Question

Five individuals, including a and b, take seats around a circular table in a completely random fashion. suppose the seats are numbered 1, . . . , 5. 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?

Answer 1

Will use A and B in place of a and b for clarity.
Let x=number of individuals away from A, including A & B

Without loss of generality, assume A is seated in seat #1.

Then B is seated at 2,3,4,5 with equal probability.
Half of the time B is seated at 2 or 5, each of which is next to A, therefore x=2
The other half of the time B is seated at 3 or 4, each of which is separated from A by one seat, then x=3.

The expected number of individuals
E[X]=sum (x*P(x))
=2*(1/2)+3(1/2)
=2.5

So the expected number of individuals to handle the message is 2.5.