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1590s, as both a verb and noun, in the argot of petty criminals, of unknown origin; probably a shortening of abet or else from obsolete beet "to make good," from Old English bætan "make better, arouse, stimulate," from Proto-Germanic *baitjanan, in which case the verb would be the original. The original notion is perhaps "to improve" a contest by wagering on it, or it is from the "bait" sense in abet. Used since 1852 in various American English slang assertions (cf. you bet "be assured," 1857). Related: Betting.
Yes; certainly; surely; bet your boots
[1857+; betcha form fr you bet you or you bet your life]