The seven dwarfs dormitory problem

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Each of the seven dwarves sleeps in his own bed in a shared dormitory. Every night, they retire to bed one at a time, always in the same sequential order, with the youngest dwarf first and the oldest last. On a particular evening, the youngest dwarf has had too much to drink and is in a jolly mood. He does not choose his own bed, but randomly chooses one of the other six beds to fall asleep on. As each of the other dwarves retires, he chooses his own bed if it is not occupied, and otherwise chooses another unoccupied bed at random. What is the probability that the oldest dwarf sleeps in his own bed? What is the expected number of dwarfs sleeping in their own bed?

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