snowball's chance in hell, no more than/not a

No chance at all. The hell in question, of course, is the fabulously hot place of tradition. This term appears to have replaced the earlier no more chance than a cat in hell without claws, an eighteenth-century locution that, according to Grose’s Dictionary, was applied to a person quarreling with or fighting against a much stronger opponent. The current cliché comes from late nineteenth-century America; in Britain and other English-speaking countries it is sometimes put as a snowflake’s chance in hell. See also Chinaman's chance.