
Crazy guy on the plane
Answer: The jumping stops if someone sits on the crazy man's seat before the 100th persons, hence, the 100th will sit in his/her seat for sure then. Similarly, if anyone sits on the 100th passengers seat at any point, then he/she won't get to sit his/her seat for sure. If at any point someone picks a random seat other than #1 or #100, then the jumping is postponed till that seat number is reached. At any jumping point there is a equal probability that seats #1 or #100 will be chosen (other choices will just delay the problem till later down the line). By symmetry, the probability of the 100th passenger sitting in their seat is 5050, since at all jump points there's an equal probability of choosing seats #1 or 100.
andrew
Monday, December 13, 2004
You've got it right. Let me simplify the language a bit:
If any passenger <100 randomly selects seat 1, then each successive passenger can sit in his/her assigned seat including passenger 100.
If any passenger <100 randomly selects seat 100, then passenger 100 is forced to choose the only other remaining seat.
For each passenger (who is forced to choose randomly), the chance of choosing 1 and 100 is equal.
(The EXACT probability depends on the number of other remaining seats, but in each case the chance of choosing 1 and 100 is the same. 1/100 and 1/100 for the 1st passenger, 1/40 and 1/40 for the 60th passenger)
Each choice that doesn't choose seat 1 or 100 simply defers the final decision to a later passenger.
Therefore the chance of passenger 100 getting his own seat is 50%.
I had a nice proof by induction in a previous thread, but that thread has apparently been pruned.
BradC
Tuesday, December 14, 2004
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