Sitting in a bus
There is a bus with 50 labelled seats from 1 to 50. There are also 50 persons labelled from 1 to 50 standing in a queue.
People board on the bus in sequence from 1 to n. The rule is, if a person ‘i’ boards the bus, he checks if seat ‘i’ is empty. If it is empty, he sits there, else he randomly picks an empty seat and sits there. Given that 1st person picks seat randomly, find the probability that 50th person sits on his place i.e. 50th seat.
Once the first person finds a seat, remaining all people will sit in sequence. so, the 49th person can either sit on the seat allocated to 50th person or somewhere else.
Based on where he sits, the last person i.e. 50th person will have 2 options – sit on the 50th seat or somewhere else.
Hence probability is 1/2.
Lets go backwards. If there where only 3 seats and the 1st person did not take your seat than the 2nd person can choose between your seat or the 1st person seat so the probability is 1/2.
If there are 4 seats than there are 3 empty seats: yours, 1st person, and the person before the last. so the 2nd person has a probability of 1/3 that he chooses your seat, or 1st person seat, or the person before the last, So if he didn’t choose your seat we come back for the probability of 1/2 for the person before you and so on.
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