A probability question

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Sender: winterlover (both husband and wife return their families ~~~~), Email: quant
Question: A probability question
Mailing station: BBS untitled Space Station (Fri Aug 31 14:04:08 2007)

100 passengers must be on the plane and 100 seats on the plane. The first passenger did not press the number and sat down randomly. The rest of the passengers were all seated.
Keep yourself in line with your seat number (the second goes to the second, the third goes to the third, so on and so forth
), Unless the seats are occupied, they will pick one. May I ask 100th passengers (the last one) to sit on their own?
What is the probability of a self number?

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The answer is 0.5

If there are n people, the probability of N people being able to do so is P (n)
Except for the two situations where the first person chooses numbers 1 and N-1 (the probability is 1/N and 1/N * 0.5 respectively ),
Are equivalent to {n-1... 3} people. That is, 1/N (P (n-1) + P (n-2)... P (3 )). P3 = 0.5,
It can be inferred that all PN values are 0.5.
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In the case of everyone randomly picks seat, the chances are 1/100

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Set a random variable X describing the seat the first person chose. So
X cocould be 1, 2,..., n with 1/N chance.
Set event a as the case the last guy has his right seat.
So conditional on X, total probability depending on N is
P_n (A) = 1/N [P (A | x = 1) + P (A | x = 2) +... + P (A | x = n)]
And P (A | x = 1) = 1, P (A | x = n) = 0, P (A | x = I) = P _ {n-I + 1} ().
As a result, a recurrence equation is formed
P_n (A) = 1/N [1 + p_2 (A) + p_3 (A) +... + P _ {N-1} (a)]
Then, after reforming the terms
P_n (A) = P _ {N-1} ()

As we know P_2 (A) = 1/2,
Then p_100 (A) = 1/2, and P_n (A) = 1/2 for any n.

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If the first person is sitting in chair 1, the probability is 1.
If the first person is in the N chair, the probability is 0,
If the first person is sitting in any other chair m,
Kick this man out with the Chair m he made,
Then, set "re-label" of "M" to "1", and the subsequent probability is the probability of N-1 individual.

So P (n) = 1/N * 1 + 1/N * 0 + (n-2)/n * P (n-1 ),
P (1) = 1

Thus, P (n) = 1/2

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