The answer is 2/3
The problem is defined as follows: if the probability of being occupied for one day is 1/3, then P(C) = 1/3, thus the probability of not being occupied is P(~C) = 2/3, let us make it easier by naming if P(F) = P(~C) {probability of being free}
The probability of being not occupied (free) for n consequence days is P(Fn) = (2/3)^n
i.e. Probability of being free for 2 consequence days = 4/9, for 10 consequence days = (2/3)^10, and for 9 days = (2/3)^9
Now the problem asks about the probability of being free for the 10th day GIVEN that it is free for 9 consequence days. that is:
P(F_10 | F_9) = P(F_10)/P(F_9) = (2/3)^10 / (2/3)^9 = 2/3
Please let me know if you find a mistake in my answer.