POSNER, Circuit Judge.
The plaintiff brought suit against her employer under Title VII of the Civil Rights Act of 1964, charging racial discrimination. The case was tried, the jury returned a verdict for the defendant, and the plaintiff appeals, arguing that a juror named Leiter should have been struck for cause. If the plaintiff is right, she is entitled to a new trial without having to show that Leiter's presence on the jury caused the jury to side with the defendant. Denial of the right to an unbiased tribunal is one of those trial errors that is not excused by being shown to have been harmless. Gray v. Mississippi, 481 U.S. 648, 668, 107 S.Ct. 2045, 95 L.Ed.2d 622 (1987); Grancorvitz v. Franklin, 890 F.2d 34, 41 (7th Cir.1989); Fietser v. Ford Motor Co., 622 F.2d 281, 286 (7th Cir.1980); Dyer v. Calderon, 151 F.3d 970, 973 and n. 2 (9th Cir.1998) (en banc).
But what of the plaintiffs failure to use any of her three peremptory challenges to strike Leiter? She says that she used up her peremptory challenges on jurors whom she considered even less likely to favor her cause than Leiter was. This acknowledgment might seem to imply-since the plaintiff is not contending that any of those [623]*623jurors had to be stricken for cause — that she can’t really think that Leiter was biased; for if Leiter was biased and those other three were not, surely the plaintiff would have used a peremptory challenge to get rid of Leiter first. That doesn’t follow. Bias is only one factor in deciding whether to challenge a juror. A lawyer might be utterly convinced that a member of the jury venire would vote against his client no matter what the evidence showed, and yet his belief might be based on a hunch that he could not articulate as a ground for a challenge for cause. He might be more eager to strike that juror than one who had an evident bias (though the judge hadn’t been convinced of this), for he might think he could overcome the hurdle posed by that bias more readily than he could persuade the stubborn but not demonstrably biased juror.
A plaintiff may have a duty to use a peremptory challenge to remove a juror whom the judge correctly or incorrectly has failed to remove for cause, in which event a plaintiff who fails to use a peremptory challenge for this purpose will have forfeited the right to appeal the denial of the challenge for cause. In Ross v. Oklahoma, 487 U.S. 81, 108 S.Ct. 2273, 101 L.Ed.2d 80 (1988), the Supreme Court in a capital-punishment case upheld against constitutional challenge a state rule imposing just such a duty. On the basis of Ross we then held that the loss of a peremptory challenge because the challenge was used to “cure” the judge’s error in failing to remove a juror challenged for cause did not impair the right to an impartial tribunal. United States v. Nururdin, 8 F.3d 1187, 1191 (7th Cir.1993). That became the law of the nation in United States v. Martinez-Salazar, 528 U.S. 304, 120 S.Ct. 774, 145 L.Ed.2d 792 (2000); see also United States v. Polichemi, 219 F.3d 698, 705-06 (7th Cir.2000); Walzer v. St. Joseph State Hospital, 231 F.3d 1108, 1111 (8th Cir.2000). All but Ross are different cases from the present one, however. In those cases the peremptory challenge had been exercised and the biased juror removed. As a result, the jury that decided the defendant’s guilt was impartial. But could a defendant preserve the issue of bias simply by failing to use his peremptory challenge to remove the biased juror? Since the use of a peremptory challenge to remove that juror would cure the judge’s error, the defendant’s failure to use a peremptory challenge to do this might well be thought to make the error a self-inflicted wound, as argued in a concurring opinion in Martinez-Salazar, 528 U.S. at 318-19, 120 S.Ct. 774. The majority opinion, however, suggests a different view — that the litigant can let the biased juror be seated and seek to reverse the adverse judgment (if one results) on appeal on grounds of bias. See id. at 314-17, 120 S.Ct. 774. The suggestion is dictum, and can be questioned as putting the litigant in a heads-I-win-tails-you-lose position: if he wins a jury verdict, he can pocket his victory, and if he loses, he can get a new trial.
But this is not the case in which to consider whether to take on the Supreme Court’s dictum, since the defendant is not arguing that the plaintiffs failure to use a peremptory challenge against Leiter prevents the plaintiff from challenging Letter’s presence on the jury. It argues the distinct point that the plaintiff cannot complain about a violation of her statutory right to three peremptory challenges. 28 U.S.C. § 1870; see also Fed.R.CivJP. 47(b). The defendant is of course right. Martinez-Salazar states that securing an impartial jury is one of the intended uses of peremptory challenges, 528 U.S. at 316, 120 S.Ct. 774; see also Walzer v. St. Joseph State Hospital, supra, 231 F.3d at 1111; United States v. Quinn, 230 F.3d 862, 865 (6th Cir.2000), and hence may be [624]*624a reason why the statutory allotment is three rather than one or two or none. (.Marbinez-Salazar was a criminal case, but we cannot think of any difference which that would make.) But the statutory question is a side issue. The important question is whether the plaintiffs constitutional right to an impartial tribunal was infringed. Let us see.
During the voir dire of the jury, the judge asked the members of the venire whether “there is something about this kind of lawsuit for money damages that would start any of you leaning for or against a particular party?” Letter raised her hand and explained that she has “been an owner of a couple of businesses and am currently an owner of a business, and I feel that as an employer and owner of a business that will definitely sway my judgment in this case.” The judge asked her whether “if I instructed you as to what the law is that you would be able to apply the law recognizing that you are a business owner?” To which she replied, “I think my experience will cloud my judgment, but I can do my best.” The judge permitted the lawyers also to ask questions of the prospective jurors and Thompson’s lawyer asked Letter, “And you said earlier that you were concerned that your position as a business owner may cloud your judgment.
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POSNER, Circuit Judge.
The plaintiff brought suit against her employer under Title VII of the Civil Rights Act of 1964, charging racial discrimination. The case was tried, the jury returned a verdict for the defendant, and the plaintiff appeals, arguing that a juror named Leiter should have been struck for cause. If the plaintiff is right, she is entitled to a new trial without having to show that Leiter's presence on the jury caused the jury to side with the defendant. Denial of the right to an unbiased tribunal is one of those trial errors that is not excused by being shown to have been harmless. Gray v. Mississippi, 481 U.S. 648, 668, 107 S.Ct. 2045, 95 L.Ed.2d 622 (1987); Grancorvitz v. Franklin, 890 F.2d 34, 41 (7th Cir.1989); Fietser v. Ford Motor Co., 622 F.2d 281, 286 (7th Cir.1980); Dyer v. Calderon, 151 F.3d 970, 973 and n. 2 (9th Cir.1998) (en banc).
But what of the plaintiffs failure to use any of her three peremptory challenges to strike Leiter? She says that she used up her peremptory challenges on jurors whom she considered even less likely to favor her cause than Leiter was. This acknowledgment might seem to imply-since the plaintiff is not contending that any of those [623]*623jurors had to be stricken for cause — that she can’t really think that Leiter was biased; for if Leiter was biased and those other three were not, surely the plaintiff would have used a peremptory challenge to get rid of Leiter first. That doesn’t follow. Bias is only one factor in deciding whether to challenge a juror. A lawyer might be utterly convinced that a member of the jury venire would vote against his client no matter what the evidence showed, and yet his belief might be based on a hunch that he could not articulate as a ground for a challenge for cause. He might be more eager to strike that juror than one who had an evident bias (though the judge hadn’t been convinced of this), for he might think he could overcome the hurdle posed by that bias more readily than he could persuade the stubborn but not demonstrably biased juror.
A plaintiff may have a duty to use a peremptory challenge to remove a juror whom the judge correctly or incorrectly has failed to remove for cause, in which event a plaintiff who fails to use a peremptory challenge for this purpose will have forfeited the right to appeal the denial of the challenge for cause. In Ross v. Oklahoma, 487 U.S. 81, 108 S.Ct. 2273, 101 L.Ed.2d 80 (1988), the Supreme Court in a capital-punishment case upheld against constitutional challenge a state rule imposing just such a duty. On the basis of Ross we then held that the loss of a peremptory challenge because the challenge was used to “cure” the judge’s error in failing to remove a juror challenged for cause did not impair the right to an impartial tribunal. United States v. Nururdin, 8 F.3d 1187, 1191 (7th Cir.1993). That became the law of the nation in United States v. Martinez-Salazar, 528 U.S. 304, 120 S.Ct. 774, 145 L.Ed.2d 792 (2000); see also United States v. Polichemi, 219 F.3d 698, 705-06 (7th Cir.2000); Walzer v. St. Joseph State Hospital, 231 F.3d 1108, 1111 (8th Cir.2000). All but Ross are different cases from the present one, however. In those cases the peremptory challenge had been exercised and the biased juror removed. As a result, the jury that decided the defendant’s guilt was impartial. But could a defendant preserve the issue of bias simply by failing to use his peremptory challenge to remove the biased juror? Since the use of a peremptory challenge to remove that juror would cure the judge’s error, the defendant’s failure to use a peremptory challenge to do this might well be thought to make the error a self-inflicted wound, as argued in a concurring opinion in Martinez-Salazar, 528 U.S. at 318-19, 120 S.Ct. 774. The majority opinion, however, suggests a different view — that the litigant can let the biased juror be seated and seek to reverse the adverse judgment (if one results) on appeal on grounds of bias. See id. at 314-17, 120 S.Ct. 774. The suggestion is dictum, and can be questioned as putting the litigant in a heads-I-win-tails-you-lose position: if he wins a jury verdict, he can pocket his victory, and if he loses, he can get a new trial.
But this is not the case in which to consider whether to take on the Supreme Court’s dictum, since the defendant is not arguing that the plaintiffs failure to use a peremptory challenge against Leiter prevents the plaintiff from challenging Letter’s presence on the jury. It argues the distinct point that the plaintiff cannot complain about a violation of her statutory right to three peremptory challenges. 28 U.S.C. § 1870; see also Fed.R.CivJP. 47(b). The defendant is of course right. Martinez-Salazar states that securing an impartial jury is one of the intended uses of peremptory challenges, 528 U.S. at 316, 120 S.Ct. 774; see also Walzer v. St. Joseph State Hospital, supra, 231 F.3d at 1111; United States v. Quinn, 230 F.3d 862, 865 (6th Cir.2000), and hence may be [624]*624a reason why the statutory allotment is three rather than one or two or none. (.Marbinez-Salazar was a criminal case, but we cannot think of any difference which that would make.) But the statutory question is a side issue. The important question is whether the plaintiffs constitutional right to an impartial tribunal was infringed. Let us see.
During the voir dire of the jury, the judge asked the members of the venire whether “there is something about this kind of lawsuit for money damages that would start any of you leaning for or against a particular party?” Letter raised her hand and explained that she has “been an owner of a couple of businesses and am currently an owner of a business, and I feel that as an employer and owner of a business that will definitely sway my judgment in this case.” The judge asked her whether “if I instructed you as to what the law is that you would be able to apply the law recognizing that you are a business owner?” To which she replied, “I think my experience will cloud my judgment, but I can do my best.” The judge permitted the lawyers also to ask questions of the prospective jurors and Thompson’s lawyer asked Letter, “And you said earlier that you were concerned that your position as a business owner may cloud your judgment. Can you tell me how?” And she replied, “I am constantly faced with people that want various benefits or different positions in the company [what Thompson was seeking from her employer, the defendant, Al-theimer & Gray] or better contacts or, you know, a myriad of issues that employers face on a regular basis, and I have to decide whether or not that person should get them.” The lawyer then asked Letter whether she was concerned “that if somebody doesn’t get them [benefits sought from their employer] .they’re going to sue you,” and she answtered, “Of course.” Asked then whether “you believe that people file lawsuits just because they don’t get something they want?”, she answered, “I believe there are some people that do.” In answer to the next and last question, “Are you concerned that that might cloud your judgment in this case?” she said, “I think I bring a lot of background to this case, and I can’t say that it’s not going to cloud my judgment. I can try to be as fair as I can, as I do every day.”
That was the end of the voir dire of Letter. After refusing to strike her for cause (though urged to do so by the plaintiffs lawyer), and releasing the jurors who had not been selected for the jury (the defendant had also exercised its three peremptory challenges, none overlapping with the plaintiffs), the judge asked the eight remaining jurors, that is, the jurors selected to hear the case, whether they would follow his instructions on the law even if they didn’t agree with them and whether they would be able to suspend judgment until they had heard all the evidence. The question was asked to the jurors at large and all either nodded their heads or said yes. The defendant, again perhaps dropping the ball, makes nothing of Letter’s failure at this stage to reiterate her doubts about her ability to exercise an unclouded judgment. The defendant is content to argue that the answers that Letter gave to the earlier questions by the judge, and the questions by Thompson’s lawyer, did not require that Letter be struck for cause.
Our review of the trial judge’s ruling with respect to a challenge for cause is deferential, Salvato v. Illinois Dept. of Human Rights, 155 F.3d 922, 927 (7th Cir.1998); Pitsonbarger v. Gramley, 141 F.3d 728, 734-35 (7th Cir.1998); United States v. Vega, 72 F.3d 507, 512 (7th Cir.1995); United States v. Blom, 242 F.3d 799, 805 (8th Cir.2001); United States v. Beasley, 48 F.3d 262, 266 (7th Cir.1995); Wolfe v. [625]*625Brigano, 232 F.3d 499, 502 (6th Cir.2000), but not completely supine, and it is pertinent to note that no issue of credibility is presented. There is no argument that Leiter was not telling the truth. The issue is interpretive: did what she say manifest a degree of bias such that the judge abused his discretion in failing to strike her for cause?
In defense of his ruling, the defendant has come up with only one case, United States v. Ricketts, 146 F.3d 492, 496 (7th Cir.1998), where we upheld the trial judge’s refusal to strike for cause jurors who, in a trial for complicity in a prison riot, said on voir dire that they would tend to believe prison guards over inmates. The defendant says in its brief that we did this even though the jurors in question “never testified to their willingness or capacity to put this tendency aside for the purposes of trial.” That is an inaccurate description of our decision, and leaves the defendant startlingly bereft of apposite case law. The trial judge in Ricketts had faulted defense counsel for failing to follow up the question about the juror’s tendency to believe guards over inmates, and as a result of that failure the jurors had never had a chance to affirm their willingness or capacity to judge impartially. The tendency was not itself a sign of bias. There is a critical difference between a “prior [belief]” in the Bayesian statistician’s sense and “bias” in the sense that requires disqualification of a juror or judge. Everyone brings to a case a set of beliefs that may incline him in one direction or another. A person told that X had been indicted, and asked whether he thought X guilty, might reply that he thought X probably was guilty because few innocent people are indicted. That would be a prior. It would be a bias only if it were irrational or unshakable, so that the prospective juror “would be unable to faithfully and impartially apply the law,” Wainwright v. Witt, 469 U.S. 412, 424, 105 S.Ct. 844, 83 L.Ed.2d 841 (1985) (emphasis added), would be, in other words, “adamant,” Fleenor v. Anderson, 171 F.3d 1096, 1099 (7th Cir.1999) — in our hypothetical if, for example, the person added, “Nothing will ever convince me that the government would indict an innocent person.” Cf. Ross v. Oklahoma, supra, 487 U.S. at 83-84, 108 S.Ct. 2273.
When Leiter said that she believed that some people sue their employer just because they haven’t gotten a promotion or a raise or some other benefit, she was not manifesting bias. She was expressing a prior belief (prior, that is, to hearing any evidence in this case) that was not only not irrational, but -was undoubtedly true— there are indeed some people who will sue them employer just because of disappointment over the failure of the employer to give them something they want. In other words, there are spurious suits, in the employment domain as elsewhere. Leiter could not be thought biased for holding a true belief, or even for holding it unshakably if it is indubitably true. The belief that some employees make bogus claims against employers is so obviously true that it could not be shaken; but inability to set aside a clearly sound belief does not make for a biased juror. It makes for a realistic one.
Suppose a member of the venire in a case involving alleged sex discrimination by a fire department stated his belief that men on average have greater upper-body strength than women. Suppose he added that this belief was unshakable in the sense that if some social scientist testified otherwise, he would conclude that he was being fed junk science. Should this juror be disqualified? Not automatically, surely. The relevant questions would be whether he could distinguish averages from individ[626]*626uals, and thus recognize the possibility that a given woman might have greater upper-body strength than a given man, and whether he was so fixated on the average sex difference in upper-body strength that he was not open to the possibility that a woman whose upper-body strength was indeed less than that of the least strong firefighter in the fire department could nevertheless be as good a firefighter, or even a better one.
The question in this case was not whether Letter’s belief that some claims against employers are spurious was true or false (it was, as we have noted, true), but whether this belief would somehow impede her in giving due weight to the evidence and following the judge’s instructions. That question was not adequately explored. The last thing Letter said before the judge refused to strike her for cause was that she couldn’t say the “background” she brought to this case wasn’t going to “cloud” her judgment. She said she would try to be fair, but she expressed no confidence in being able to succeed in the attempt. She may have realized that because of bad experiences in the past, she might have difficulty separating the logically distinct propositions that some claims against employers are bogus and that this claim must be bogus because it is a claim against an employer.
Had she said she could not be fair, the judge would of course have had to strike her for cause. She did not say that, and so the judge (the defendant, though citing Ricketts, does not argue that the plaintiffs lawyer was at fault in failing to follow up his question whether Letter’s background would cloud her judgment) should have followed up by asking her, as he later asked the jury en masse, whether she would follow his instructions on the law and suspend judgment until she had heard all the evidence.
Instead the matter was left dangling, just as it had been in Martinez-Salazar. That juror whom the defendant in that case used a peremptory challenge to excuse after the judge refused to excuse him for cause, when asked “whether, if he were a defendant facing jurors with backgrounds and opinions similar to his own, he thought he would get a fair trial,” answered: “I think that’s a difficult question. I don’t think I know the answer to that." 528 U.S. at 308, 120 S.Ct. 774 (emphasis added). And when asked whether he “would feel more comfortable erring on the side of the prosecution or the defense,” he said he “would 'probably be more favorable to the prosecution.” Id. (emphasis added). When the judge then scolded him for reversing the presumption of innocence, the juror said, “I understand that in theory." Id. (emphasis added). The judge nevertheless refused to excuse the juror for tíause because “he said ... he could follow the instructions, and he said ... 1 don’t think I know what I would do,’ et cetera.” Id. at 309, 120 S.Ct. 774. The Supreme Court held that in these circumstances the judge had erred in not allowing the challenge for cause. It is just like our case. The judge didn’t push hard enough to determine whether Letter could relinquish her prior beliefs for purposes of deciding the case.
Had the judge pushed Letter and had she finally given unequivocal assurances that he deemed credible, his ruling could not be disturbed. See Lockett v. Ohio, 438 U.S. 586, 595-96, 98 S.Ct. 2954, 57 L.Ed.2d 973 (1978); Salvato v. Illinois Dept. of Human Rights, supra, 155 F.3d at 926-27; Pitsonbarger v. Gramley, supra, 141 F.3d at 735; Art Press, Ltd. v. Western Printing Machinery Co., 791 F.2d 616, 619 (7th Cir.1986); United States v. Blom, supra, 242 F.3d at 805. But he failed to do that. The venire contained 20 prospective ju[627]*627rors, and more than enough were left to make up a full jury of 8 when he refused to excuse her. A candid and thoughtful person, if one may judge from the transcript, Letter would probably have made an excellent juror — in another case.
When a prospective juror manifests a prior belief that is both material and contestable (for, to repeat an earlier point, it is not bias to cling to a belief that no rational person would question), it is the judge’s duty to determine whether the juror is capable of suspending that belief for the duration of the trial. When as in this case the record contains no assurances that the belief is “shakable,” that the prospective juror can exercise a judgment unclouded by that belief, the verdict cannot stand. See Art Press, Ltd. v. Western Printing Machinery Co., supra, 791 F.2d at 619; Wolfe v. Brigano, supra, 232 F.3d at 502-03; United States v. Gonzalez, 214 F.3d 1109, 1113-14 (9th Cir.2000); United States v. Padilla-Mendoza, 157 F.3d 730, 733-34 (9th Cir.1998); United States v. Salamone, 800 F.2d 1216, 1226-27 (3d Cir.1986). “When a juror is unable to state that she will serve fairly and impartially despite being asked repeatedly for such assurances, we can have no confidence that the juror will ‘lay aside’ her biases or her prejudicial personal experiences and render a fair and impartial verdict.” United States v. Gonzalez, supra, 214 F.3d at 1114. That’s this case. Missing are those “unwavering affirmations of impartiality” that permitted the district judge in United States v. Garcia, 936 F.2d 648, 653 (2d Cir.1991), to find the challenged juror unbiased.
Reversed and Remanded.