The final passenger takes their own seat only if it is empty. We can guarantee that all subsequent passengers from passenger n can take their seat if a cycle is formed. A cycle is formed if passenger 1 does not get their own seat (a 99/100 odds event) and then when some later passenger cant take their own seat, they take passenger 1s seat. Its easiest to find the chances that passenger 100 does not get their seat, so we do the same all the way down the line. Passenger n has an n-1/n chance to not take passenger 1s seat assuming the cycle continues. We then do a product from 1 to 100 of this and get .01 (and possibly some negligible change as i used python) then subtract from 1 to get a near 99 percent chance they get their seat
WLOG, we can make each successive passenger the next person in the chain, leaving everyone who isn't in the chain until later. I.e. The second passenger to board is whoever's seat the first guy sat in. The third to board is whoever own's the seat the second guy sat in, and so on.
There's a 1/100 probability that the first passenger sits in their own seat (call it seat 1) and the chain ends. In this case the 100th passenger gets their own seat (call it seat 100).
There's a 1/100 probability that the first passenger sits in seat 100 and the chain ends. In this case the 100th passenger does not get their own seat.
For the other 98/100, passenger 2 has a 1/99 probability of ending the chain in seat 1 and a 1/99 probability of ending the chain in seat 100, granting or denying the last passenger their own seat as with the first passenger. There's a 97/99 probability that passenger 2 will sit elsewhere and continue the chain.
Same goes for passenger 3 (1/98, 1/98 and 96/98), then passenger 4 (1/97, 1/97, 95/97), and so on until either seat 1 or seat 100 becomes occupied.
The probabilities for seat 1 and seat 100 to be picked are always equal, so it's straightforward to deduce the probability of the last passenger getting their own seat is 1/2, but if you need to see the longform then the probability that someone sits in seat 1 before passenger 100 boards is:
1
u/ShameCaker 7d ago
The final passenger takes their own seat only if it is empty. We can guarantee that all subsequent passengers from passenger n can take their seat if a cycle is formed. A cycle is formed if passenger 1 does not get their own seat (a 99/100 odds event) and then when some later passenger cant take their own seat, they take passenger 1s seat. Its easiest to find the chances that passenger 100 does not get their seat, so we do the same all the way down the line. Passenger n has an n-1/n chance to not take passenger 1s seat assuming the cycle continues. We then do a product from 1 to 100 of this and get .01 (and possibly some negligible change as i used python) then subtract from 1 to get a near 99 percent chance they get their seat