This is actually isomorphic to the absent-minded driver problem. If you precommit to going straight, there is a 50/50 chance of being at either one of the two indistinguishable points on the road. If you precommit to turning left, there is a nearly 100% chance of being at the first point on the road (Since you wouldn't continue on to the second road point with that strategy.) It seems like probability can be determined only after a strategy has been locked into place.