The probability of the last passenger getting his assigned seat is 1/2.

This is one of those results in mathematics where it seems easier to prove a more general result:  When there are n ³ 2 passengers in line and n seats on the plane, the probability that the last passenger gets his assigned seat is 1/2.

The result is clearly true if there are two passengers and two seats: If the first passenger takes his assigned seat, so will the second; if he does not, neither will the second one.  

Now suppose the result hold for fewer than n passengers and seats, and let's prove by induction that the result holds for n passengers and seats.  If the first passenger takes his assigned seat, which happens with probability 1/n, then so will everyone else, in particular, the last passenger will, too.  If the first passenger takes the seat assigned to the last person, which again happens with probability 1/n, then everyone else except the last passenger will, too.  In the rest of the cases, which occur with probability (n - 2)/n, the first passenger takes the seat assigned to neither the first nor the last passenger.  The second passenger now enters the plane and, finding his seat occupied, must select one of the other seats at random; that is, there are now n - 1 seats and n - 1 passengers, and we have exactly the same situation originally faced by the first passenger!  By induction, the last passenger gets his seat with probability 1/2.  Putting this all together, the total probability of the last passenger getting his seat is therefore 

= 1/2.

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ã2006 Alberto L. Delgado