Green’s Dictionary of Slang

no luck, no chance whatsoever or the slightest possible chance; cite 1912 suggests a ‘slim chance’.

1904		Sydney Sportsman (Surry Hills, NSW) 1 June 1/2: Certainly our biggest horse-owner [...] has Chinaman’s luck with his protests over in the dark continent.
1911		Green Book Mag. VI 984/2: From what I hear, he hasn’t a Chinaman's chance of winning his suit.
1912		C. Mathewson Pitching in a Pinch 202: Still we had a Mongolian’s chance with them only one run ahead of us.
1914		T.A. Dorgan Indoor Sports 27 Apr. [synd. cartoon] The poor boob hasn’t got a chinaman’s chance.
1915		‘Ian Hay’ First Hundred Thousand (1918) 48: The Service Battalions simply must be led by the officers who have trained them if they are to have a chinaman’s chance when we go out.
1924		H.L. Wilson Professor How Could You! 317: She ain’t got a Chinaman’s chance.
1933		H. Asbury Barbary Coast (2002) 207: Once a sailor was actually in the clutches of a boarding-house master, he hadn’t even the proverbial Chinaman’s chance of regaining his liberty.
1946		S. Jackson Indiscreet Guide to Soho 94: He hasn’t a Chinaman’s chance of winning at one of these parties.
1956		A. Ginsberg ‘America’ Howl and Other Poems 32: Asia is rising against me / I haven’t got a chinaman’s chance.
1966		K. Porter ‘Still More Ethnic and Place names as Derisive Adjectives’ Western Folklore XXV:1 37: Chinaman’s chance. No chance at all.
1979		Maledicta III:2 155: Chinaman’s chance n [DAS ca 1850] Extremely poor chance; from the Chinese miners’ practice of working the ground white miners had already covered; or, from the intense hostility against the Chinese that made fair legal proceedings unlikely.
1989		H. Rawson Dict. of Invective (1991) 80: Not a chinaman’s chance, meaning no chance at all, or a slim one at best, also comes from the nineteenth century.
1998		US Naval Historical Center (Oral Histories) 🌐 Those guys down in them compartments dogged down – they didn’t have a chance. They didn’t have a Chinaman’s chance to get out of there.