Assume that there are two rooms and there are equal number of people in each room. One random person from each room leaves their own room and enters to the other room. If we repeat this process N times, what is the probability that all the people in each room will be where they started originally? (answer in terms of N) What happens to this probability as N → ∞?

A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
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Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Assume that there are two rooms and there are equal number of people in each room. One random person
from each room leaves their own room and enters to the other room. If we repeat this process N times,
what is the probability that all the people in each room will be where they started originally? (answer in
terms of N) What happens to this probability as N → ∞?

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