All good answers above. There's a reason it being a short straw, rather than a long one. If the loser had the long straw, there would be an incentive to surreptitiously break your straw to make it shorter.

To work out who has to do an unpleasant task, the group "draw straws". A bunch of straws is held in someone's fist, however one is short (no-one knows which). Each person takes a straw, and the person who draws the shortest straw is the one who must do the unpleasant task. So we say "draw the short straw" to mean you're picked for the unpleasant job... ie. you're unlucky.

draw the short straw (informal) to be the member of a group who has to do an unpleasant job in other words to lose

Another variant is draw the short straw, meaning to be the one so selected for a particular task. cowboy...