game of chance played with dice, 1843, American English, unrelated to the term for excrement, instead it is from Louisiana French craps "the game of hazard," from an 18c. continental French corruption of English crabs, which was 18c. slang for "a throw of two or three" (the lowest throw), which perhaps is from crab (n.2), the sense in crab apple. The 1843 citation (in an anti-gambling publication, "An Exposure of the Arts and Miseries of Gambling") calls it "a game lately introduced into New Orleans." To shoot craps is by 1885.
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