Question

Q6 The state of Georgia is divided up into 159 counties. Consider a population of Georgia residents with mutually independent and equally likely home locations. If you have a group of n such residents, what is the probability that two or more people in the group have a home in the same county?

Answer 1

We break the problem into 2 cases :

CASE 1 :

By Pigeon Hole Principle, there exists atleast two people who live in the same country. Thus, in this case, the probability is 1.

CASE 2 : $n \leq 159$

We find the number of ways in which all the people live in a different country. First, we choose the required number of countries. This can be done in $\binom{159}{n}$ ways. Next we assign them randomly to the n people in n! ways. So, total number is $\binom{159}{n} \times n!$ .

Total number of outcomes is $159^n$ as each of them has 159 choices of countries. Thus, the probability here is :

$\frac{159^n - (\binom{159}{n} \times n!)}{159^n}$