Question

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

Given:

Counties=159

The probability that a person is living in one of the county is (1/159)

P(1 person) = (1/159)

Now, for another person, the probability that he is also living in the same county where the first person lives is (1/159)².

Therefore, for n persons, the probability that all are living in the same county is (1/159)ⁿ.

P(n person) = (1/159)ⁿ

Given problem is to find the probability of 2 or more people living in the same county that is, probability(n>=2).

P(n>=2) = 1 - P(n<1)

P(n>=2) = 1 - ⁿC₁(1/159)

Thus, the required probability is (1 - ⁿC₁(1/159)).