In United States v. Carroll Towing Co., 159 F.2d 169 (2d Cir. 1947), Judge Learned Hand held that a barge owner's duty to guard against a vessel breaking free and causing damage is a function of three variables: the probability the barge will break away (P), the gravity of the resulting injury (L), and the burden of taking adequate precautions (B). A defendant is negligent when the burden of precaution is less than the probability of harm multiplied by its gravity — B < PL. Because the barge owner left the barge unattended with no bargee aboard during daylight business hours, and the burden of keeping someone aboard was slight relative to the risk, the owner was found contributorily negligent and its recovery reduced.

This is the case that turned the fuzzy "reasonable person" standard into something that looks like math. You will be asked about the Hand Formula, and you will be asked to say the words "B less than P times L" out loud. Here is everything you need to survive that moment and to use the case correctly on your Torts exam — where, fair warning, the trap is bigger than most students realize.

What is the Carroll Towing case about?

United States v. Carroll Towing Co. was decided by the United States Court of Appeals for the Second Circuit in 1947, in an opinion written by Judge Learned Hand — one of the most influential American judges never to sit on the Supreme Court. The dispute grew out of a wartime harbor accident: a barge, the "Anna C," broke loose from a pier in New York Harbor, drifted into a tanker, was punctured by its propeller, and sank along with its cargo of flour (owned by the United States). The legal question was how to allocate the loss among the several parties whose boats and crews had a hand in the accident. Hand used the occasion to articulate a general standard for evaluating negligence — the algebraic formula that now bears his name.

What were the facts of United States v. Carroll Towing?

In a crowded New York Harbor during World War II, a tug (the Carroll) and its crew were rearranging the mooring lines of several barges tied together at a pier. In the course of shifting the lines, the crew negligently refastened them so that one barge — the Anna C, owned by Conners Marine Co. and chartered to carry a cargo of flour belonging to the United States — broke free from the pier.

The Anna C drifted down the harbor, collided with a tanker, and the tanker's propeller punched a hole in her hull. She began to leak and, ultimately, sank with her cargo. Critically for the legal analysis, the Anna C had no bargee (the person responsible for tending the barge) aboard at the time she broke loose. The bargee had left the barge and had been gone for roughly 21 hours. Had he been aboard, the argument ran, he could have signaled for help or taken steps to keep the barge afloat after the collision, limiting or preventing the loss.

What was the procedural history?

The case arose in admiralty. Multiple parties — the barge owner, the charterer, the tug owner, and the United States — sought to sort out liability for the damage to the barge and the loss of the cargo. The district court apportioned the loss. On appeal, the Second Circuit reviewed the allocation of fault, and in particular whether Conners (the barge owner) could recover fully for the loss of its barge, or whether Conners's own failure to keep a bargee aboard made it contributorily negligent and thus reduced its recovery. Judge Hand wrote the opinion resolving that question.

What was the legal issue?

The precise question: was the barge owner negligent — specifically, contributorily negligent — for leaving the Anna C unattended, without a bargee aboard, such that this failure was a legal cause of the barge sinking and should reduce the owner's recovery? Framed more broadly for doctrine: how does a court decide whether the failure to take a particular precaution was unreasonable?

What did the court hold, and what is the Hand Formula?

Hand held that there is no general rule requiring a bargee to be aboard at all times; whether the absence is negligent depends on the circumstances. To decide that, he articulated the test that made the case famous. A party's duty to guard against harm is a function of three variables:

• P — the probability that the vessel will break away (or that the harm will occur);
• L — the gravity of the resulting injury if it does occur;
• B — the burden of taking adequate precautions to prevent it.

The rule: liability turns on whether B is less than L multiplied by P — that is, B < PL. If the burden of avoiding the harm is less than the probable loss (probability times gravity), then failing to take the precaution is negligent. Hand put it in his own words: liability depends on whether B < PL.

Applying it, Hand concluded that under these facts — a busy harbor in wartime, full daylight and business hours, other boats being moved around, and a bargee who had been absent for about 21 hours with no adequate excuse — the burden of having someone aboard was small relative to the risk and the potential loss. So Conners was contributorily negligent, and its recovery for the loss of the barge was reduced accordingly.

What was the court's reasoning?

Hand's move was to admit openly what courts had always been doing implicitly. The "reasonable person" standard is famously vague; Hand tried to give it analytical structure. His insight was that reasonable care is not the same as taking every possible precaution. It would be absurd to require a bargee aboard 24 hours a day forever, at any cost. Instead, the reasonableness of a precaution depends on what it costs (B) weighed against what it prevents — the size of the harm (L) discounted by how likely that harm is (P).

Notice the balance built into the formula. If a harm is catastrophic but wildly unlikely, PL may still be small, and a heavy burden of prevention may be unreasonable to demand. If a harm is likely and severe, even a substantial precaution is required. Hand was careful to note that these are not quantities a court can actually measure precisely — the values are estimates, sometimes little more than intuitions. But the framework tells you what the reasonableness inquiry is about: it is a comparison of the cost of caution against the expected cost of the risk.

On the facts, Hand did not say a bargee must always be aboard. He said the opposite — there's no such fixed rule. But given that it was daytime, that the harbor was full of activity that made breaking loose foreseeable, and that a 21-hour absence during business hours was unexcused, a reasonable barge owner would have kept an attendant aboard. The burden was low; the probability and gravity, under these conditions, were not.

Why does Carroll Towing matter?

This case is the doctrinal home of the cost-benefit conception of negligence. It anchors the breach element — the question of whether the defendant fell below the standard of reasonable care. Every negligence analysis has four elements (duty, breach, causation, damages), and Carroll Towing lives in "breach." Where the reasonable-person standard tells you who we compare the defendant to, the Hand Formula tells you how to decide whether that person would have acted differently.

The formula also became the intellectual foundation of the law-and-economics approach to torts — the idea, associated with scholars like Richard Posner, that negligence law efficiently allocates the burden of accident avoidance to the party who can prevent the harm most cheaply. That is why casebooks pair Carroll Towing with the standard-of-care and reasonable-person cases (Vaughan v. Menlove, the Blyth v. Birmingham definition), and often with policy material on the economics of tort law.

Cold-call prep: the questions your professor will ask

This is the section that gets you through class. Expect these, roughly in this order.

"State the Hand Formula." Answer: A defendant is negligent when the burden of taking precautions (B) is less than the probability of harm (P) multiplied by the gravity of the loss (L) — B < PL. Know the letters cold and know what each stands for. Do not mix up L and P.

"What was the precaution at issue here?" Answer: Keeping a bargee aboard the Anna C. B is the burden of having an attendant present; the professor may push you on whether that burden is trivial. Concede it is not zero — the bargee has to be paid and can't be everywhere — but it is small relative to the risk of a loose barge in a crowded harbor.

"Did the court hold that a bargee must always be aboard?" This is the trap. Answer: No. Hand expressly said there is no general rule requiring a bargee aboard at all times. The holding was fact-specific — under these circumstances (daytime, busy harbor, 21-hour unexcused absence) the absence was unreasonable. Students who say "the rule is you must always have a bargee" get corrected.

"Whose negligence are we even talking about?" Answer: This is a contributory-negligence holding about the barge owner (Conners), the plaintiff seeking recovery for its own barge — not a straightforward defendant-liability holding. The tug was also at fault. The point of the Hand analysis in this case was to reduce the barge owner's recovery because of its own carelessness. Getting the posture right signals you actually read it.

"Can you actually plug numbers into B, P, and L?" Answer: Almost never in practice — Hand himself acknowledged the values are estimates and often unquantifiable. The formula is a way of organizing judgment, not a calculator. Say this and you sound like you understand the case rather than just memorized it.

"How does this relate to the reasonable person?" Answer: It is a translation of the reasonable-person standard into cost-benefit terms. The reasonable person takes precautions whose burden is justified by the expected harm avoided — which is exactly B < PL.

How Carroll Towing shows up on the exam

On an issue-spotter, the Hand Formula is your framework for arguing breach once you've established duty. When a fact pattern gives you a defendant who did or didn't take some safety measure, structure the breach argument around B, P, and L: how burdensome was the precaution, how likely was the harm, how bad would it have been? Use the facts the professor buried in the problem — a cheap fix that was ignored, a known risk, a severe potential injury — to argue B < PL and therefore breach.

The trap. Weaker exam answers treat the Hand Formula as if it were a rule you literally compute, assigning made-up numbers and declaring a winner. Professors hate this. The stronger move is to use B, P, and L as a rhetorical structure and argue both sides: the plaintiff argues the burden was small and the expected harm large (B < PL, breach); the defendant argues the precaution was costly or impractical and the risk remote (B > PL, no breach). The formula is where you show your analysis, not where you announce a verdict.

Second trap. Don't forget the formula is only one way courts assess breach. Custom, statutes (negligence per se), and res ipsa loquitur are alternative or supplementary routes. On an open-ended breach question, mention that reasonable care can be shown through the Hand cost-benefit analysis and through these other doctrines — that breadth earns points.

Related cases you'll see alongside it

• Vaughan v. Menlove — establishes the objective reasonable-person standard the Hand Formula operationalizes.
• Blyth v. Birmingham Waterworks — the classic definition of negligence as failing to do what a reasonable person would do.
• The T.J. Hooper — another Learned Hand admiralty opinion, on custom and the duty to adopt available safety measures (radios on tugs); frequently taught in the same breach unit.
• Cases on the economics of tort law — casebooks often follow Carroll Towing with Posner-influenced material developing the efficiency rationale.

Bottom line

Carroll Towing gives you the analytical engine for the breach element: reasonable care means taking precautions whose burden is less than the expected harm, B < PL. Know the letters, know that Hand rejected any fixed rule about bargees, know that the case is about the barge owner's own contributory negligence, and know that on the exam the formula is a way to argue — not a calculator to run. Get those four things right and this is one of the easiest cold calls of your semester.