Chinaman's chance means little or no chance at all, freighted with a particularly anti-Asian racism. It is an American idiomatic expression, popular in the 19th and 20th centuries. The original phrase was that one had only a Chinaman's Chance in Hell, but it morphed through usage into its current state.
The historical context of the phrase comes from the old railroad and gold rush days of California, where many Chinese came to work as railroad workers for the First Transcontinental Railroad, especially the Central Pacific Railroad. In this employ, they were sought out for the demanding and dangerous jobs involving explosives, often for half the pay of the Mexican workers. Yet the Chinese faced higher taxes, denials of citizenship, and could not testify in court for violence against them.
According to the California Supreme Court, the Chinese "recogniz[ed] no laws ... except through necessity, [brought] with them their prejudices and national feuds, in which they indulge[d] in open violation of law." The court also noted that their "mendacity is proverbial; [that they were] a race of people whom nature has marked as inferior, and who are incapable of progress or intellectual development beyond a certain point ... [and they would not be granted] the right to swear away the life of a citizen, ... [or] the ... privilege of participating with us in administering the affairs of our Government."
The Chinaman's chance originated from the middle of the 19th century, potentially from several events. One explanation is that during construction of the Transcontinental Railroad (1863-1869) Chinese migrant workers in the U.S. carried unstable bottles of nitroglycerine to tunnel and mountain passes under construction. (The stable form of nitroglycerine—called dynamite—was not invented until 1867 by Alfred Nobel, and only was available in the final stages of the project.)
They were also lowered over cliffs by rope and boatswain's chairs to set the nitroglyercine in place. In this work, if they were not lifted back up before the blast, serious injury or death would result. Therefore, the phrase a "A Chinaman's Chance" may have been coined in this context.
Another explanation for the phrase is the California Gold Rush 1849. The travel time for news of the gold rush to reach China was quite long, and by the time Chinese from China arrived to prospect, many of the rich mines were already taken. These Chinese immigrants who missed out time-wise had to work with only those lands which had already been exploited or which were rejected by others, meaning these Chinamen had a slim chance of success. The historical record, however, indicates that many Chinese combined efforts with each other and did very well in the goldfields, introducing mining techniques then unknown to non-Chinese.
According to Nothing Like It in the World by Stephen Ambrose, his book on building the railroad, the phrase was cemented by murders of Chinese that were condoned by state law. "In 1854, in a case heard in Nevada County, George W. Hall was convicted of murdering a Chinese man. On appeal to the State Supreme Court the decision was overturned because all of the evidence against him was from Chinese individuals." 
- "Asia is rising against me. / I haven't got a chinaman's chance. / I'd better consider my national resources." — Allen Ginsberg, America
The phrase is used in Fritz Lang's You Only Live Once, Frank Capra's Lost Horizon and also in Rob Cohen's Dragon: The Bruce Lee Story, when the character of Bruce Lee is on his way to America in a ship. This phrase is also uttered by James Dunn (Eddie Collins) in Bad Girl and Randolph Scott (Wyatt Earp) to Cesar Romero (Doc Halliday) in Frontier Marshal and in the film The People vs. Dr. Kildare (Alma Kruger as Molly Byrd).
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- Patti Waldmeir (June 22, 2015). "The quest for the meaning of Chineseness from Beijing to US sitcoms". Financial Times. Retrieved January 1, 2016.
- People v. Hall (1854) 4 Cal. 399, 404.
- People v. Hall, 4 Cal. at 405.
- The California Supreme Court currently has three Justices of Asian descent, two of them Chinese.
- A Chinaman's Chance: The Chinese on the Rocky Mountain Mining Frontier, Liping Zhu
- The History of Chinese in California.
- Ginsberg, Allen (1988) . "America". Collected Poems 1947-1980. Harper Perennial. ISBN 978-0-06-091494-3. Retrieved December 7, 2011.