Eight individuals, including A and B, take seats around a circular table in a completely random fashion. Suppose the seats are numbered 1, …, 8. Let X = A’s seat number and Y = B’s seat number. If A sends a written message around the table to B in the direction in which they are closest, how many individuals (including A and B) would you expect to handle the message?
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If you renumber the seats after everyone is seated, starting with 1 at A's seat, this doesn't affect anyone's position at the table. So, without loss of generality, you can put A in seat 1 and B in a random seat from 2 through 8, each with probability 1/7.
Then make a table showing the number of "handlers" for each situation:
B n
2 2
3 3
4 4
5 5
6 4
7 3
8 2
The expected value of n (number of handlers) is then the sum over all B of n(B) * p(B). SInce all the p(B) values are 1/7, then E(n) = (2 + 3 + 4 + 5 + 4 + 3 + 2)/7 = 23/7, or 3 2/7.
Then make a table showing the number of "handlers" for each situation:
B n
2 2
3 3
4 4
5 5
6 4
7 3
8 2
The expected value of n (number of handlers) is then the sum over all B of n(B) * p(B). SInce all the p(B) values are 1/7, then E(n) = (2 + 3 + 4 + 5 + 4 + 3 + 2)/7 = 23/7, or 3 2/7.
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3