[Caution: This post is a bit more... uhh... geeky. But its good for you to know. Really. =)]
So the chance of winning a Tim Hortons Roll Up The Rim prize is 1 in 9, just over 11%. So you can go every day, and tell yourself right before your coffee that today is a new day, and you will win your donut. And you continue in this little mental drama for the next few weeks, until Tim Hortons either run out of cups or until you actually win something better than food.
But statistics schamistics aside – all you really care about how lucky you are in the end, right? That inner urge to find out whether or not you got lucky more often than the next guy. Well, its actually quite simple – you use binomial distribution and Microsoft Excel! If you’re totally lost, binomial distribution is the:
Princeton – a theoretical distribution of the number of successes in a finite set of independent trials with a constant probability of success.
Lets go with an example – Lets say this year you only buy 9 cups of coffee, and you win once. So you know that you have conducted nine trials, your outcomes of success has been one, and your odds are 1/9 (11%). So open up Excel and punch this into a cell:
=BINOMDIST(1,9,1/9,FALSE)
Your answer comes out to 39%. This means that the odds of buying nine cups of coffee and winning only once would be 39%. In fact, someone would win nothing 35% of the time, and someone would win two food prizes 19.5% of the time!
Lets use a more realistic example – Most people will buy a few dozen coffees during the course of this contest. Last year, I bought 45 cups and only won 3 times. By using the formula above, the odds of my situation occuring was 14%. Even worse, the probability of my case or worse occuring, was a low 24% (You can get this number by changing FALSE to TRUE, which gives you the cumulative sum of that point and everything below it)! A nice curve below visualizes where you sit on the curve.
Tim Hortons Roll Up The Rim 2005 binomial distribution
I’ve always wanted to implement this into some sort of web-application, but I never got the chance – so now you can do it yourself. 
Sweet! I’m doing something similar this year, but not quite so, umm… mathematical. More pragmatic. Buy coffee. Buy Super-7. Pretend to buy random stocks on the TSX. Watch which one does better….
Thanks for commenting!
Depends on how you’re calculating the return on the TSX, of course. =) I’d take the money over my coffees any day…
In regards to your experiment… You’ve probably heard a study before, where a group of researchers took a few darts and threw it at a newspaper – wherever it landed was the stocks the researchers were going to buy. They then compared the ending portfolios of their random one against professionally maintained portfolios. The result was that the random one was, on average, performed better than the professionally maintained ones!
The take-away? Invest in dart-making companies. =)
It is the simplest case of binomial distribution where the trials follow bernaulli trial.