# Collecting Prizes from Cereal Boxes A variation of Basilio's Problem

Sampling with replacement

"The boxes of Corn Flakes marketed by the Crackle-and-Pop Cereal Company contain 20 different prizes. How many boxes we have to purchase to have a 50/50 chance of getting all the prizes?"

If we assume that the number of Crackle-and-Pop Corn Flakes available in supermarkets is so large in comparison with the number we can possibly purchase, then the probability of getting any one of the 20 prizes is only infinitesimally affected by our choices. Thus we may simulate our picking a box of Crackle-and-Pop Corn Flakes from the shelves of a supermarket, opening it an getting the prize, by the spinning of a fortune wheel with 20 equally probable outcomes. We may of course generalize the problem by changing the number k of different prizes that may be found in these boxes. In particular, when k=6 we can simulate this cereal box experiment with an ordinary die. The question then becomes: how many times to we have to cast the die to have 50/50 chance of all the six numbers to show up?

In the following Applet you enter the number k of different prizes and the number x of boxes you want opened. Upon pressing the ''Go'' button the Applet will simulate the random selection of x of these boxes, opening them and counting the number of different prizes found. Repeat this experiment as often as you wish starting say with k=20 and x=35 and see how often it gets all the k different prizes. Progressively increase the value of x until you reach a point where all k prizes are obtained roughly half of the time.