The Cereal Box Problem:

As you do more and more trials, you will find that the number approaches 14.7. This number is given by:
6/1 + 6/2 + 6/3 + 6/4 + 6/5 + 6/6 = 14.7

Similarly, it can be shown that for an eight sided die, the theoretical number (expected value) of rolls needed to get all eight sides is:
8/1 + 8/2 + 8/3 + 8/4 + 8/5 + 8/6 + 8/7 + 8/8 = 21.7

Use this pattern to find the expected number of rolls of a 12-sided die to get all 12 sides. Then use a 12-sided die, a spinner, or some other appropriate model and conduct 50 trials. Compare your estimated number of rolls of the die with the theoretical value given by this formula"*

Now that you have modelled the problem with a die and seen the theoretical expected value, try this online demonstration of the problem.

For a full analytic explanation of the mathematics behind this problem, see "The Cereal Box Problem" by Jay Wilkins (PDF file).

*From Using Statistics, by Travers, Stout, Swift, and Sextro. Addison-Wesley Publishing Company. 1985.

• The Cereal Box Problem: Introduction
• Go on a simulated shopping trip to get the prizes
• Teacher's Notes
• Model the problem in your classroom
• Further Expected Value Questions

Interested in the mathematics of the cereal box problem?
Check out the following article by Jesse "Jay" Wilkins, Wilkins, J. L. M. (1999).
The cereal box problem revisited . School Science and Mathematics, 99(3), 117-123.
Click on the title to download the article in Adobe Acrobat format.

Special thanks to School Science and Mathematics, for allowing us to distrubute the article in this manner.

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