Question

Problem 3 A group of n couples composed of n men and n women enter a restaurant and sit randomly at n tables, two on each. i. Find the expected number of tables occupied by two people of different gender. ii. Find the expected number of tables occupied by couples. iii. If the n women enter the restuarant first and each sits at a separate table and then the men enter and sit randomly at the n tables. Find the expected number of tables occupied by couples iv. Verify the above in case n 3 by listing all possible outcomes.

Answer 1

i. Let X; -1 if ith table is occupied by two people of dif ferent gender = 0 otherwise 2 then P(X, = 1) =-(2n)_一2n(2n-1)/2 2n-1 X = total no. of tables occupied by tuo different people of different gender 2 ii. Let Y, ifith table is occupied by a couple 0 otherwise (2)21 n(n-1) =n-1 then P(Y = 1) = 2n-2n(2n-1)/2-2n-1 Y = total no. of tables occupied by couples 7l E(Y) Σ E(X) = n(n-1)

iii. Let Zi-1 if ith table is occupied by a couple 0 otherwise then P(Z: 1) (since n women can be sitted in any position among two positions o f ith table in2! ways and then her husband can be sitted in the remaining position of the ith table in 1 way) 7t 2! 2n(2n - 1)/2 2n -1 Z total no. of tables occupied by couples 7t 7L