Venn Diagrams, Coupon Collections, Bingo Games and Dirichlet Distributions
Milton Sobel, K. Frankowski · Birkhäuser Boston eBooks · 1997
This chapter is an extension of the coupon collectors problem which arose from the free baseball cards that used to be inserted in a penny package of chewing (bubble) gum. The setting is that of a bingo game with a number caller. Each player has a list of numbers and waits for all his numbers to be called. These lists may be overlapping to any degree; hence, the need for Venn diagrams. The contest usually ends with the first winner, but need not. If not, we have to specify whether or not we use curtailment, i.e., stop as soon as the results are known. The calculations include many concepts including: 1. Expected waiting time for the first winner (in terms of numbers called) and for the entire contest (if it is different), 2. Probability of a single winner, i.e., without ties, 3. Expected number of integers not yet observed when the contest ends. In addition, for “with replacement”, we also find: 1. Expected maximum frequency among the numbers called, 2. Expected number of singletons among the numbers called, 3. Expected number of integers repeated among the numbers called. In summary, it is generally not realized how useful the Dirichlet integrals and their hypergeometric analogues can be to solve challenging, non-trivial problems; this chapter illustrates one more important use of these integrals.