Recall Example 3.4, in which 2n individuals, who are paired as n couples, are randomly seated around a table.

(a) Use simulation to determine the probability that s of the 2n people have left and right table neighbors of the same gender. Plot Pr(S = s) for several values of n. What appears to be the limiting distribution?

(b) Probl`eme des m´enages This is the more popular (and more difficult) problem associated with sitting next to one’s spouse at a round table. There are n married couples at a dinner party, and the women sit themselves in alternating seats around a table with 2n seats. Then, each man is randomly assigned to an open seat. The probleme des m ` enages is concerned with the probability ´

that no one sits next to his/her spouse. More generally, let Wn be the number of husbands who are seated next to their wives. Then

and zero otherwise. See Holst (1991), and the references therein for derivation of (3.27).3 You should: (i) program the exact probabilities (3.27), (ii) construct a program to simulate the result, (iii) plot both for several n and confirm that they are arbitrarily close for a large enough number of replications, and (iv)
compare (3.4) and (3.27). As the probabilities for both must sum to one, how might you expect the distribution of the probabilities to change?

2n ("-") wr!2n-r (") Pr (W = w)=(-1)- () w= 0, 1,...,n,