The Tim Horton's Lottery

The Canadian doughnut and coffee shop, Tim Horton's has been running its Roll-Up-The-Rim-To-Win Contest for about a month. According to their website, there is about a 1 in 9 chance of winning something, most likely a donut, but possibly a cookie or muffin or coffee, and maybe a bigger prize like a plasma television or an SUV.

Food Prizes: Eligible "winning" RIM TABS have been distributed as
follows: approximately 111 contest cups per medium case of 1,000, approximately 111 cups per large case of 1,000, approximately 111 cups per extra large case of 1,000 are printed with "winning" RIM TABS. ... For the “winning” RIM TABS for food prizes, 50% are donuts, 20% are coffees, 15% are muffins, and 15% are cookies.

Last year, when completing our large art project, my son and I wanted to make sure that we had absolutely no winning cups in our sculpture, and so, along with his daughters, we checked every single cup that we had collected. The result was that we found approximately 40 more winners among the 3200 cups we collected. These were all discarded cups -- cups that people had bought coffee in. We concluded they were either cups that people had not bothered to check or cups for which people had not bothered to collect the prize.

This, year, we have bought maybe 35 cups of coffee from Tim Horton's while the contest has been on. We have not won a single thing. At first, I thought the odds of not winning even a doughnut must be pretty small with that many trials, but I guess it isn't. The purchases were pretty close to random, independent events, since we made them at different times at different Tim Horton's outlets.

The probability of not winning any prizes in 35 purchases? (8/9)**35, which is roughly one in sixty.