POSNER, Circuit Judge.

The high crime rate in the United States has interacted with expanding notions of tort liability to make suits charging hotel owners with negligence in failing to protect their guests from criminal attacks increasingly common. See Annot., 28 A.L.R.4th 80 (1984). Dula McCarty, a guest at the Pheasant Run Lodge in St. Charles, Illinois, was assaulted by an intruder in her room, and brought suit against the owner of the resort. The suit charges negligence, and bases federal jurisdiction on diversity of citizenship. The parties agree that Illinois law governs the substantive issues. The jury brought in a verdict for the defendant, and Mrs. McCarty appeals on a variety of grounds.

In 1981 Mrs. McCarty, then 58 years old and a merchandise manager for Sears Roebuck, checked into Pheasant Run — a large resort hotel on 160 acres outside Chicago — to attend a Sears business meeting. In one wall of her second-floor room was a sliding glass door equipped with a lock and a safety chain. The door opens onto a walkway that has stairs leading to a lighted courtyard to which there is public access. The drapes were drawn and the door covered by them. Mrs. McCarty left the room for dinner and a meeting. When she returned, she undressed and got ready for bed. As she was coming out of the bathroom, she was attacked by a man with a stocking mask. He beat and threatened to rape her. She fought him off, and he fled. He has never been caught. Although Mrs. McCarty's physical injuries were not serious, she claims that the incident caused prolonged emotional distress which, among other things, led her to take early retirement from Sears.

Investigation of the incident by the police revealed that the sliding glass door had been closed but not locked, that it had been pried open from the outside, and that the security chain had been broken. The intruder must have entered Mrs. McCarty's room by opening the door to the extent permitted by the chain, breaking the chain, and sliding the door open the rest of the way. Then he concealed himself somewhere in the room until she returned and entered the bathroom.

Mrs. McCarty argues that the judge should have granted her motion for judgment notwithstanding the jury's verdict for the defendant. But she failed to move for a directed verdict on the issue of the defendant's negligence, and that is a prerequisite to judgment n.o.v. Fed.R.Civ.P. 50(b). It is true that she made a motion for a directed verdict on the issue of her contributory negligence, which was denied, and that the defendant made a motion for a directed verdict on the issue of its negligence, which was also denied, but these motions were not equivalent to the motion she failed to make. Even if she had been innocent of contributory negligence as a matter of law, this would not have made the defendant guilty of negligence as a matter of law; in many accidents, neither injurer nor victim is at fault, and then there is no liability. Similarly, all that the denial of the defendant's motion for a directed verdict showed was that the defendant was not innocent of negligence as a matter of law; it could of course be guilty of negligence as a matter of law. Thus, neither motion for directed verdict presented the question whether the issue of the defendant's negligence should be withdrawn from the jury and resolved in the plaintiff's favor. She could not present that issue for the first time in her motion for judgment n.o.v.

The modern rationale for the rule that a motion for directed verdict is a prerequisite to judgment n.o.v. is that the opposing party should have a chance to rectify (or at least seek the court's leave to rectify) deficiencies in his evidence before it is too late, that is, before the case goes to the jury. McKinnon v. City of Berwyn, 750 F.2d 1383, 1388 (7th Cir. 1984); see also Benson v. Allphin, 786 F.2d 268, 273-74 (7th Cir. 1986). That rationale is applicable to this case. After both motions for directed verdict (the plaintiff's on contributory negligence, and the defendant's on negligence) were denied, the defendant had no reason to think it hadn't put in enough evidence to get to the jury on the issue of liability. If the plaintiff thought otherwise she had to move for a directed verdict on that issue.

As an alternative ground for denying the motion for judgment n.o.v., the district judge correctly pointed out that the case was not so one-sided in the plaintiff's favor that the grant of a directed verdict or judgment n.o.v. in her favor would be proper. Her theories of negligence are that the defendant should have made sure the door was locked when she was first shown to her room; should have warned her to keep the sliding glass door locked; should have equipped the door with a better lock; should have had more security guards (only two were on duty, and the hotel has more than 500 rooms), cf. Nordmann v. National Hotel Co., 425 F.2d 1103, 1107 (5th Cir. 1970); should have made the walkway on which the door opened inaccessible from ground level; should have adopted better procedures for preventing unauthorized persons from getting hold of keys to guests' rooms; or should have done some combination of these things. The suggestion that the defendant should have had better procedures for keeping keys away from unauthorized persons is irrelevant, for it is extremely unlikely that the intruder entered the room through the front door. Compare Danile v. Oak Park Arms Hotel, Inc., 55 Ill. App.2d 2, 203 N.E.2d 706 (1964). The other theories were for the jury to accept or reject, and its rejection of them was not unreasonable. Cf. Courtney v. Remler, 566 F. Supp. 1225, 1233-34 (D.S.C. 1983).

There are various ways in which courts formulate the negligence standard. The analytically (not necessarily the operationally) most precise is that it involves determining whether the burden of precaution is less than the magnitude of the accident, if it occurs, multiplied by the probability of occurrence. (The product of this multiplication, or "discounting," is what economists call an expected accident cost.) If the burden is less, the precaution should be taken. This is the famous "Hand Formula" announced in United States v. Carroll Towing Co., 159 F.2d 169, 173 (2d Cir. 1947) (L. Hand, J.), an admiralty case, and since applied in a variety of cases not limited to admiralty. See, e.g., United States Fidelity Guaranty Co. v. Jadranska Slobodna Plovidba, 683 F.2d 1022, 1026 (7th Cir. 1982); Maryland Cas. Co. v. City of Jackson, 493 So.2d 955, 960 n. 3 (Miss. 1986) (dictum); People Express Airlines, Inc. v. Consolidated Rail Corp., 100 N.J. 246, 266-67, 495 A.2d 107, 117-18 (1985); Micallef v. Miehle Co., 39 N Y2d 376, 386, 384 N.Y.S. 2d 115, 348 N.E.2d 571, 577-78 (1976); Phillips v. Croy, 173 Ind. App. 401, 404-05, 363 N.E.2d 1283, 1285 (1977); Benlehr v. Shell Oil Co., 62 Ohio App.2d 1, 9 and n. 5, 402 N.E.2d 1203, 1208 and n. 5 (1978); Golden v. McCurry, 392 So.2d 815, 819 (Ala. 1980) (separate opinion); 3 Harper, James Gray, The Law of Torts § 16.9, at pp. 467-68 (2d ed. 1986); Prosser and Keeton on the Law of Torts § 31, at p. 173 (5th ed. 1984); cf. East River S.S. Corp. v. Transamerica Delaval, Inc., 476 U.S. 858, 106 S.Ct. 2295, 2302, 90 L.Ed.2d 865 (1986); Prentis v. Yale Mfg. Co., 421 Mich. 670, 687, 365 N.W.2d 176, 184 (1984).

We are not authorized to change the common law of Illinois, however, and Illinois courts do not cite the Hand Formula but instead define negligence as failure to use reasonable care, a term left undefined. See, e.g., Hardware State Bank v. Cotner, 55 Ill.2d 240, 247-48, 302 N.E.2d 257, 262 (1973); Denniston v. Skelly Oil Co., 47 Ill.App.3d 1054, 1067, 6 Ill.Dec. 77, 87, 362 N.E.2d 712, 722 (1977). But as this is a distinction without a substantive difference, we have not hesitated to use the Hand Formula in cases governed by Illinois law. See EVRA Corp. v. Swiss Bank Corp., 673 F.2d 951, 958 (7th Cir. 1982); Davis v. Consolidated Rail Corp., 788 F.2d 1260, 1263-64 (7th Cir. 1986). The formula translates into economic terms the conventional legal test for negligence. This can be seen by considering the factors that the Illinois courts take into account in negligence cases: the same factors, and in the same relation, as in the Hand Formula. See Hendricks v. Peabody Coal Co., 115 Ill.App.2d 35, 45-46, 253 N.E.2d 56, 61 (1969); Bezark v. Kostner Manor, Inc., 29 Ill.App.2d 106, 111-12, 172 N.E.2d 424, 426-27 (1961). Unreasonable conduct is merely the failure to take precautions that would generate greater benefits in avoiding accidents than the precautions would cost.

Ordinarily, and here, the parties do not give the jury the information required to quantify the variables that the Hand Formula picks out as relevant. That is why the formula has greater analytic than operational significance. Conceptual as well as practical difficulties in monetizing personal injuries may continue to frustrate efforts to measure expected accident costs with the precision that is possible, in principle at least, in measuring the other side of the equation — the cost or burden of precaution. Cf. Conway v. O'Brien, 111 F.2d 611, 612 (2d Cir. 1940) (L. Hand, J.), rev'd on other grounds, 312 U.S. 492, 61 S.Ct. 634, 85 L.Ed. 969 (1941). For many years to come juries may be forced to make rough judgments of reasonableness, intuiting rather than measuring the factors in the Hand Formula; and so long as their judgment is reasonable, the trial judge has no right to set it aside, let alone substitute his own judgment.

Having failed to make much effort to show that the mishap could have been prevented by precautions of reasonable cost and efficacy, Mrs. McCarty is in a weak position to complain about the jury verdict. No effort was made to inform the jury what it would have cost to equip every room in the Pheasant Run Lodge with a new lock, and whether the lock would have been jimmy-proof. The excluded exhibits (of which more later) were advertisements for locks, and Mrs. McCarty's lawyer expressed no interest in testing the claims made in them, or in calculating the expense of installing new locks in every room in the resort. And since the door to Mrs. McCarty's room was unlocked, what good would a better lock have done? No effort was made, either, to specify an optimal security force for a resort the size of Pheasant Run. No one considered the fire or other hazards that a second-floor walkway not accessible from ground level would create. A notice in every room telling guests to lock all doors would be cheap, but since most people know better than to leave the door to a hotel room unlocked when they leave the room — and the sliding glass door gave on a walkway, not a balcony — the jury might have thought that the incremental benefits from the notice would be slight. Mrs. McCarty testified that she didn't know there was a door behind the closed drapes, but the jury wasn't required to believe this. Most people on checking into a hotel room, especially at a resort, are curious about the view; and it was still light when Mrs. McCarty checked in at 6:00 p.m. on an October evening.

It is a bedrock principle of negligence law that due care is that care which is optimal given that the potential victim is himself reasonably careful; a careless person cannot by his carelessness raise the standard of care of those he encounters. Davis v. Consolidated Rail Corp., supra, 788 F.2d at 1265. The jury may have thought it was the hotel's responsibility to provide a working lock but the guest's responsibility to use it. See Brewer v. Roosevelt Motor Lodge, 295 A.2d 647, 652 (Me. 1972). We do not want to press too hard on this point. A possible explanation for the condition of the door as revealed by the police investigation is that Mrs. McCarty on leaving the room for the evening left the door unlocked but with the safety chain fastened, and she might have been reasonable in thinking this a sufficient precaution. But it would not follow that the hotel was negligent, unless it is negligence to have sliding doors accessible to the public, a suggestion the jury was not required to buy. We doubt whether a boilerplate notice about the dangers of unlocked doors would have altered the behavior of the average guest; in any event this too was an issue for the jury. Cf. Rosier v. Gainsville Inns Associates, Ltd., 347 So.2d 1100, 1102 (Fla.App. 1977); Otwell v. Motel 6, Inc., 755 F.2d 665, 667 (8th Cir. 1985) (per curiam).

Now it is true that in Illinois an innkeeper, which in contemplation of law this defendant is, is required to use a high (not merely the ordinary) standard of care to protect its guests from assaults on the innkeeper's premises. Mrzlak v. Ettinger, 25 Ill.App.3d 706, 712-13, 323 N.E.2d 796, 800 (1975); Danile v. Oak Park Arms Hotel, Inc., supra, 55 Ill.App.2d at 8-9, 203 N.E.2d at 709. This is not the general rule, see, e.g., Kveragas v. Scottish Inns, Inc., 733 F.2d 409, 413 (6th Cir. 1984); Peters v. Holiday Inns, Inc., 89 Wis.2d 115, 123-24, 278 N.W.2d 208, 212 (1979); Phillips Petroleum Co. v. Dorn, 292 So.2d 429, 431-32 (Fla.App. 1974), though it has some ambiguous support in Louisiana, see Kraaz v. La Quinta Motor Inns, Inc., 410 So.2d 1048 (La. 1982) — ambiguous because while the court said that "a guest is entitled to a high degree of care and protection," it promptly added that "the innkeeper has a duty to take reasonable precautions against criminals" ( id. at 1053, emphasis added). Conceivably, as suggested in Dorn, it is no longer the rule in Illinois either, though Yamada v. Hilton Hotel Corp., 60 Ill.App.3d 101, 112, 17 Ill.Dec. 228, 237, 376 N.E.2d 227, 236 (1977), decided after Dorn, suggests it is. The rule may simply be an inadvertent extrapolation from the principle (see Restatement (Second) of Torts, § 314A and comment e (1965); Kveragas v. Scottish Inns, Inc., supra, 733 F.2d at 412) that an innkeeper, like a common carrier but unlike a mere bystander, has a duty to prevent (or rescue from) dangers created by third parties. See Fortney v. Hotel Rancroft, Inc., 5 Ill. App.2d 327, 331, 125 N.E.2d 544, 546 (1955), seeming to equate these distinct propositions.

The rule, if it is a rule, may be defensible however; and whether it is or is not defensible is relevant to whether it is a genuine rule or a mere inadvertence. Ordinarily the innkeeper knows much more about the hazards of his trade than the guest, and can take reasonable (=cost-justified) steps to reduce them, while ordinarily the guest can do little to protect himself against them. See Banks v. Hyatt Corp., 722 F.2d 214, 226-27 (5th Cir. 1984). Pheasant Run, Inc. knows more about the danger of break-ins to guest rooms at its lodge than the guests do, and more about the alternative methods for preventing such break-ins, as well. Maybe this asymmetry in the parties' position should make the defendant's standard of care higher than it would be in, say, an ordinary collision case. See Danile v. Oak Park Arms Hotel, Inc., supra, 55 Ill.App.2d at 6-8, 203 N.E.2d at 708-09. But it does not make the defendant's liability strict. In this case there was evidence of negligence but not so much as to establish liability as a matter of law or (the plaintiff's alternative argument) to require a new trial. And the rule, based as it seems to be on an asymmetry in the parties' abilities to prevent mishaps, has a certain hollowness in a case such as this, where the victim may have failed to take an elementary precaution — locking the sliding door before leaving the room.

The next issue that Mrs. McCarty seeks to raise is whether the judge should have instructed the jury to decide whether she had been contributorily negligent. She argues that there was no evidence of her contributory negligence. Pheasant Run is not in the middle of a large city and it might not occur to guest that a safety chain on a sliding door to the outside was an inadequate protection against nocturnal marauders. On the other hand Mrs. McCarty was an experienced business traveler, so maybe she should have known better; and most people don't consider a safety chain an adequate substitute for a lock. But even if there was no evidence of contributory negligence, there was no prejudicial error in giving an instruction on it. The jury was clearly and correctly instructed that contributory negligence in Illinois is not a complete defense; it just cuts down the amount of damages that the plaintiff would otherwise be entitled to. This is the principle of comparative negligence, and at the time of the trial of this case it existed in Illinois in its pure form, meaning that the plaintiff is entitled to some damages even if he was more negligent than the defendant. See Alvis v. Ribar, 85 Ill.2d 1, 25-28, 52 Ill.Dec. 23, 421 N.E.2d 886, 897-98 (1981). (The rule has since been modified. See Ill.Rev.Stat. ch. 110, ¶¶ 2-1107.1, 2-1116; Davis v. United States, 824 F.2d 549, 551 (7th Cir. 1987).) Since the jury returned a verdict for the defendant, rather than a verdict for the plaintiff with truncated damages, it probably thought that the defendant had not been negligent at all or that its negligence had not caused the mishap; in either case the plaintiff's contributory negligence or lack thereof would be moot. It is unlikely that the mere giving of the instruction somehow signaled to the jury the judge's belief that the verdict should be for the defendant.

The remaining questions concern the judge's exclusion of evidence that the plaintiff sought to put before the jury. The exclusion of evidence about proper key-control procedures was proper for a reason we have already indicated: such evidence was not relevant to any plausible theory of the defendant's negligence. Also proper or at least defensible was the judge's decision to exclude evidence of previous criminal activity at Pheasant Run that did not involve breaking into a room through the sliding glass door. The judge admitted evidence of the nine previous break-ins that did. The principal evidence in the previous-crimes category that he excluded was of two alleged sexual assaults and eleven alleged thefts from rooms. This evidence was of limited relevance, at best. One of the so-called assaults involved a complaint from a man who said that he saw a man and woman having intercourse in a hallway and that he sprained his ankle pursuing the man; it is entirely unclear whether the intercourse was coerced or what the relationship of the complainant to the couple was. The circumstances of the other alleged assault are equally shadowy. Neither involved an intrusion into a room. The eleven reports of theft appear to include cases where a guest lost or mislaid an item as well as cases of genuine theft, but in any event are remote from the issues in this case; among other things, none involved forcing the sliding glass door.

A trial judge has broad discretion in administering Rule 403 of the Federal Rules of Evidence, which authorizes him to exclude relevant evidence if its probative significance is substantially outweighed by its prejudicial, confusing, or cumulative effect. Where as here the judge explains the reasoning process behind his exclusions, they will rarely be overturned. See United States v. Beasley, 809 F.2d 1273, 1278-79 (7th Cir. 1987). Pheasant Run is a large place, and it is not to be supposed that it would be free of criminal activity no matter how careful (within the bounds of reason) the management was. There is no indication that its experience with criminal activity was abnormal or indicative of a need to take additional precautions. Maybe the jury should have been allowed to figure this out for itself, but a jury's ability to digest statistical evidence is limited, especially when no comparison was attempted by the plaintiff's counsel between the frequency of criminal activity at Pheasant Run and at comparable resort hotels, cf. Anderson v. Malloy, 700 F.2d 1208, 1211-12 (8th Cir. 1983), and no effort was made to show that precautions which would have averted crimes not involving the forcing of the sliding glass doors would also have averted the attack on Mrs. McCarty.

She also complains about the exclusion from evidence of advertisements for locks for sliding glass doors. These locks are designed to foil intruders, as the advertisements make clear, and Mrs. McCarty argues with some show of reason that the advertised locks appear to be more effective than the locks on the sliding glass doors at Pheasant Run. The problem is the absence of a causal relationship between the failure to have fancy locks and the attack on Mrs. McCarty. There is no evidence that Mrs. McCarty's assailant jimmied the Lock. The door was unlocked. The world's fanciest lock — a lock to foil a Houdini — would thus have done her no good, and the failure to install a precaution that would not have avoided this accident (the accident that is the basis of the suit) is not actionable. Kveragas v. Scottish Inns, Inc., supra, 733 F.2d at 415. Her complaint about the exclusion of evidence of inadequate maintenance by the defendant of its sliding glass doors fails for the same reason; there is no indication that her failure to lock the door was due to improper maintenance. Finally, it is merely speculation that if the door had been equipped with a lock that locked automatically when the door was slid closed, the door would not have been left open with merely the safety chain fastened.

AFFIRMED.