Re: Airplane puzzle: Slick solution?

From: Ted Hwa (hwatheod_at_xenon.Stanford.EDU)
Date: 10/17/04 

Date: Sun, 17 Oct 2004 21:52:22 +0000 (UTC)

In sci.math Stephen J. Herschkorn <herschko@rutcor.rutgers.edu> wrote:
: N (> 1) passengers are about to board, one at a time, an airplane with
: N seats. Each passenger has an assigned seat. On the way to his seat,
: the first passenger to board loses his boarding pass, so he selects a
: seat at random Each subsequent passenger sits in his/her assigned seat
: if it is not already taken; otherwise, s/he selects at random an empty
: seat. What is the probability that the last passenger to board (i.e.,
: the Nth passenger) sits in her assigned seat?

: Below, I give a solution to this problem. However, this problem
: appeared in a popular, non-mathematical forum, and my solution strikes
: me as a bit too mathematically/probabilistically sophisticated for the
: (wo)man on the street. Does anyone have an explanation that is less
: sophisticated?

The last passenger sits in his assigned seat if and only if the
first passenger's seat is taken before the last passenger's seat.

As long as both of these seats are empty, they are either both available
(with equal probability), or both not available, to any particular
passenger. (Note that this includes the first passenger, who
sits in a random seat.) Therefore, both seats have the same
probability of being taken first -- so the answer is 1/2.

Ted

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