### Matters of Size

Someone with more resources than me has tackled a problem related to the subject of an earlier post about Tim Hortons' Roll Up the Rim contest. Do the odds of success increase with cup size?

The analysis done by the authors is incomplete, in my opinion. The findings of their experiment are consistent with popular suspicions. The largest cup has the highest success rate. The smallest cup has the lowest success rate. The two middle sizes have a success rate of exactly 1 in 6. So success is roughly ordered in terms of cup sizes. Does their experiment actually prove that the game is rigged in favour of larger cups, though?

This is after all a relatively small number of cups. There are millions of cups distributed during the contest, but they didn't even check 100. How well does the data on this particular sample reflect the distribution of cups across the country (and the small portion of the US with Tim Hortons)?

Again, as in my earlier post, we can consider the likelihood that one of these would actually happen. If something extremely unlikely happened, it's fair to accuse Tim Hortons of skulduggery (something we should do as often as we can, if only because of the greatness of the word). If we can expect these events to happen fairly often, we should reserve our judgment.

How likely is it, then, for you to win only 4 times in 38 tries, as in the case of the small cups in the article? Without getting into details, that number is 11.6% (rounded to 1 decimal place). Compare this with the 1/6=16.7% chance of winning on a single cup. So your chance of winning on a single cup is only about 50% greater than winning only 4 times out of 38. You're on the wrong side of the odds, but not so far that it proves the likelihood of conspiracy. Roughly 1 in 10 people will have the same experience as you.

What about at the upper end? What is the probability that you would win 5 times with a mere 18 cups? 10.3%. This is also close to 1 in 10, nearly the same probability as winning only 4 times in 38 tries. Again, it's not an extremely unlikely event, and not enough to prove foul play.

My own analysis, of course, is not sufficient to prove that there is no conspiracy either. My only complaint here is that nor is the Huffington Post's analysis sufficient to prove that there is. Unfortunately, in my opinion, the manner in which their findings were presented might serve to suggest that they have proven the popular perception.

There are better ways of discerning how reliable our inferences from Huffington Post data are. This is where my expertise runs out. The math professors in the article might have been more knowledgeable. If they weren't, one of them most certainly could have found someone who was. It's a shame that this wasn't done, or if it was done, that it was left out. It may leave the readers with the wrong impression.

One of the main responsibilities of the media, whether print or electronic, should be to properly inform the public. In my opinion, the Huffington Post only did half a job in that regard, and in this case, to the extent that the odds winning free stuff from a coffee cup matters, it might be worse than doing nothing at all.

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As an addendum, we could ask why it would be wrong if large sizes gave more prizes. Presumably people object on the grounds of fairness. But is it fair to give the same odds of winning to people who pay more for a higher price for their extra large as the people who pay the lower price for a small, especially when you know that if people win a free coffee off the small, they're likely to opt for a larger size when they collect than what they would normally get? In what other context would we think it fair for someone who paid more to have the same chance of winning?

For public relations reasons, Tim Horton's is probably right to make it even. But I think they'd be justified if it weren't, as long as they only adjusted the odds to compensate for the different prices paid.