Interesting Probability Questions about the Tim Horton's "Roll Up the Rim to Win" Contest

2014 (JavaScript application for the 2014 game) (Official rules)

2013 (JavaScript application for the 2013 game)

• The joke around my office is that i'm rather unlucky when i comes to winning at this contest.  Right now i am 0 for 15 this year (if i'm correct i'm only about 6.5% likely to be in that situation).  Last year i went 1 for 28.  Combined i'm 1 for 43.  I have some questions in regards to how i would go about calculating the odds for some of the following:

1) As the events are not completely independent, do my odds of winning increase every time i roll up a loser?  If so, what are my odds of winning on my next cup?

• While the events are not completely independent, the large number of cups in the contest (130025160 in Ontario) means that you will improve your chance by very little. For example, let say you are already in the middle of the contest, and you roll up one more loser. Then your chance of winning the next one will increase slightly by about 0.0000008% and therefore the events are essentially independent.

2) The contest odds this year were the same as last year (1 for 6) how would i combine my stats for these two years, or can I even do this with any certainty?

• For the reason I explained in question 1, you can combine these two years together. In your case, the probability of going 0 for 15 is (5/6)^15=6.49% and the probability of going 1 for 43 or 0 for 43 are 0.34% and 0.04% respectively.  So assuming 10% people in Canada (3500000) join the competition and have at least 43 coffees in these two years,  then about 11851 people share the same experience with you and 1378 people do worst than you.

3) I would assume that is this case the events are in fact independent, but what are the odds of me going 1 for 28 (or worse) every year?

• The chance is 0.04^n where n is the number of years.

4) How much of a factor does the cup distribution play at a municipal level (in order words, do my odds change by buying a coffee at a downtown location vs a suburban location)?

• While we do not have any information about it, it probably should not matter for the small prices; there is no reason to believe that Tim Horton's want to make people angry. But for the large prices, there is a significant difference, even though the chance of winning is very slim anyway.
  

• I'm curious though, i understand the formula to calculate going 0 for X, but i can't seem to figure out how you calculated going 1 for 43?  (or any number of winners for any number of cups for that matter) 

• Assuming independence, the distribution of the number of winners in X cups is actually Binomial(X,1/6). This means that the probability of getting K winners is given by
  

which is the product of the number of ways to choose K winners from X, times the probability of getting K winners in a particular arrangements.

• In the video here, it seems that the chance of winning with a small cup is way too small (4 out of 38). Is there enough evidence to conclude that the true odds of winning for small cups is actually smaller than 1 in 6, as one suspects? Don't they want to encourage you to buy a bigger cup so that they can make more money?

• No. In fact, assuming that the true odds for small cups is in fact 1 in 6, the probability that you have less than or equal to 4 winners out of 38 is 21.78%, which is not small enough to suspect that the assumption is wrong (or "reject the null hypotheses" in statistics, which typically requires a "p-value" less than 5%). What that means is that assuming that Tim Horton's did not lie and that say 10000 people have 38 small cups, you will find around 2178 of them sharing the same experience (or worse), which is not a small number.
  

• 1. Does 1 in 6 odds mean you can expect to win something for every six cups you buy? Why not? How do you explain the Timmies customer with 20 cups and no wins?

• No. It only means that in the long run if you buy a large number of coffee, say 600 cups, you are expected to win around 100 times. However, anything can happen if you only buy 6. Nevertheless, you can still calculate the chance of winning any number of times and see how lucky or unlucky you are. For example, there is a probability of 2.61% that you have no wins in 20 cups. With so many people buying more than 20 cups during the contest, you will surely find a fair number of unlucky people.

2. Are your odds of winning better if you buy a bigger cup? Why or why not?

• They have not disclosed this information for years now (it was available in the 2006 contest). So we don't know the actual odds anymore.

3. Are your odds better if you live in certain parts of Canada? Why or why not?

• It depends on what odds you are talking about. If you are talking about small prizes (coffee, donuts, etc), they don't disclose it but it is reasonable to assume that the odds are 1 in 6 everywhere. But for large prizes, there are some differences. For example, your chance of winning a car is highest in Atlantic Province, which almost doubles that in Alberta, Saskatchewan, Manitoba, Northwest Territories & Yukon. USA has the best odds for the MasterCard (about 10% better than the worst in Quebec) and Tim Cards (slightly). For the Grills, Ontario has a slight edge.

2006 (Java applet for the 2006 game)

(All Questions pertain to Western & Southern Ontario only.)

• How many coffees would you have to buy before you would expect to win one Toyota RAV4 on average. 
  

• 11,353,500 

• What is the average cash worth of a Roll Up the Rim cup? 

• Approximately 16-18 cents, depending on when the "coffee prices" are redeemed (earn another chance to win if redeemed before the end of the contest). However, if you are like me, who don't consider a coffee or a donut a "real prize", then the worth drops dramatically to about 2 cents only. 

• True or False: you are more likely to win a RAV4 on an extra-large cup than any other.
  

• False. Your chances of winning a RAV4 are lowest on an Extra-Large (about 1 in 16 million). You are 36% more likely to win on a Large than an Extra-Large, and 72% more likely to win on a Medium than an Extra-Large. However, note that the average cash worth of a cup is only about 0.2 cents better on a Medium than on an Extra-Large. (My thoery to this advantage / unfairness is that they might have a slightly higher profit margin when they sell a medium cup and this is a strategy to motivate someone (like me) to buy a medium. Imagine you drank an extra large coffee in the morning, how likely are you going to drink another one in the afternoon?) 
  

• If I won a coffee prize, based on the answer of the previous question, should I then redeem it for a medium coffee over an extra-large coffee?

• The answer of this question depends on whether you really enjoy coffee or you simply want a "real prize". Since  the average cash worth of a cup is only about 0.2 cents better on a Medium than on an Extra-Large, if you really enjoy coffee, an extra-large coffee is definitely worth more than the 0.2 cents (or the tiny extra winning chance of a medium cup). However, if you don't really enjoy coffee, you should take that extra winning chance of a medium cup, no matter how small it is!
  

• What if the contest has three days left and there is one more RAV4 to be won in Southern Ontario? How do your chances of winning a RAV4 compare to at the beginning of the contest? 

• Your probability of winning a RAV4 on a cup is now slightly better than 1 in 4 million. You are now about 3 times as likely to win a RAV4 as at the beginning of the contest. 

• How do my chances of winning a RAV4 on a single cup compare to winning the Lotto 6/49 Jackpot? 

• The probabilities are very similar: about 1 in 14 million for the Lotto 6/49 Big Jackpot, and about 1 in 11 million for the RAV4. You are thus about 23% more likely to win a RAV4 on a single play than a Lotto 6/49 Jackpot. 

• What is the probability of buying 100 cups and not winning any prize?
  

• The probability is about 1 in 130,000 - so you can count yourself very unlucky if this happens!

• What is the probability of winning a "real prize" (i.e. a BBQ, $1000 cash, Plasma TV, or RAV4) if you buy 10,000 cups? 

• The probability is still about 1/3 (34.7%) of not winning. 

• How many cups would you have to buy to have a greater than 50% chance of winning a "real prize"?

• You would have to buy 16,246 cups.

• If you bought a coffee for yourself and three co-workers, what is the probability that all four of you would win a prize?

• The probability is about 0.000152, or 1 in 6577.

    The calculations for the the 2006 contest were done with the help of my student Tomas Farrar.