## Thursday, November 16, 2006

### Pepsi Math

I bought a Diet Pepsi yesterday and the label described a bottle top promotion: One in six wins "Buy One Get One Free". Suppose you drink a large amount of Diet Pepsi, what is your asymptotic expected discount?

If you buy six bottles then you expect to have one winning bottle top enabling you to get the next two bottles for the price of one, or eight bottles total for the price of seven. But bottles seven and eight have bottle tops too which may be winners. One can continue this process and take the limit but is there a simpler argument?

Soda bottles are interchangeable, a free soda in the future can be exchanged for your current soda. So you can assume that when you get the "Buy One Get One Free" bottle top, your current soda is free. That means you get one free soda from every six, a discount of 16 2/3%.

You get the same discount if the bottle top said "Buy Two Get One Free" or simply "Get One Free". But if the bottle top said "50% off next purchase" which seems equivalent to the original promotion, the discount is only 8 1/3%.

1. "50% off next purchase"

It doesn't seem equivalent to me. 100% off next purchase seems equivalent...

2. The implication is that "buy n, get 1 free" is an equivalent discount for all n, but it seems to break down for large n. (n>6, in this case?).

Call me risk averse if you like, but I'd like to go the merchant and say "I waive my rights to any free sodas, now please sell me soda at a 10% discount, and you'll be better off in the long run". Somehow, I don't think they would buy it.

Graham

3. Here in New York, I don't know of any merchant that actually honors these discounts. So you end up with a 0% discount.

4. Lance your first argument 16.7 percent discount is right.

Another arguments is that 6 caps are spent to get 1 drink and 1 cap.

Therefore a cap is worth 0.2 drinks.

5. A simple argument:

Suppose you have a total of n bottles, where n is large. Then 1/6 of them have winning caps, which means 1/6 of the bottles were given for free.

Noam

6. One the other hand, if you sum the series, you get:

1) Get 6 sodas - get 1 winning cap
2) Get 2 sodas (one free) - get 1/3 winning cap
3) Get 2/3 sodas (1/3 free) - get 1/9 winning cap
...

The sum is you get 1 1/4 sodas free out of 8 1/2. That's 5/34.

7. One the other hand, if you sum the series, you get:
1) Get 6 sodas - get 1 winning cap
2) Get 2 sodas (one free) - get 1/3 winning cap
3) Get 2/3 sodas (1/3 free) - get 1/9 winning cap
...
The sum is you get 1 1/4 sodas free out of 8 1/2. That's 5/34.

That's just bad arithmetic. Notice that the total # of sodas is
6(1+1/3+1/9+...) whereas the number of free sodas is (1+1/3+1/9+...). Hmm. I wonder what that ratio is?

8. On the other hand, putting "buy 10 get 1 free" is not the same as "get 1 free", because with high probability it will leave you with unused discount caps when the promotion is over. Much like pyramid games...

9. I think you didn't calculate it right.
Actually, I think that, in average, you pay the price of six bottles while getting seven bottles which means a discount of 1/7 or less than 15 percent.