Coupon collectors problem CS 658 PoChing Liu Question

Coupon collector’s problem CS 658 Po-Ching Liu

Question • A person trying to collect each of b different coupons must acquire approximately x randomly obtained coupons in order to succeed. • find x

Balls and bins • 1 st question: If n identical balls are tossed randomly into b bins, then how many balls will fall in a given bin? • Ans:

Second • Second question: Find the expected number of balls we need to toss (one ball for each toss) until a given bin contains a ball. The number of bins = b • Ans: E = b

proof • p=pro(success) = 1/b=1 -q • q=pro( without success)=1 -1/b=1 - p • E[# of tosses until a given bin contains a ball]

Third • the third question: Find the expected number of balls we need to toss (one ball for each toss) until every bin contains at least one ball? The number of bins = b • Ans: E = b(lnb+O(1))

proof • There are b stages • The ith stage consists of the tosses after the (i-1)th hit until the ith hit. • how many balls do we have to toss in order to move from the (i-1) th stage to the ith stage there are (i-1) bins that contain balls and b-(i-1) empty bins. for each toss in the ith stage, the probability of obtaining a hit is (b-i+1)/b.

proof • E[# of tosses in the ith stage]= • E[total # of tosses in the 1 st to the bth stage] =

proof • We find the two following questions are the same: • Q: Find the expected number x of balls we need to toss (one ball for each toss) until every bin contains at least one ball? The number of bins = b • Q: A person trying to collect each of b different coupons must acquire approximately x randomly obtained coupons in order to succeed.

Code by java compile: javac Coupon. Collector. java Run: java Coupon. Collector • • • • • • • public class Coupon. Collector { public static void main(String[] args) { int N = 50; // number of different card types boolean[] found = new boolean[N]; // found[i] = true ==> if card i has been collected , int cardcnt = 0; // total number of cards collected int valcnt = 0; // number of distinct cards false ==> the new type of card // repeatedly choose a random card and check whether it's a new one while (valcnt < N) { int val = (int) (Math. random() * N); // random card between 0 and N-1 cardcnt++; // we collected one more card ==>total number +1 if (!found[val]) valcnt++; // it's a new card type found[val] = true; // update found[] } } } // print the total number of cards collected System. out. println(cardcnt);

references • Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, Clifford Stein(1990), Introduction to Algorithms, The MIT Press, Cambridge, Massachusetts London, England, pp. 109 -110. • http: //wwwstat. stanford. edu/~susan/surprise/Collector. h tml