A Coffee and a Chance with Tim

Sitting in a Tim Horton’s coffee shop recently, I was enjoying my afternoon coffee and pondering the questions of the here-and-now. Questions such as: What are the good prizes in Tim’s roll-up-the-rim contest? What is the chance that the cup in my hands will win any of those prizes?

The first question is subjective, but the second one can succumb to the cold hard crunching of numbers. The probability of winning each of the prizes can be calculated from Tim’s contest website:

Prize	Available to be Won	Odds of Winning
Coffee	about 35.4 million	1 in 8
Donut	about 11.8 million	1 in 24
$100 Tim Card	25,000	1 in 11,308
$5,000 Visa prepaid Card	100	1 in 2,826,944
Toyota Corolla	50	1 in 5,653,887

It’s difficult to have an intuitive feeling for just how large those large numbers are, how distant those prizes may be from your hands. So let’s try another approach. If I regularly buy coffee from Tim Hortons, how often can I expect to win the prizes? I computed the following table for someone who buys two cups of coffee a day, seven days a week, 52 weeks of the year:

Prize	Available to be Won	Expect to Win*
Coffee	about 35.4 million	once every 4 days
Donut	about 11.8 million	once every 11 days
$100 Tim Card	25,000	once every 13.5 years
$5,000 Visa prepaid Card	100	once every 3.4 millennia
Toyota Corolla	50	once every 6.8 millennia

*Expectation with buying two cups of coffee per day. For the last three prizes, I’m assuming that the contest runs continuously forever and that you are immortal. I’m also including the cups from the coffee prizes in calculating the expectations.

Yes indeed, if the contest ran continuously forever, and if you were immortal, you would have to play on average for as long as all of recorded history to win a Corolla. (With 50 Corollas on offer, these numbers also hint at the amazing quantity of coffee that must flow through all of Tim Horton’s coffee shops.) No wonder the guy who threw away a Tim cup without rolling it, only to find out later that it had won a car, was upset. That guy hadn’t just thrown away the chance of a lifetime, he had thrown away the chance of 90 lifetimes.