Green’s Dictionary of Slang

no chance at all; also as excl.

[1850		Western Times 9 Feb. 6/3: Salter said— ‘I may as well hold my tongue and say nothing — no good telling the truth here.’ Wilkie also delivered it as her opinion that ‘any one had a fat lot of chance to speak the truth there.’ They were committed for trial].
1904		Brownsville Dly Herald 23 Apr. 2/1: Now he is suing for damages, with a fat chance of getting them.
1905		T.A. Dorgan in Zwilling TAD Lex. (1993) 35: The sports outside the ropes had a fat chance of seeing anything.
c.1905		J. Furphy Buln-Buln and the Brolga (1948) Ch. i: 🌐 ‘Thought you was goin’ to take me kneelin’,’ observed Wes indifferently. ‘So I am. One hand, kneelin’.’ ‘Fat likely!’ he retorted.
1913–14		Van Loan ‘No Business’ in Taking the Count 148: ‘Didn’t see you at the fight last Tuesday night, Tirney.’ ‘A fat chance!’ said the proprietor [...] ‘A fine, fat chance!’.
1918		J.E. Rendinell diary 21 Sept. in One Man’s War (1928) 167: ‘If you will give me pass, I will go up & find them.’ He said, ‘Fat chance.’.
1922		S. Lewis Babbitt (1974) 54: I’d be tickled to death if she’d really go making love with somebody. Fat chance!
1928		Thurman & Rapp Harlem in Coll. Writings (2003) 327: Hot chance I gotta vamp, what with Basil messin’ ’round an’ —.
1933		W.R. Burnett Dark Hazard (1934) 3: Fat chance of him ever getting away.
1943		R. Llewellyn None But the Lonely Heart 50: A fat chance she had of catching Him.
1944		E.S. Gardner Case of the Crooked Candle (1958) 16: Mason laughed. ‘Fat chance.’.
1955		B. Schulberg On the Waterfront (1964) 157: ‘If they lay a hand on any of you I’ll see they go to jail.’ [...] ‘Fat chance.’.
1962		H.S. Thompson letter 28 Aug. in Proud Highway (1997) 350: Fat chance, considering the fucking mails here.
1973		A. Buckeridge Speaking of Jennings (1989) 7: A fat chance we’ve got of going fishing.
1987		T. Wolfe Bonfire of the Vanities 536: ‘He wants to throw me out of the building.’ ‘Fat chance.’.
1998		Guardian Media 7 Dec. 2: Fat chance. I knew I had to act fast.
2002		D.H. Sterry Chicken (2003) 20: Where will I go? What will do? My mom? Fat chance. She doesn’t want me around.
2012		A. Bergen ‘Lazarus Slept’ in C. Rhatigan and N. Bird (eds) Pulp Ink 2 [ebook] ‘Sorry, I haven’t domesticated her yet.’ ‘Fat chance’.