The idea of drawing straws is used as a "fair" way to choose one person from a group to do an unsavory activity. I recently saw this in a movie and was wondering if this was actually fair.

With n people there are n straws, n-1 normal straws and 1 short straw. The person who picks the short straw loses, and there is no replacement of straws. So the first person trying to pick any of the n-1 long straws from the n straws, person 2 picks n-2 from the n-1 straws and so on. Of course the next person only goes if the person before them succeeds, picks a long straw.

First off is this fair, aka does the order you pick in not matter, secondly if it is not fair is there a formula for the probability of each spot (I have a feeling it is something like the probability of success of all previous persons x prob of drawing a long straw) ?