159 F.2d 169, (2d. Cir. 1947)

Background
In the early evening on January 3, 1944, in New York Harbor, a bargee moored the barge "Anna C" at the pier along with several other barges. The next day, the tug "Carroll" was sent to the pier to "drill out" another barge at Pier 52. In other words, the "Anna C" was tied to a barge tied to another barge tied to a barge that was eventually tied to the pier. The harbormaster boarded the Anna C without permission because there was no captain or deckhand present. During this process, the "Anna C", with six other barges, broke away. The "Anna C" hit a tanker, and then sank. The United States, lessee of the "Anna C", was sued by Carroll Towing Co., owner of the "Carroll" in an indemnity action.

Ruling
The judge, Judge Learned Hand, stated that there was no general rule with which to deal with liability when the barge of an absentee bargee breaks free and causes damage. Consequently, the Judge Hand proposed an algebraic formula to determine if the standard of care has been met.
Since there are occasions when every vessel will break from her moorings, and since, if she does, she becomes a menace to those about her; the owner’s duty, as in other similar situations, to provide against resulting injuries is a function of three variables: (1) The probability that she will break away; (2) the gravity of the resulting injury, if she does; (3) the burden of adequate precautions. Possibly it serves to bring this notion into relief to state it in algebraic terms: if the probability be called P; the injury, L; and the burden, B; liability depends upon whether B is less than L multiplied by P: i.e., whether B < PL. Simply put, the test says:
If (Burden < Cost of Injury x Probability of occurrence), then the accused will not have met the standard of care required.
If (Burden > Cost of injury x Probability of occurrence), then the accused may have met the standard of care.
On the facts, the Court ruled that leaving a barge unattended during the daylight hours poses significant risk such that it would be fair to require a bargee to be aboard the ship. Thus, the accused was found liable for negligence for being absent from the ship without excuse.
This case is most famous for its first expression of Judge Hand's formula, C > GL or in the more common shorthand, BPL. This formula was first suggested, however, in The T.J. Hooper 60 F.2d 737 (2d Cir.), cert. Denied, 287 U.S. 662 (1932), another tugboat case.