# Thread: Probability question relating to harmonic series

1. ## Probability question relating to harmonic series

Hello all,

I'm stuck on another simple question, I think the solution is based somewhere on reversing Euler's approximation for harmonic series, yet I'm unable to make the link between the math and the question.

Question:
McBurgerQueen wants to run a competition where the grand prize is a very expensive car. In order to enter the competition customers must posses 1 of each coupon that comes attached in a random manner to their McBurger.

However McBurgerQueen wants to ensure that everyone entering the competition purchase on average at least 12 burgers.

At least how many different coupons should there be?

I've run a simulation iterating from 2 to 10 coupons over 10^7 selections in each, and it seems that ~6 coupons is the smallest number required, to average 12 burgers and to posses all 6 unique coupons (5 averages ~11 and 7 averages ~14). I'd really like to know what the pure mathematical way is to solve this problem. any assistance will be greatly appreciated.

Other ways to think of the problem:

1. How many times on average should a 6 sided dice be rolled so as to get each of the 6 sides at least once?

2. I have a fair n-sided dice, which on average takes 12 rolls to get each side at least once, how many sides does the dice have?

2. The question asks you to ensure that everyone buys at least 12 burgers. I think you'll need 12 coupons for that.

If you want to target the average number of burgers (or something other than certainty) then the number of coupoons required will depend on the relative frequency of each coupon. For example you could do it with 2 coupons if the second coupon only appeared on 1 in 100 burgers.

3. Hello SpringFan25,

Thanks for the reply, however that is definetly NOT the correct solution - I'm not sure if you have access to Fifty Challenging Problems in Probability with Solutions (Amazon.com: Fifty Challenging Problems in Probability with Solutions (9780486653556): Frederick Mosteller: Books) but if you do, this problem is essentially the reverse of Problem 14 "collection coupons". Where it says a cereal brand has 5 different coupons, on average at least how many boxes of cereal must you purchase to to have all 5, the answer is roughly: n*ln(n) + 0.577n + 1/2 where n is the number of coupons which for n=5 is ~ 11.43

4. The difference between the type of question you just posted and the one in your first post is the word average.

"How do I ensure that everyone buys at least 12 burgers" is not the same question as "how many coupons are required so that the average number of burgers bought is at least 12".

Moving on to your intended question:

It seems you already have the solution for a closely related problem "if I have n coupons, what is the average number of burgers that will be bought", so a convenient approach is to use trial and error to find the smallest number of n where the average is at least 12.

5. Hi SpringFan25,

You're right, I've added the word average to the question. The issue is I'm not interested in a trial-n-error approach (as I've already run a numerical simulation and also done a trial-n-error using the above mentioned formula, what I would like a is more consistent numerical approach.

btw I'm going to add the following two similar examples, that might help explain the type of problem this is:

Other ways to think of the problem:

1. How many times on average should a 6 sided dice be rolled so as to get each of the 6 sides at least once?

2. I have a fair n-sided dice, which on average takes 12 rolls to get each side at least once, how many sides does the dice have?

6. Originally Posted by Sherrie
Hello all,

I'm stuck on another simple question, I think the solution is based somewhere on reversing Euler's approximation for harmonic series, yet I'm unable to make the link between the math and the question.

Question:
McBurgerQueen wants to run a competition where the grand prize is a very expensive car. In order to enter the competition customers must posses 1 of each coupon that comes attached in a random manner to their McBurger.

However McBurgerQueen wants to ensure that everyone entering the competition purchase on average at least 12 burgers.

At least how many different coupons should there be?

I've run a simulation iterating from 2 to 10 coupons over 10^7 selections in each, and it seems that ~6 coupons is the smallest number required, to average 12 burgers and to posses all 6 unique coupons (5 averages ~11 and 7 averages ~14). I'd really like to know what the pure mathematical way is to solve this problem. any assistance will be greatly appreciated.

Other ways to think of the problem:

1. How many times on average should a 6 sided dice be rolled so as to get each of the 6 sides at least once?

2. I have a fair n-sided dice, which on average takes 12 rolls to get each side at least once, how many sides does the dice have?
Coupon Collector's Problem

CB