Randomly arrange N couples on N tables each table has 2 peo

Randomly arrange N couples on N tables , each table has 2 people.

Let X = number of couples sitting together on a table. Find E(X). Solve it using indicator (0-1 type) random variable. Define your indicator random variable to get full credit. Otherwise, I will take out at least 2 points even if your answer is right.

You need a value for N to solve this proble. Assume N = 15

Solution

N coupres and N tables.

Can be arranged in N! ways as each couple has N choice, and ii couple N-1 choices, etc.

Now couple can sit in any of the two orders among themselfes

Hence total ways = 2(N!)

X = no of couples sitting to gether

X can take values from 0 to15 as N =15

There are two outcomes for a couple either sit together or not sit together.

All trials are independent.

p for a couple to gether

p = A husband or wife selects his spouse to sit near

= 1/699

q = 698/699

Hence Expected value = np = 15(1/699) = 15/699