Actually, you win if you are able to choose a princess other than Random—you do not need to know which of the two remaining ones is Random. Otherwise, this would clearly be impossible since the answer provides only one bit and there are three possibilities. (And that’s not even considering that under sensible interpretations of the rules, you don’t get any information if you happen to ask Random—i.e., you’re not allowed to ask e.g., “Is it true that (you are False) OR (you are Random and you’ve decided to answer truthfully this time)”, which, if allowed, would be answered in the affirmative iff the one you asked is Random.)
Actually, you win if you are able to choose a princess other than Random—you do not need to know which of the two remaining ones is Random. Otherwise, this would clearly be impossible since the answer provides only one bit and there are three possibilities. (And that’s not even considering that under sensible interpretations of the rules, you don’t get any information if you happen to ask Random—i.e., you’re not allowed to ask e.g., “Is it true that (you are False) OR (you are Random and you’ve decided to answer truthfully this time)”, which, if allowed, would be answered in the affirmative iff the one you asked is Random.)