Solution Found!
There are n distinct types of coupons, and each coupon
Chapter 3, Problem 29STE(choose chapter or problem)
There are n distinct types of coupons, and each coupon obtained is, independently of prior types collected, of type i with probability \(p_i\), \(\sum_{i=1}^np_i=1\).
(a) If n coupons are collected, what is the probability that one of each type is obtained?
(b) Now suppose that \(p_1 = p_2 = \cdots = p_n = 1/n\). Let \(E_i\) be the event that there are no type i coupons among the n collected. Apply the inclusion–exclusion identity for the probability of the union of events to \(P(\cup_iE_i)\) to prove the identity
\(n !=\sum_{k=0}^{n}(-1)^{k}\left(\begin{array}{l}n \\k\end{array}\right)(n-k)^{n}\)
Questions & Answers
QUESTION:
There are n distinct types of coupons, and each coupon obtained is, independently of prior types collected, of type i with probability \(p_i\), \(\sum_{i=1}^np_i=1\).
(a) If n coupons are collected, what is the probability that one of each type is obtained?
(b) Now suppose that \(p_1 = p_2 = \cdots = p_n = 1/n\). Let \(E_i\) be the event that there are no type i coupons among the n collected. Apply the inclusion–exclusion identity for the probability of the union of events to \(P(\cup_iE_i)\) to prove the identity
\(n !=\sum_{k=0}^{n}(-1)^{k}\left(\begin{array}{l}n \\k\end{array}\right)(n-k)^{n}\)
ANSWER:Solution: Step 1 of 2: Given that there are n distinct types of coupo