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Smokey Mountain Express. Say there are 80 people in line to board the Smokey Mountain Express, which has exactly 80 seats. The first person in line realizes she lost her ticket, which has her seat number on it, so when she boards the train she just takes a random seat instead. Every person who boards the train after her will either take their assigned seat, or, if that seat is occupied, a random seat instead. What is the probability that the last person who boards the train will end up in his or assigned seat?
math   Statistics-And-Probability   
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The answer is that the probability that the last person ends in up in his proper seat is exactly The reasoning goes as follows: First observe that the fate of the last person is determined the moment either the first or the last seat is selected! This is because the last person will either get the first seat or the last seat. Any other seat will necessarily be taken by the time the last guy gets to choose. Since at each choice step, the first or last is equally probable to be taken, the last person will get either the first or last with equal probability: .

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