The chances are one in two. Proof by induction. The base case, N = 2, is immediate. In general, the absent-minded professor might choose her assigned seat with probability 1/N, in which case the last woman surely gets her assigned seat. The absent-minded professor might choose the last woman's seat with probability 1/N, in which case the last woman definitely loses her seat. In each of the remaining N-2 cases, by induction, the chances of the last woman getting her assigned seat are one in two. Combining these cases, we see that the overall probability is .