Question

Six individuals, including A and B, take seats around a circular table in a completely random fashion. Suppose the seats are numbered 1, …, 6. 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

Solution:

From the given information, six individuals take seats around a table. Let x = A's seat number Let y = B's seat number The seating arrangement is shown below: 4 1 2 2 6 2 5 3 3 3 4 4 From the diagram, it is observed that the six individuals are named 1, 2, 3, 4, 5, and 6. 2 seats are available from 1 and 2. 3 seats are available from 1 and 3. 4 seats are available from 1 and 4. 3 seats are available from 1 and 5. 2 seats are available from 1 and 6.

Similarly, the available seats are summarized in the below table: Consider the following table of values of h(x,y) for all possible joint values of X and Y: - y h(x,y) 1 2 3 4 5 6 2 3 4 3 2 2 2 2 3 4 3 3 2 2 3 4 4 4 3 2 2 3 5 3 4 3 2 2 6 2 3 4 3 2 - دعا There are 84 individuals are available including A and B. From the table, p(x, y) 1 30 for each possible (x, y).

Use the following formula to compute the expected value: E[h(x,y)]= E2h(x,y)p(x, y) {"(x, y) 30 (2+3+4+3+2+2+2+3+4+3 1 +3+2+2+3+4+4+3+2+2+3 30 +3+4+3+2+2+2+3+4+3+2 = = 84 30 = 2.8 23 [Rounded to an integer] The expected number of individuals to handle the message is 3).