Chapter 1.4, Problem 19E

Extend Example 1.4-6 to an n-sided die. That is, suppose that a fair n-sided die is rolled ii independent times. A match occurs if side us observed on the ith trial, i = 1,2,. .. n.

(a) Show that the probability of at least one match is $1-{\left(\frac{n-1}{n}\right)}^n=1-{\left(\frac{n-1}{n}\right)}^n$.

(b) Find the limit of this probability as n increases without bound.