3. There are 13 students in a class (one of them being the monitor) and 13 seats in the classroom. Every day, the 13 students line up in random order and then enter the classroom one by one. Except for the monitor, each student will randomly choose an unoccupied seat and sit down. The monitor, however, prefers the seat next to the door and chooses it if possible. What is the probability that the monitor can choose his favourite seat?
(1 mark)
某班有 13 名學生（其中一人為班長），課室內有 13 個座位。每天上課時，這 13 名學生均按隨意的次序排隊進入課室。除班長外, 每人進入課室後均會隨意選一個空位坐下, 唯班長特別喜歡近門口的一個座位, 若未有人選擇的話他便會選那個座位。班長能選到他喜歡的座位的概率是多少？
(1 分)
3. $\frac{7}{13}$
3. The probability that the monitor can choose his favourite seat depends on his position in the queue. If he is first in the queue, he must be able to get his favourite seat; if he is the second in the queue, there is a probability of $\frac{12}{13}$ that his favourite seat has not been chosen by the first person in the queue. Similarly, if he is $k$-th in the queue, the probability that he gets his favourite seat is $\frac{14-k}{13}$. Since the monitor is equally likely to occupy each position in the queue, the answer is $\frac{1}{13}\left(1+\frac{12}{13}+\frac{11}{13}+\cdots+\frac{1}{13}\right)=\frac{7}{13}$.