Repeat the preceding problem when the seating is random but

Repeat the preceding problem when the seating is random but subject to the constraint that the men and women alternate

A total of 2n people, consisting of n married couples, are randomly seated (all posible orderings being equally likely) at a round table. Let C_i denote the event that the members of couple i are seated next to each other, i = 1,...,n

(a) Find P(C_i)

(c) Approximate the probability, for n large, that there are no married couples who are seated next to each other.

Repeat the preceding problem when the seating is random but subject to the constraint that the men and women alternate A total of 2n people, consisting of n married couples, are randomly seated (all posible orderings being equally likely) at a round table. Let Ci denote the event that the members of couple i are seated next to each other, i = 1,...,n (a) Find P(Ci) (b) For , find P(Cj l Ci). (c) Approximate the probability, for n large, that there are no married couples who are seated next to each other.

Solution

2n people with n couples

When couple are seated to each other treat each couple as a single unit.

There are n units and they can be seated in a circle in (n-1)! ways.

among the two they can sit in 2 ways

Hence no of ways for couples sitting to gether = 2(n-1)!

Total no of ways for 2n people in circle = (2n-1)! WAYS

Thus

a) prob (cij) =

-----------------------------------------------

b) j not equal to i

couples do not sit together

1-

c) when n is large, it is almost equal to 1.