Chapter 4, Problem 4.16STPE

Each of n boys and n girls, independently and randomly, chooses a member of the other sex. If a boy and girl choose each other, they become a couple. Number the girls, and let $G_i$ be the event that girl number i is part of a couple. Let $P_0=1-P\left({\displaystyle {\cup }_{i=1}^n{G}_i}\right)$ be the probability that no couples are formed.

a. What is $P\left(G_i\right)$?

b. What is $P\left(G_i|G_j\right)$?

c. When n is large, approximate $P_0$

d. When n is large, approximate $P_k$ the probability that exactly k couples are formed.

e. Use the inclusion—exclusion identity to evaluate $P_0$