Of American-English origin, the colloquial phrase a Chinaman’s chance denotes a negligible likelihood; it is often used in negative constructions, (not) a Chinaman’s chance meaning (not) the least chance—synonyms: not a cat in hell’s chance and a snowball’s chance (in hell).
The meaning of the earliest instance that I have found is not really clear; it is from one of the unconnected paragraphs of the second page of The West Alabamian (Carrolton, Alabama) of Wednesday 21st June 1893:
Col. Ainsworth, who had charge of the government building in Washington, known as Ford’s theater, stated that all he wanted was a “Chinaman’s chance.” The investigation before the deputy coroner looked more to the trial of Col. Ainsworth than an investigation to know the truth. Justice Bingham decided that a deputy could not hold an inquest and referred the matter to the proper officer. The government clerks did not demean themselves very well while the inquest was proceeding before the deputy.
Ford’s Theater, in Washington, D.C., was the site of the assassination of Abraham Lincoln (1809-65), 16th president of the USA (1861-65). In the 1880s, Ford’s Theater became a War Department pensions office; according to Atlas Obscura,
In 1893 Col. Fred Ainsworth, the chief of the pension bureau, ordered construction to install electric lights in the building, “for the comfort and convenience of the employees,” the New York Times reported at the time. The electric light plant required extensive renovation to the basement, and employees began getting edgy when they noticed plaster falling from the ceiling on the upper floors.
On Friday, June 9th, during the workday, a beam in the basement collapsed, and a huge portion of the interior of the building collapsed, killing 22 people and injuring more than 100, most of them War Department employees.
The earliest unambiguous instance of a Chinaman’s chance that I have found is from The World (New York City, N.Y.) of Tuesday 3rd October 1893:
The novel idea, suggested by The World, of a contest between a bicyclist and a thoroughbred race-horse in training was universally discussed yesterday by turfmen. […]
“The bicyclist would win sure,” said David Gideon, one of the shrewdest judges of horses in the East. “I don’t think the horse would have a Chinaman’s chance to beat him. It would be a “cinch” for the wheelman. The World’s suggestion, if carried out, would furnish a great race, however. Only I would like to take the bicycle end of the betting.”
The phrase might refer to the fact that persons of Chinese descent in the USA had little prospect of obtaining reparations for the racial discrimination of which they were victims.
That was mentioned for example by The Boston Daily Globe (Boston, Massachusetts) of Tuesday 9th January 1883:
Among the many curious ideas of Pacific coast life prevalent in this section of the country is one that Chinamen are universally abused in California, and whenever a hoodlum assaults a washerman on the street the Eastern papers are apt to wax very indignant and lecture the Californians for their intolerance and barbarity, as though the average citizen of California were accustomed to beat and maltreat a Mongolian now and then for diversion. A little experience and observation of the criminal courts in this city will convince one that the Boston hoodlum is the same cowardly, brutal fellow who disgraces San Francisco, and that a Chinaman’s chances of being assaulted in the streets of this self-righteous burg are fully as good as though he were among the wicked people of the sandlots. Locate 40,000 Chinese in Boston, and it would require the whole police force of the city to give them reasonable protection, judging from the manner in which the few who now live here are treated by the haughty South Cove Caucasian. Long Shee and two companions were passing along Beach street Sunday evening, when they came in contact with John Callahan and two other South Covers, whose patriotism and courage had just been stimulated by six drinks apiece. Long Shee mortally offended Callahan, first by being a Chinaman and second by minding his own business, and he moreover plainly showed his vile determination not to assimilate by shamelessly walking the streets in a state of heathen sobriety. The proud Caucasian, with six drinks of South Cove liquor in his superior Christian stomach, could not brook the palpable insult of a sober and industrious Chinaman passing him on the street without getting off the sidewalk, and he, or one of his companions, resented it by calling public attention to the alleged canine origin of Long Shee. The depraved and turbulent Mongolian had the audacity to turn around and look angry, when he was promptly knocked down by Callahan, according to his own story as told in the court yesterday. He got up and retreated and Callahan pursued him, and the rest of the affair was seen by Officer Franks, who testified that the Chinaman ran across the street and back, followed by Callahan, who finally knocked him down, and before the officer could reach the spot the Chinaman clinched with his assailant and both rolled on the ground. At this point the scene was enlivened by the sudden arrival of a fifty-pound dog of a pugnacious disposition. The dog was not personally interested, but a free fight had attractions for him in a general way, and he naturally favored fair-play; so he took a mouthful of Callahan, who was on top, and became quite enthusiastic in his efforts to discourage the whole of Callahan. Franks called off the dog and arrested Callahan. In court Callahan could not remember the dog episode, but was quite positive that Long Shee began the fight by taking hold of his coat, and the court actually swallowed the story that he had to chase a Chinaman all over the street, and knock him down from behind to protect himself from the ferocious heathen. Counsel for the defence made an able argument in behalf of the dog, and succeeded in clearing him of any blame in the matter by showing that he was not present at all, Callahan having denied that the dog took hold of him. It was only justice to the dog to prove an alibi for him under such circumstances, as it would be a reflection upon the character of a respectable dog to show that he was present and did not bite Callahan. Counsel for the prosecution called the court’s attention to the fact that Callahan had taken six drinks of Cove sherry, and left the probable effects of six glasses of sherry to the experience of the court. As for a motive for the assault, counsel thought liquor and cowardly contempt for the Chinese would account for the affair. The court commented with gravity intended for wisdom, and although it had a lurking idea that everybody has an antipathy to Chinamen, it could not judiciously find a fact to be true, and therefore could not attribute to Callahan the motive of prejudice. The defendant might have been in liquor, but the court could not judiciously find that everybody in liquor commits assaults, and therefore was obliged to reject Callahan’s partial intoxication as an explanation of the attack. Because no good motive could be shown for Callahan’s conduct, the court found that he did not commit an assault which an officer and two Chinamen plainly saw, and the defendant was discharged. The logical conclusion to be drawn from this decision is that hoodlums may assault Chinamen on the streets with impunity, and to escape punishment they have only to prove that the assault was wholly unprovoked and cowardly. Any Californian paper that now chooses to state that Boston courts uphold drunken hoodlums in abusing peaceable Chinese on the public streets cannot be justly accused of wilful misrepresentation.
On Thursday 22nd March 1883, The Newberry Herald (Newberry, South Carolina) also mentioned that difficulty in getting legal reparations:
The Chinese Merchants who were driven out of the town of Waynesboro, Ga., several weeks ago, have begun suit against eighteen of the most prominent citizens of that place, charging them with damages to the amount of $115,000. Loo Chang & Co., as a firm, claim $50,000; Tom Loo Chang, $40,000; and Ah Sing, $25,000. The suits are brought in the U.S. Circuit Court in Savannah. It is understood that the Chinese minister at Washington has engaged the services of distinguished attorneys, and intends to make a test case to see what a Chinaman’s chances are in this Country. Very large damages are claimed; it is likely that very, very small damages will be recovered.