In a bus with $n$ seats, all tickets are sold to $n$ passengers. The first to enter the bus is the Absent-Minded Scholar, who, without looking at the ticket, takes the first available seat. Subsequently, passengers enter one by one. If an entering passenger sees that their seat is free, they take their seat. If their seat is occupied, the entering passenger takes the first available free seat. Find the probability that the last passenger to enter will take the seat according to their ticket?
Let's number all passengers starting with the Scientist in the order in which they entered the bus. The last passenger has the number $n$. For simplicity, let's number the seats in the same way. Suppose all but the last passenger have already entered and taken their seats. There is one free seat left. If this were the second seat, the second passenger (or the Scientist) would have already taken it. The same is true for seats numbered $3,4,5, \ldots, n-1$. Therefore, this seat must belong to either the last passenger or the Absent-Minded Scientist.

It is clear that it has an equal chance of belonging to either the first or the last. In the first case, the last passenger will not sit in their own seat, and in the second case, they will. Therefore, the probabilities of both events are equal to $1 / 2$.

## Answer

$1 / 2$.