New pepsi contest odds

Factual Questions
1

Pepsi has a new contest where you get an NFL team name under each cap. Match three caps and you win a t-shirt of the NFL team. This got me thinking about the odds.

First, what are your odds of winning any t-shirt if you buy N bottles of Pepsi?

Next, what are your odds of winnning a particular team’s t-shirt if you buy N bottles of Pepsi?

Finally, how many bottles of Pepsi would you need to buy to have a better than 50% chance of winning ALL the teams’ shirts?

To complicate matters, 1 in 6 wins a free 20 oz pepsi, but I won’t complain if you choose to leave that out of your calculations.

And on a side note, is it legal to buy Pepsi, and then take your entire group of one or two matching caps and sell them on ebay?

2

There’s really not quite enough information here to now. How are the caps distibuted? Does each team have the same number of caps, or do some have more than each other? Or maybe there are different distributions regionally?

Having said that, I imagine most people would be more interested in the scenario where the caps are distributed evenly among all the teams. I’m afraid I don’t have time to look into that at the moment, but I did want to mention the issue of not knowing the distribution.

3

I’ll chip away at your questions and for simplicity I won’t show my work.

For all N<3, then your chance of winning a particular t-shirt is zero and you chance of winning any t-shirt is zero.

:smiley:

4

Usually for questions like this, their website is the place to go to first, as IIRC, contests like this have to give the odds of winning.

http://www.pepsi.com/pepsipromoscur/national_rules/index.php#1

5

Does anyone know how to do this calculation?

6

I’m going to ignore the 1 in 6 chance of a free Pepsi because that makes the numbers easier. I’m also going to assume that the team names are identically distributed.

There are, what, 16 NFL teams? So if you buy 33 bottles of Pepsi, you will win some team’s t-shirt. Off the top of my head, I don’t know how to calculate the odds of winning with any number less than that, but I’ll tell you that for N = 3, it’s pretty close to 0, and for N = 32, it’s pretty close to 1.

The question of winning a particular team’s t-shirt is a little easier–it’s the probability that a binomial (n, 1/16) random variable is at least 3. Obviously we have to assume that n is also at least 3. That’s equal to 1 - sum( [sub]n[/sub]C[sub]k/sub[sup]k/sup[sup]n - k[/sup], 0 < k < 2 ), and while there may be a way to simplify that, I don’t see it.

The last question in the OP–how many bottle caps to collect to have a 50% chance of winning all the t-shirts–is fairly similar to the collector’s problem, but just different enough that you can’t use that problem’s solution here.

7

There are 30 proper teams, plus San Francisco and Miami. :wink:

The rules that Ferret posted imply that you can pick any logo to be on your shirt, no matter which team your three caps are for. Anybody know if this is the case?

8

For r teams, you need 2r + 1 caps to be guaranteed one t-shirt, and the probability of getting any one team’s t-shirt with n bottle caps (n > 3) 1 - sum( [sub]n[/sub]C[sub]k/sub[sup]k[/sup](1 - 1/r)[sup]n - k[/sup], 0 < k < 2 ).

9

(Assuming all 32 teams are distubuted evenly and there are no free pepsis)

With buying 3 bottles:

Odds of getting all 3 to be Miami= 32 x 32 x 32 = 1:32,768

Odds of getting all 3 to be any team= 32 x 32 = 1:1,024

10

Ultrafilter’s formula generates the probability of getting 3 or more bottlecaps with one particular team’s logo, I think. Multiply it by r to get the probability of getting 3 bottlecaps of any team’s logo. This assumes, of course, that teams’ logos are evenly distributed. If some team’s logos are more common than others where you’re buying your sodas, then your odds are even better of getting a t-shirt.

I have no idea how Pepsi came up with the 1:3,455 number. Maybe they asked Jessica Simpson.

11

Think about this. Is it really going to work? What if the probability is greater than 1/r?

12

Look, plug in the numbers n = 3 and r = 32. As Hampshire said, the probability of having the three bottlecaps match each other is 1:1,024. Just plugging in those numbers into your formula gives 1/32,768, 1/32 of what the odds actually are.

13

First off: you’re mixing odds and probability. Don’t do that.

Second: If the probability of something is 1/1024, the odds are 1:1023, not 1:1024.

Third: The formula I gave is for the probability of getting three bottlecaps that have a fixed team (say the Redskins), not three bottlecaps that have the same team. So three Miami caps wouldn’t count as a success in that logic if you were looking for three Redskins caps. A fine distinction, but that’s what probability theory is built on, so be careful not to get confused (especially if I don’t explain it well at first–mea culpa).

14

Oh, and by this formula, the odds with 3 bottles is, as said before, 1/1,024. With 10 bottles it’s 873,825,392,567/8,796,093,022,208 (about 1/10) and with 20 bottles it’s 29,626,639,209,920,638,243,832,708,565/39,614,081,257,132,168,796,771,975,168 (about 3/4).

Oh, and to answer your question, ultrafilter, these fractions are the expected number of t-shirts you’ll win. So with n = 65 bottles, you should expect to win about 10 shirts. The way the contest is written is that it’s any three caps with the same logo for your choice of t-shirts, so if you get 3 Ravens caps you can still get that 49’ers shirt you so desire.

Point taken about the 1:1,023 odds versus 1/1,024 probability.