P(X > s + t | X > s) = P(X > t).
Only the geometric (discrete) and exponential (continuous) distributions have it - a uniqueness result that is itself an occasional interview question.
The interview form. "A fair coin has come up tails ten times. Expected further flips to a head?" Two, exactly as before. Hesitating here suggests the gambler's fallacy.
The related trap - the inspection paradox. Buses arriving on average every 10 minutes leave you waiting 10 minutes if you arrive at random, not 5, because you are more likely to land inside a long gap than a short one.