there was nothing in the agreement which enhanced the government's ability to detect that the witness was lying or suggested that the government had special knowledge of the witness' veracity.[10]Id. (citing United States v. Henderson, 717 F.2d 135, 138 (4th Cir.1983) cert. denied, 465 U.S. 1009, 104 S.Ct. 1006, 79 L.Ed.2d 238 (1984)).
Evidence of a plea bargain can be both favorable and unfavorable to the government. On the one hand, it may impeach the witness' credibility by showing the witness' incentive for providing testimony favorable to the government solely for the witness' own personal gain. Henderson, supra, 717 F.2d at 137. On the other hand, such evidence may tend to suggest to the jury that the witness has relevant and truthful information, the disclosure of which the prosecutor forces through the terms of the agreement. Id. (citing United States v. Roberts, 618 F.2d 530, 536 (9th Cir.1980), cert. denied, 452 U.S. 942, 101 S.Ct. 3088, 69 L.Ed.2d 957 (1981)). However, "introduction of the entire plea agreement permits the jury to consider fully the possible conflicting motivations underlying the witness' testimony and, thus, enables the jury to assess more accurately the witness' credibility." United States v. Townsend, 796 F.2d 158, 162 (6th Cir.1986). Given this two-sided nature of the issues raised by such agreements, which must be resolved ultimately by the jury, admission of the agreements specifying the consequences of perjury are not considered impermissible bolstering of the witness' testimony. See United States v. Edelman, 873 F.2d 791, 795 (5th Cir.1989) (citing United States v. Martino, 648 F.2d 367, 389 (5th Cir.1981), cert. denied, 456 U.S. 949, 102 S.Ct. 2020, 72 L.Ed.2d 474 (1982)).
In this case, defense counsel mentioned in opening statement to the jury that the testimony of the government's witnesses, Leo Spriggs and Michael Allison, could not be relied upon because they would receive favorable treatment in their own criminal cases in exchange for their testimony. During Allison's direct testimony, the prosecutor read provisions of Allison's plea agreement with the government, which explained the charges and that the government would seek a lesser sentence for him if it determined that he had provided substantial assistance in a case involving another person. The prosecutor also read, over defense objection, the provision which forbade perjury. In a bench conference which preceded this testimony, defense counsel objected to the inquiry generally on the ground that the government would be vouching improperly for its own witness, and specifically to the perjury provision. When Leo Spriggs testified, the prosecutor elicited similar information concerning his plea agreement with the government by asking that he acknowledge various terms as his *1367 agreement. Defense counsel did not object, but when the government moved the complete agreements into evidence at the close of its case, the defense objected to their admission. Defense counsel cross-examined both Allison and Spriggs about the agreement with the government and its advantages to them.
We are persuaded by the reasoning of the majority of the circuits which have held "that elicitation during direct examination of a plea agreement containing a promise to testify truthfully does not constitute impermissible bolstering of the witness' credibility." Townsend, supra, 796 F.2d at 163 (citations omitted); Spriggs, supra, 996 F.2d at 325. Indeed, we have held that disclosure of a witness plea agreement, "including the requirement therein that he testify truthfully, [does] not constitute improper prosecutorial vouching for [the witness'] veracity." Ford v. United States, 616 A.2d 1245, 1254 n. 21 (D.C.1992) (citing Felder v. United States, 595 A.2d 974, 979 (D.C.1991)). We stated that
we have considerable difficulty with the proposition that such supposed "vouching" would have substantially swayed an impartial jury where the witness in question had already been introduced by the prosecutor and had sworn in the presence of the jury to tell the truth.
Id. (citation omitted). The opinion in Ford does not state at what point portions of the plea agreement were mentioned to the jury. However, that makes no difference to the outcome of this case. Here, defense counsel raised the issue of the two witnesses' motivations not to tell the truth in opening statement. This previewed the challenge that would be made to their testimony based upon the plea agreements. Prior to the admission of the agreements in their entirety, defense counsel cross-examined the witnesses concerning the agreements, thereby making the nature of the agreements an issue in the case. See Edelman, supra, 873 F.2d at 795. Under such circumstances, we find no error in their admission. See id.
Smith argues for the first time on appeal the prejudicial nature of two other specific provisions of the agreement, i.e., the willingness of the witnesses to take a polygraph examination and the witness protection provisions. Smith contends that the polygraph provision vouched for the witness' credibility and that the witness protection provision suggested that he must be a dangerous person. We review Smith's claim for plain error.[11]See Harris v. United States, 602 A.2d 154, 159-60 (D.C.1992) (en banc) (error must be readily apparent and "so prejudicial as to jeopardize the fairness and integrity of the trial" to warrant reversal). We are not persuaded that the court's failure to exclude the evidence on these grounds amounted to plain error. The provisions pertaining to the polygraph commitment were not commented upon by the prosecutor or the witnesses, and they were but one of many provisions of the lengthy agreements, which were but two items of evidence in the trial. See United States v. Rosa, 891 F.2d 1063, 1068 (3d Cir.1989) (low profile and marginal significance of a polygraph commitment not prejudicial error where government's case is strong); United States v. Herrera, 832 F.2d 833, 835 (4th Cir.1987) (references to polygraph in plea agreement harmless error); Porter, supra, 821 F.2d at 974 (references in plea agreement to witness' polygraph commitment harmless beyond a reasonable doubt where not used to bolster credibility of crucial witnesses). In addition, there was no evidence that either witness ever took a polygraph test. See Herrera, 832 F.2d at 835. Under the circumstances we find no plain error in the admission of this evidence.
Similarly, we reject Smith's argument that the admission of the witness protection provisions of the agreements was unduly *1368 prejudicial or amounted to plain error. This provision was not highlighted. Moreover, the defense cross-examined Allison extensively about the reason for his failure to come forward voluntarily, and Allison explained what happens to people in his neighborhood who help the police. Such evidence is not inadmissible. See Carpenter v. United States, 635 A.2d 1289, 1294 (D.C.1993) (witness' testimony about what happens in general to snitches relevant and admissible to explain fear of going to the police). The need for a witness protection provision is explained to some extent by Allison's testimony in this regard. Therefore, we are not persuaded that such a provision as one of many terms in the plea agreement, which was not highlighted during the trial, was plain error. Since the circumstances concerning what happens to snitches generally were already in evidence with respect to Allison, we see no undue prejudice in the introduction of Spriggs' agreement containing the same term. Accordingly, we find no plain error.
For the foregoing reasons, the judgment appealed from hereby is
Affirmed.
SCHWELB, Associate Judge, concurring:
I adhere to the views expressed in my dissenting opinion in Proctor v. United States, 685 A.2d 735 (D.C.1996), with respect to what I regard as unnecessary language in Butler v. United States, 646 A.2d 331 (D.C. 1994) and Foreman v. United States, 633 A.2d 792 (D.C.1993). In all other respects, I join the opinion of the court.
FARRELL, Associate Judge, concurring.
I join the opinion of the court, and have only a few observations concerning the reasonable-doubt instruction given in this case. The departure here from the Redbook instruction is, of course, further reason why the court en banc should revisit the issue of what instruction is necessary on reasonable doubt. See Proctor v. United States, 685 A.2d at 740-42 (D.C.1996) (separate remarks of Associate Judge FARRELL). I am not of the view, apparently shared by two or more federal appellate courts, that the jury should receive no explanation of that concept. Given the "vital role" which it plays in our scheme of criminal procedure, Cage v. Louisiana, 498 U.S. at 39-40, 111 S.Ct. at 329, 112 L.Ed.2d 339 (1990) (quoting In re Winship, 397 U.S. 358, 363, 90 S.Ct. 1068, 1072, 25 L.Ed.2d 368 (1970)), juries should not "be left without some elaboration of what reasonable doubt means." Butler v. United States, 646 A.2d 331, 337 (D.C.1994). The issue in the present case arises only from the appearance of two words ("real possibility") in the federal pattern instruction which the judge gave and which I believe provides a model for this court to adopt and mandate in future cases. Proctor, 685 A.2d at 741-42, (separate remarks of Associate Judge FARRELL). Appellant argues that "real possibility" could be taken by the jury to mean a "serious" or "substantial" possibility  on the order of "substantial evidence" in the civil administrative context  which is more than reasonable doubt requires. But the court explains convincingly why, taken as a whole, the instructions here do not realistically allow of that possibility. For future purposes, however, the arguable ambiguity in these words can be remedied easily by the addition of a few words, i.e., "a real possibility as distinct from an imaginary or fanciful one." With that minor supplementation, the federal pattern instruction remains a strong improvement over the Redbook explanation of reasonable doubt.
NOTES
[1]  The government argues that Smith's instructional challenge is subject to plain error review because he did not object to the instruction on the basis of the impropriety of the definition of reasonable doubt. Although Smith objected most to a portion of the instruction proposed by the court, he clearly objected to any deviation from standard jury instruction no. 2.09, CRIMINAL JURY INSTRUCTIONS FOR THE DISTRICT OF COLUMBIA (4th ed. 1993) (Redbook). After the court stated that it did not use the standard Redbook jury instruction, trial counsel stated his objection as the following transcript excerpts show.

Counsel: I do object to this, Your Honor.
Court: Make your record. I thought you would.
Counsel: I prefer the standard instruction. The portion of this proposed instruction that I object most firmly to is the last line, "if on the other hand you think there is a real possibility that he is not guilty," I don't believe the law requires them to find a real possibility that he's not guilty.
* * * * * *
I object to the pattern instruction and as a whole I prefer the original as well.
After the luncheon recess, in further discussing the instructions, there was the following colloquy between the court and defense counsel about the Federal pattern instruction.
Counsel: I take it the court is going to read the proposed instruction over objection?
Court: Yes.
Counsel: Rather than the traditional?
Court: Yes. I think it's proper and legal....
Counsel: I under[stand] the Court's decision now. In that case, I'd ask the whole thing be read and not 
Court: All right. And so 
Counsel: I withdraw my request for the last line to be eliminated.
Court: I'm going to give that [the pattern jury instruction] over objection. I understand that. But I am going to give all of it.
Faced with the prospect that the trial court would give the Federal pattern instruction over his objection, defense counsel simply expressed the preference that it be given in its entirety. The trial court discerned from all that transpired, as it stated to defense counsel when it considered reinstruction during deliberations, "You wanted the Redbook Instruction and I think  I do think your objection is preserved." Smith filed a motion for new trial on the ground that the trial court erred in instructing the jury on reasonable doubt, and the government argued in opposition that the objection was untimely under Super. Ct.Crim. R. 30. The trial court disagreed. We agree with the trial court that Smith preserved his objection and reject the government's argument for plain error review. See United States v. Loriano, 302 U.S.App.D.C. 158, 996 F.2d 424 (1993).
[2]  The instruction given by the court read as follows:

Every defendant in a criminal case is presumed to be innocent. This presumption of innocence remains with the defendant throughout the trial unless and until he's proven guilty beyond a reasonable doubt.
The burden is on the government to prove the defendant guilty beyond a reasonable doubt. The burden of proof never shifts throughout the trial. The law does not require the defendant to prove his innocence or to produce any evidence.
If you find the government has proven beyond a reasonable doubt every element of the offense with which the defendant is charged, it is your duty to find him guilty. On the other hand, if you find the government has failed to prove any element of the offense beyond a reasonable doubt, you must find the defendant not guilty.
As I have said many times, the government has the burden of proving the defendant guilty beyond a reasonable doubt. Some of you may have served as jurors in civil cases where you were told that it is only necessary to prove that a fact is more likely true than not true.
In criminal cases the government's proof must be more powerful than that. I[t] must be beyond a reasonable doubt. Proof beyond a reasonable doubt is proof that leaves you firmly convinced of the defendant's guilt.
There are very few things in this world that we know with absolute certainty. And in criminal cases the law does not require proof that overcomes every possible doubt. If based on your consideration of the evidence you are firmly convinced that the defendant is guilty of the crime charged, you must find him guilty. If, on the other hand, you think there's a real possibility that he's not guilty, you must give him the benefit of the doubt and find him not guilty.
[3]  The standard Redbook instruction, which embodies these points, reads:

Reasonable doubt, as the name implies, is a doubt based on reason, a doubt for which you can give a reason. It is such a doubt as would cause a juror, after careful and candid and impartial consideration of all the evidence, to be so undecided that he or she cannot say that he or she has an abiding conviction of the defendant's guilt. It is such a doubt as would cause a reasonable person to hesitate or pause in the graver or more important transactions in life. However, it is not a fanciful doubt, nor a whimsical doubt, nor a doubt based on conjecture. It is a doubt based on reason. The government is not required to establish guilt beyond all doubt, or to a mathematical certainty or a scientific certainty. Its burden is to establish guilt beyond a reasonable doubt.
CRIMINAL JURY INSTRUCTIONS FOR THE DISTRICT OF COLUMBIA, No. 2.09 (4th ed.1993).
[4]  In Sullivan, the Supreme Court explained why a constitutionally deficient reasonable doubt instruction must be reviewed differently than other constitutional errors which may be subject to a harmless error analysis under Chapman v. California, 386 U.S. 18, 24, 87 S.Ct. 824, 828, 17 L.Ed.2d 705 (1967) (reversal not required where it is shown "beyond a reasonable doubt that the error complained of did not contribute to the verdict.").

Consistent with the jury-trial guarantee, the question [Chapman] instructs the reviewing court to consider is not what effect the constitutional error might generally be expected to have upon a reasonable jury, but rather what effect it had upon the guilty verdict in the case at hand.... The inquiry, in other words, is not whether, in a trial that occurred without the error, a guilty verdict would surely have been rendered, but whether the guilty verdict actually rendered in this trial was surely unattributable to the error. That will be so, because to hypothesize a guilty verdict that was never in fact rendered  no matter how inescapable the findings to support that verdict might be  would violate the jury trial guarantee....
Since ... there has been no jury verdict within the meaning of the Sixth Amendment, the entire premise of Chapman review is simply absent. There being no jury verdict of guilty-beyond-a-reasonable-doubt, the question whether the same verdict of guilty-beyond-a-reasonable-doubt would have been rendered is utterly meaningless.... The most an appellate court can conclude is that a jury would surely have found petitioner guilty beyond a reasonable doubt  not that the jury's actual finding of guilty beyond a reasonable doubt would surely not have been different absent the constitutional error. That is not enough.
Sullivan, 508 U.S. at 279-80, 113 S.Ct. at 2081-82 (citations omitted).
[5]  The Supreme Court has held that the Constitution does not require or prohibit trial courts from defining the term, proof beyond a reasonable doubt. Victor, supra, 511 U.S. at 5, 114 S.Ct. at 1243. However, when a definition is given, the instruction as a whole must convey properly the concept. See Holland v. United States, 348 U.S. 121, 140, 75 S.Ct. 127, 138, 99 L.Ed. 150 (1954). Some courts view the concept of reasonable doubt to be self-evident. Butler, supra, 646 A.2d at 336 (citing Murphy v. Holland, 776 F.2d 470, 475 (4th Cir.1985), vacated on other grounds, 475 U.S. 1138, 106 S.Ct. 1787, 90 L.Ed.2d 334 (1986)). The D.C. Circuit has expressed the opinion "that the greatest wisdom may lie with the Fourth Circuit's and Seventh Circuit's instruction to leave to juries the task of deliberating the meaning of reasonable doubt." United States v. Taylor, 302 U.S.App.D.C. 349, 997 F.2d 1551, 1558 (1993). However, this court has taken the position that the jury should be instructed on the meaning of proof beyond a reasonable doubt. Butler, 646 A.2d at 336, 337.
[6]  See Proctor v. United States, 685 A.2d 735 (D.C. 1996) (reversing convictions on basis of Butler and Foreman, supra, where trial court substituted "an abiding or deep-seated belief" for "abiding conviction" and also deleted the "hesitate or pause" sentence contained in the Redbook instruction).
[7]  In Foreman, we approved the substitution of the words "firmly convinced" for the language in the Redbook instruction which defines reasonable doubt as an inability of one to say that he or she has an abiding conviction of the defendant's guilt. See Foreman, 633 A.2d at 795 & 795 n. 5.
[8]  See note 2, supra, quoting the trial court's instructions.
[9]  In reaching this conclusion, we have considered all of the omissions which Smith cites.
[10]  In Spriggs, the prosecutor inquired during the witness' direct examination about the witness' cooperation with the government and brought out that the terms of the agreement included: (1) that he could be prosecuted for perjury for false testimony; (2) that he had testified against the defendant before the grand jury which returned the indictment; and (3) that he hoped to receive a lighter sentence as a result of his cooperation. 996 F.2d at 323. Although the issue was subject to plain error review, the court found no error, and clearly no plain error. Id. at 324.
[11]  We reject Smith's claim that the general objection to the admission of the plea agreements and the specific objection only to the term requiring truthful testimony was sufficient to apprise the court of the specific objections he now raises.
