A variation of the birthday problem - a reader responds

"A variation of the birthday problem" by Mario Cortina Borja is very much like a more complex version of the "Coupon Collector Problem". The latter was reviewed and discussed by Brian Dawkins in "Siobhan's Problem: The Coupon Collector Revisited".1

Coupon collector and generalizations of the birthday problem come up in promotional games. They are fun to analyze because, unlike real gambling, players don’t lose money. They just get a game piece when they buy a soda or hamburger. I’ve analyzed many such games for clients. Generally speaking, the clients print a disclosure telling people their odds of winning a particular prize given a particular number of purchases.

Some games involved rare coupons, large prizes, and the multinomial distribution without replacement. However, the most enjoyable one that I analyzed was Pepsi’s “Match Three for a Tee”. Pepsi wanted a few million happy winners of T-shirts with football team logos, but tens of millions of winners would not have been practical.

We went through several iterations before selecting a game wherein two thirds of the bottles of single serve Pepsi had a football team logo on the inside of the bottle cap. All teams were equally represented. One-sixth of the caps gave instant wins of a free Pepsi and one sixth gave a 10% discount on the NFL web site. If one collected a triplet of the same team, one could mail the caps in for a T-shirt with the logo of one’s choice. I was told that a game that predated my involvement only required a matching pair (the classic birthday problem except that there are fewer than 365 football teams); its results were why they sought out a statistician for this game.

Like Professor Cortina Borja, I used simulation2 to estimate the probability of getting a triplet as a function of the number of caps collected. I repetitively performed several simulations involving different numbers of bottles per customer. Figure 1 shows the percentage of winners among 1,000 simulated customers versus the number of caps collected. The clustering of the points demonstrates the adequacy of the simulation size. Collecting 25 caps gives one about a 40% chance of obtaining a triplet. Thirty caps gives one about a 58% chance. We had just about the desired number of happy winners.

Figure 1. Percentage of 1,000 customers who obtained a triplet

Players can increase their chances of winning some such games by trading game pieces, but my clients have been happy with that because that sort of activity is thought to promote sales of the product.

  • Emil M. Friedman is "actively retired" and available as a consultant (www.statisticalconsulting.org). He received his BS and PhD in chemistry from MIT and Princeton. He did synthetic rubber R&D at Goodyear until they sent him to Case Western to earn an MS in Applied Statistics. After 29 years at Goodyear he did independent consulting, online teaching and contract work until becoming MannKind Biopharmaceutical's non-clinical statistician. He is now a contractor doing non-clinical statistics at Bristol-Myers Squibb.


  1. Dawkins, B. (1991). Siobhan's Problem: The Coupon Collector Revisited. The American Statistician, 45:1, 76-82 ^
  2. Anirban DasGupta gives an analytical solution for a triplets variation of the classical birthday problem, but only 2/3 of the caps had logos in our game. (“The matching, birthday and the strong birthday problem: a contemporary review”, Journal of Statistical Planning and Inference130 (2005) 377 – 389. (doi:10.1016/j.jspi.2003.11.015) ^