Chapter 1.3, Problem 9E

An urn contains four balls numbered 1 through 4. The balls are selected one at a time without replacement.

A match occurs if the ball numbered m is the, mth ball selected. Let the event A- denote a match on the ith draw $i=1,2,3,4$.

(a) Show that $P\left(A_i\right)=\frac{3!}{4!}$ for each i.

(b) Show that $P\left(A_i\cap A_j\right)=\frac{2!}{4!},i\ne j$.

(c) Show that $P\left(A_i\cap A_j\cap A_k\right)=\frac{1!}{4!},i\ne j,i\ne k,j\ne k$.

(d) Show that the probability of at least one match is

$P\left(A_1\cup A_2\cup A_3\cup A_4\right)=1-\frac{1}{2!}+\frac{1}{3!}-\frac{1}{4!}.$

(e) Extend this exercise so that there are 1, halls in the urn.

Show that the probability of at least one match is

$\begin{array}{l}P\left(A_1\cup A_2\cup \mathrm{...}\cup A_n\right)\\ =1-\frac{1}{2!}+\frac{1}{3!}-\frac{1}{4!}+\mathrm{...}+\frac{{\left(-1\right)}^{n+1}}{n!}\\ =1-\left(1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\mathrm{...}+\frac{{\left(-1\right)}^n}{n!}\right).\end{array}$

(f) What is the limit of this probability as n increases without bound?