Tim Horton's 100 challenge

Recently a few people have put the Tim Horton's roll up the rim challenge to the test. Apparently they are buying 100 cups of Tims coffee in an experiment.

I wonder what they were trying to discover. lol Tims is rigged? maybe they thought with 100 tries things would converge and they would have exactly 20 winners. In Tim Horton's roll up the rim they state that 1 in 5 cups wins something. The something is typically an inexpensive food prize such as a free coffee or doughnut.

I thought offhand 100 isn't a real large sample size and there would be some variance and probably surprises from people actually doing a sample of 100 times and recording the results. First I thought to write a script to use a random number generator to build a large set of samples of 100 trials (taking Tims at their word that indeed 1 in 5 cups randomly is a winner), then see what kind of data emerged.

Then thinking about it some more, I realized the chance of each of the outcomes, from 0 wins to 100 wins, can be computed exactly. This is the equation, where x is the number of times to win.

The probability of winning x times over a sample of 100 where each attempt has a 1 in 5 chance of winning is

This formula can be readily entered into Excel and we can determine the chance of each outcome. I entered it into a spreadsheet and these are some observations of the results.

The chance of losing all 100 times is about 1 in 4.9 billion. So any regular who tells you they never win is probably selectively forgetting a few stray wins here and there.

The chance of winning all 100 times is about 1 in 10⁷⁰, or 1 in 7,888,609,052,210,030,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.

There is about a 1.26% chance of winning fewer than 12 times.

There is about a 1.12% chance of winning more than 29 times. So regulars who think they win about half the time are likely overestimating how often they win.

With a 1 in 5 chance each time, the expected would be of course 20 wins. There is actually a 9.93% chance of winning exactly 20 times, or more than 90% to get something other than 20. Most of the action is around 20, there is an 83.2% chance of coming in between 15 and 25 wins over the random sample of 100 cups.