Sitting on a bench and getting up without apologising

Question

There's a long table and behind it - a long bench. The only way to sit and get up and leave from the bench is through one side. Behind the table, there are 14 students and they're writing an exam. Any of them, who finishes their work, gets up, asks for their friends to let them through one by one and leaves. What is the possibility that 9th student (counting from the exit side) won't have to 'apologise' to their friends while leaving?

The way I see it, the possibility is $1/256$, since there are $8$ people blocking the exit, so the student will have to either apologise or not apologise ($2$ outcomes) to $8$ people, that makes $256$ total outcomes. I know for a fact that this solution is incorrect, but don't really know how to solve this correctly. Any tips?

Answer 1

Using a binary decision with equal probability for each person is incorrect, because the considered events are not independent. If you finish earlier than your first neighbor, for example, the probability of finishing earlier than his or her other neighbor increases. Luckily, there is a more straightforward way to solve this problem.

Considering that there are eight people in front of you, you don't have to apologize if you are the last one to finish. If each person has the same probability of handing in last, the probability of not having to apologize simply equals:

$$\frac{1}{9}$$