opinion: RICHARD L. WILLIAMS, District Judge:
  This case involves a six count indictment brought against four defendants for bid rigging a sewer construction project in Lancaster County, South Carolina. The Farmers Home Administration funded the four part sewer project.
  Defendant Herbert P. Lee, III is an employee of defendant Smith Grading and Paving Inc. [Smith Grading] and is responsible for the company’s bids on all utility-related projects. Defendant Theodore C. Dellinger is the president of defendant Del-linger, Inc. and is responsible for his company’s bids. All four defendants were charged, in Count 1 of the indictment, with conspiring with unnamed co-conspirators to rig bids on the Lancaster County, South Carolina sewer project in violation of § 1 of the Sherman Act, 15 U.S.C. § 1. Counts 2 and 3 charged Dellinger and Dellinger, Inc. with submitting false statements in their bids in violation of 18 U.S.C. § 1001. Counts 4, 5 and 6 charged Lee and Smith Grading with filing false statements.
  At trial the jury found the defendants guilty on all counts. On appeal defendants claim eleven reversible errors.
  FACTS
  Viewing the evidence in the light most favorable to the government, the prosecutor’s case at trial established the following facts. On December 19, 1978 eleven companies submitted bids on one or more of the four parts of the Lancaster County sewer project. Frank Carpenter, representing Dickerson, Inc., was the low bidder on part 1, while B.S. Zeigler, representing Boozer and Wharton, Inc., successfully bid on part 2. Defendants Dellinger Inc. and Smith Grading were, respectively, the low bidders on parts 3 and 4 of the construction project.
  Carpenter and Zeigler turned states evidence and testified as to the bid rigging scheme. According to their testimony, all four successful bidders stayed at the Carriage Inn in Lancaster County the night before the bid submission. After dinner, the four met in Zeigler’s room, at which time Carpenter proposed that they rig the sewer construction project. Each man was to be the low bidder on one of the four parts of the job. According to both Carpenter and Zeigler, defendants Lee and Dellinger concurred with the plan. The men exchanged figures and contacted other potential bidders. Telephone records and the amount of the bids corroborated the testimony.
  DISCUSSION
  Defendants’ most significant claim on appeal involves the admissibility of extrinsic evidence of defendant Dellinger’s prior bid rigging activities. On cross-examination the prosecutor asked Dellinger if he previously rigged a project in Kensington, South Carolina. Dellinger denied the allegation. The district court then allowed the government to present a rebuttal witness who testified that Dellinger participated in the Kensington bid rigging scheme. On appeal defendants argue that the trial court violated Federal Rule of Evidence 608(b) when it admitted the rebuttal evidence. Rule 608(b) prohibits a party from offering extrinsic evidence to prove specific instances of conduct for the purpose of attacking or supporting a witness’s credibility.
  In response, the government argues that the trial court properly admitted the evidence of Dellinger’s past misconduct under Federal Rule of Evidence 404(b). Rule 404(b) allows a trial court to admit evidence of other crimes or wrongs if (1) it is offered for a purpose other than to prove a defendant’s criminal character or propensity to commit the crime charged, and (2) the probative value of the evidence outweighs its prejudicial effect. United States v. Tate, 715 F.2d 864 (4th Cir.1983). In this case, the government argues that the rebuttal evidence of Dellinger’s participation in the Kensington bid rigging scheme is relevant to the defendant’s intent to enter a conspiracy to rig bids as well as his understanding of the nature of the scheme. It further argues that the disputed evidence is more probative than prejudicial.
  As a preliminary matter, we acknowledge that the evidence at issue falls within the scope of both rules 608(b) and 404(b). The government’s proof of Dellinger’s participation in the earlier bid rigging scheme took the form of extrinsic evidence attacking a witness’s credibility. The government did not have to accept the defendant’s denial of participation. Yet, likewise, the evidence of Dellinger’s prior bid rigging is relevant to his intent and knowledge of the charged conspiracy, which are material issues raised in Count 1 of the indictment. United States v. United States Gypsum Co., 438 US. 422, 98 S.Ct. 2864, 57 L.Ed.2d 854 (1978). Under an abuse of discretion standard, the trial court did not err when it determined that the disputed evidence was more probative than prejudicial, for the close relationship between the prior illegal activity and the charged offense is probative of Dellinger’s knowledge. United States v. Marques, 600 F.2d 742, 751 (9th Cir.1979).
  Given the fact that the evidence of Dellinger’s earlier bid rigging falls within the scope of both rules, we must decide whether rule 404(b) takes priority over rule 608(b). All of the circuit courts presented with the conflict between the rules have held, at least implicitly, in favor of rule 404(b). These courts have allowed the admission of evidence of past bad acts, offered for a purpose other than to prove a defendant’s bad character or propensity to commit a crime, despite the fact that the evidence came in after the defendant denied the misconduct on cross-examination. See United States v. Jacobson, 578 F.2d 863 (10th Cir.1978); United States v. Batts, 558 F.2d 513 (9th Cir.1977) vacated 573 F.2d 599 (9th Cir.1978), cert. denied 439 U.S. 859, 99 S.Ct. 178, 58 L.Ed.2d 168 (1979); United States v. Herzberg, 558 F.2d 1219 (5th Cir.1977).
  We agree with the other circuits. Rule 608(b) should not be read so broadly as to disallow the presentation of extrinsic evidence that is probative of a material issue in a case. Undoubtedly rule 404(b) will undermine the purpose of rule 608(b) in particular circumstances, such as those presented in this case. However, the ultimate goals of the rules of evidence are best served by not reading any rule in isolation. In light of the need to introduce probative evidence of material issues and, even more importantly, in light of the goal of ascertaining the truth in judicial proceedings, we affirm the trial court’s admission of rebuttal testimony of prior bad acts, over the defendant’s denial, given that the testimony satisfies the requirements of rule 404(b).
  One consequence of our decision is it allows the government to present rule 404(b) evidence on rebuttal. We want to make clear, however, that the better practice is for the prosecutor to introduce this type of evidence in his or her case in chief. In most instances, evidence of a defendant’s past bad acts will surface prior to trial on a motion in limine. At that time, on a proffer by the government, the trial court must determine whether the evidence is relevant, is admissible under rule 404(b), and whether the evidence is more probative than prejudicial.
  However, if the government attempts to introduce past bad acts through cross-examination of the defendant or on rebuttal, the trial court should also determine whether the evidence is cumulative or necessary to prove an essential element of the crime charged. If the evidence is cumulative, the trial court should more closely scrutinize its prejudicial effect. If the evidence is necessary to prove an element of the charge, the trial judge may have erred in not sustaining the defendant’s motion for a directed verdict of acquittal. Moreover, a defendant may have a valid objection, under federal rule 611(b), that the government’s cross-examination exceeds the scope of the direct. These added considerations are necessary not only to limit the impact of rule 404(b) on rule 608(b), but also to limit the prejudicial effect of allowing the government to have the last say through evidence of a defendant’s prior misconduct.
  With these guidelines, we affirm the trial court’s admission of rule 404(b) evidence over the defendants’ 608(b) objection and now turn to the ten other claims raised by the defendants.
  Defendants next contend that the trial court improperly admitted evidence of defendant Dellinger’s prior bad acts during the government’s case in chief. On direct examination, government witness Carpenter testified that he and Dellinger engaged in bid rigging on three other projects, two prior to and one after the Lancaster County sewer project. Carpenter did not remember his exact conversation with Del-linger. Defendants claim that the evidence was more prejudicial than probative.
  As mentioned previously, prior bad acts of a defendant are admissible under rule 404(b) as long as they are offered for a purpose other than to prove bad character and are more probative than prejudicial. In this case, the trial court properly admitted, during the government’s case in chief, Carpenter’s testimony. Dellinger’s other bid rigging was probative of his knowledge of entering a bid rigging conspiracy. Moreover, the trial judge did not abuse his discretion by determining that the evidence was more probative than prejudicial as far as its effect on all four of the defendants. On two separate occasions during trial, the court gave the jury a comprehensive instruction on the limited use of rule 404(b) evidence, which is the suggested procedure of this circuit. See United States v. Teague, 737 F.2d 378, 381 (4th Cir.1984).
  Defendants’ third claim of error focuses on the government’s alleged failure to disclose exculpatory evidence under Brady v. Maryland, 373 U.S. 83, 83 S.Ct. 1194, 10 L.Ed.2d 215 (1963). Defendants contend they were denied due process of law because the government did not disclose evidence provided by the engineer who originally estimated the cost of the Lancaster County sewer project. On cross-examination, the engineer testified that he deliberately under-estimated the project’s cost and expected the bids to exceed his estimate. Prior to his testimony, the defendants were unaware of the information. They argue that the testimony was exculpatory because it illustrates that their bids were not excessively high and, therefore, should have been disclosed under Brady.
  Even if we assume that the engineer’s testimony is exculpatory, its belated disclosure does not constitute reversible error. No due process violation occurs as long as Brady material is disclosed to a defendant in time for its effective use at trial. United States v. Higgs, 713 F.2d 39 (3rd Cir.1983). In this case, the exculpatory information was put before the jury during cross-examination of the very first trial witness. The information was available for use in the defendant’s cross-examination of all further government witnesses as well as in the defendants’ case in chief. The disclosure of this exculpatory evidence, at trial, does not rise to the level of a constitutional violation.
  Defendants’ fourth claim of reversible error is that they were denied due process of law when the trial court refused to clarify an alleged inference of attorney misconduct. While on cross-examination, a government witness asked Dellinger’s defense counsel whether he was the attorney paying him $500.00 per day. (The payment was a witness fee.) A few minutes later, the trial court recessed for the day, but in the presence of the jury asked Dellinger’s counsel to remain for a moment. The next morning the Court instructed the jury on the propriety of witness fees, but did not instruct the jury as to why the Court asked to speak to counsel.
  Defendants argue that the trial court’s failure to give the requested instruction left an inference of attorney misconduct, which in turn violated their right to a fair trial. The alleged error does not constitute a due process violation. The trial court explained to the jury that witness fees were proper, adequately dispelling any notion that Dellinger’s defense counsel acted improperly.
  Defendants next argue that their convictions should be reversed because the trial court improperly admitted testimony of Smith Grading’s ownership interest in two other contracting companies. During defendant Lee’s cross-examination, the government elicited testimony that Smith Grading owned part of the contracting firms RWF, Inc. and Bingham Construction. RWF, Inc. also bid on the Lancaster County sewer project, while Bingham Construction, along with Smith Grading, was a successful bidder on a different job known as the Hampton project. Defendants claim the admission of this evidence was both irrelevant and improper.
  Although the relevance of the evidence was limited, the inclusion of Lee’s testimony was not an abuse of the trial court’s discretion. Smith Grading’s ownership interest in both RWF, Inc. and Bingham Construction was probative of defendants’ participation in the Lancaster County bid rigging scheme. RWF, Inc. only bid on part 2 of the project — the one section Smith Grading did not bid upon. Moreover, Smith Grading’s ownership interest in Bingham Construction helped explain particular telephone records, corroborating the defendants’ participation in the scheme.
  Defendants’ sixth claim of reversible error focuses on the trial court’s conspiracy instruction. In its charge, the trial court instructed the jury that the defendants must have knowingly participated in a conspiracy in order to be found guilty of violating § 1 of the Sherman Act. The court went on to say that a conspiracy to rig bids is per se illegal and, therefore, the government need not prove defendants’ specific intent to unreasonably restrain trade. Defendants argue that the court’s conspiracy instruction should have included a specific intent instruction.
  As indicated in both United States v. Portsmouth Paving Corp., 694 F.2d 312 (4th Cir.1982) and United States v. Society, of Independent Gasoline Marketers, 624 F.2d 461 (4th Cir.1979), cert. denied 449 U.S. 1078, 101 S.Ct. 859, 66 L.Ed.2d 801 (1981), this circuit recognizes per se antitrust violations. Therefore, the trial court correctly instructed the jury that the government must prove the defendants’ intentional participation in the conspiracy to rig bids. The government did not have to prove the defendants’ specific intent to unreasonably restrain trade. See also United States v. Brighton Building, 598 F.2d 1101 (7th Cir.1979), cert. denied 444 U.S. 840, 100 S.Ct. 79, 62 L.Ed.2d 52 (1979).
  Defendants’ seventh claim is that the trial court erred in its reasonable doubt instruction by not including language that a reasonable doubt is the type of doubt that would make a reasonable person hesitate to act. The trial court’s instruction included the sentence, “Proof beyond a reasonable doubt is established if the evidence is such as you would be willing to rely and act upon in the more important matters of your own life.”
  In United States v. Moss, 756 F.2d 329 (1985), this Court recently considered the efforts, by trial judges, to define reasonable doubt as well as the variations of language used in the process. The rule from Moss is that the term reasonable doubt is self-explanatory. Attempts to further define it are ill advised. Yet, when a district judge does attempt to explain reasonable doubt, his instruction does not constitute reversible error unless it adds such confusion as to be prejudicial to a defendant. In this case, the absence of the “hesitating” language in the trial court’s reasonable doubt instruction is not reversible error. Read in its entirety, the instruction was neither incomprehensible nor sufficiently misleading. It therefore did not deny the defendants’ right to due process of law.
  Defendants’ eighth claim of reversible error focuses upon a comment made by a U.S. magistrate who was presiding over the case during jury deliberations. The jury asked to rehear portions of the testimony of government witnesses Carpenter and Zeigler. The jury reheard Carpenter’s testimony, which was quite lengthy, and then recessed for dinner. When the jury returned, the presiding magistrate explained that counsel had endeavored to streamline Zeigler’s testimony in order to save time, but had not reached agreement. The magistrate stated, “I am advised that they have been unable to come to any firm agreement on that because apparently the next witness’s testimony goes more directly into the circumstances that you want to hear than the previous witness’s testimony.” The jury heard Zeigler’s testimony and then recessed for the evening.
  Defendants contend that the magistrate’s statement implied 1) Zeigler’s testimony was more probative than Carpenter’s testimony and 2) even defense counsel thought this was true. Based on these concerns, the magistrate spoke to the jury the next morning, before they resumed deliberations. The magistrate explained that he did not intend to imply that one witness was more important than another. All he meant to convey was that Zeigler’s testimony had less introductory material.
  The magistrate’s comment is not grounds for the reversal of defendants’ convictions. His explanatory instructions rendered his earlier statement harmless. In addition, the jury knew that the magistrate had not been present at trial, and therefore, he did not know the content of the witnesses’ testimony. The magistrate did not violate the defendants’ right to a fair trial.
  In their ninth claim of error, defendants argue that there was insufficient evidence presented at trial to support their conspiracy convictions. This claim is also meritless. A jury’s verdict must be sustained if any rational trier of fact could find guilt beyond a reasonable doubt when the evidence is viewed in the light most favorable to the government. Jackson v. Virginia, 443 U.S. 307, 99 S.Ct. 2781, 61 L.Ed.2d 560 (1979); Portsmouth Paving Corp., supra at 317. When viewed in this light, the prosecutor’s case included sufficient evidence to support the jury’s conviction on Count 1 of the indictment. The jury believed Carpenter and Zeigler, the defendants’ co-conspirators. Moreover, the bid submissions and telephone records support the government’s case.
  Next defendants argue that they were denied due process of law because the jury verdict was against the weight of the evidence. This claim also deserves short consideration. The applicable question before us is whether the district court abused its discretion by not granting the defendants’ motion for a new trial. United States v. Lincoln, 630 F.2d 1313 (8th Cir.1980). There was no such abuse. The evidence in this case does not sufficiently preponderate against the verdict to render that verdict a miscarriage of justice.
  Finally, defendants argue for a reversal of their convictions on grounds that the cumulative effect of all the above mentioned errors denied them their right to a fair trial. Again, the defendants’ claim lacks merit. Whether considered singularly or together, all the defendants’ arguments fail to justify reversal. This Court AFFIRMS the convictions below.
  . Federal Rule of Evidence 608(b) states,
  “Specific instances of conduct. Specific instances of the conduct of a witness, for the purpose of attacking or supporting his credibility, other than conviction of crime as provided in rule 609, may not be proved by extrinsic evidence. They may, however, in the discretion of the court if probative of truthfulness or untruthfulness, be inquired into on cross-examination of the witness (1) concerning his character for truthfulness or untruthfulness, or (2) concerning the character for truthfulness or untruthfulness of another witness as to which character the witness being cross-examined has testified.
  “The giving of testimony, whether by an accused or by any other witness, does not operate as a waiver of his privilege against self-incrimination when examined with respect to matters which relate only to credibility.”
  . Federal Rule of Evidence 404(b) states,
  “Other crimes, wrongs, or acts. Evidence of other crimes, wrongs, or acts is not admissible to prove the character of a person in order to show that he acted in conformity therewith. It may, however, be admissible for other purposes, such as proof of motive, opportunity, intent, preparation, plan, knowledge, identity, or absence of mistake or accident.”
  . The purpose of the no extrinsic evidence rule found in 608(b) is to prevent undue delays of trial, surprise and prejudice. Rule 608(b) seeks to avoid proof of the underlying behavior, which may confuse or sidetrack the jury. Moreover, the rule is designed to protect a witness, by prohibiting extrinsic evidence offered solely for the purpose of attacking that witness’s credibility. See Weinstein & Berger, Weinstein's Evidence, § 608(5).
  . The purpose of the Federal Rules of Evidence is found iri rule 102. It states, “These rules shall be construed to secure fairness in administration, elimination of unjustifiable expense and delay, and promotion of growth and development of the law of evidence to the end that the truth may be ascertained and proceedings justly determined.”
  . Defendants Lee and Smith Grading argue that the admission of Dellinger’s past bad acts should have been excluded because it prejudiced them by association. This argument is also without merit. The trial judge properly admitted evidence of Dellinger’s prior misconduct and twice gave a cautionary instruction to the jury that the evidence was not to be considered against the other defendants.
  . In addition, the fact that disclosure came from a source other than the prosecutor is of no consequence. When determining the constitutional validity of a belated Brady disclosure, the relevant inquiry is solely whether the defendant was able to effectively use the exculpatory information. See United States v. McCrary, 699 F.2d 1308 (11th Cir.1983).
  . This is one sentence within a lengthy reasonable doubt instruction. The trial judge stated, "... a reasonable doubt is not a fanciful doubt or a slight doubt. Neither does proof beyond a reasonable doubt require that the prosecution prove a defendant guilty beyond all doubt. A reasonable doubt is a fair doubt, based upon reason and common sense and arising from the state of the evidence.
  "We recognize that it is rarely possible to prove anything to an absolute certainty, and surely the prosecution is not required to do that. Proof beyond a reasonable doubt is established if the evidence is such as you would be willing to rely and act upon in the more important matters of your life. A defendant is not to be convicted on mere suspicion or conjecture or speculation.
  "____ A reasonable doubt exists in any case when after a careful and impartial consideration of all the evidence, you jurors do not feel convinced to a reasonable moral certainty that a defendant is guilty of the charges against him.”
  . United States Magistrate Charles W. Gambrell presided over the case during the jury deliberations at the request of the presiding trial judge, Charles E. Simons, Jr. According to the record, Judge Simons left Rock Hill, South Carolina because of a prior commitment.