The Cereal Box Problem

1
The Cereal Box Problem

• By Narineh Movsessian

2
Random or Chance Outcomes

• How much precipitation is expected next year in
  the Kansas wheat belt?
• How many people in your town are expected to get
  the flu this winter?
• How much of an increase is expected in the value
  of a particular stock in the next two weeks?

3
Predicting the Future

• Suppose each cereal box has one of 6 different
  colored pens.
• Orange Yellow Blue
• Pink Red Green
• Assume chances of getting any of the 6 colored
  pens are equal.
• How many boxes of cereal would you expect to have
  to buy to get a complete set of all six colored
  pens?

4
Method One

• Conduct an experiment.
• Go on a shopping trip and repeatedly buy cereal
  boxes until you get a pen of each color.
• This ends one shopping trip.
• Repeat this process.

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Results of One Shopping Trip
Shopping Trip
Orange
Yellow
Pink
Red
Green
of Boxes
Blue
1
///
/
//
////
///////
////
21
We see that we bought 21 boxes of cereal before
we had a complete set.
6
Method Two

• Use a six sided die as a physical model for
  buying cereal boxes.
• Randomly let
  
• 1orange 2yellow 3blue
• 4pink 5red 6green
• One toss of the die will correspond to the
  purchase of one box.

7
Results of Shopping Trips
Shopping Trip
Orange
Yellow
Blue
Pink
Red
Green
of Boxes
1
///
/
//
////
///////
////
21
2
//
///
///
/
//
/////
16
3
/
//
////
//
//
///
14
4
//
////
///
//
/
/
13
5
////////////
/////
/
/
//
///////
28
8
Our experiment gives us an average of
The Statistic of Interest

• So on average one would have to buy about
  19 boxes of cereal in order to get all 6
  colors of pens.

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Method Three

• Using computer generated random numbers to
  simulate the experiment.
• The actual expected value was found using method
  three over a large number of trials, about
  10,000.
• The actual average is 14.7.

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Can use graphs to organize the data.
Graphs

• Histogram 1

• Histogram 2

11
Summary
Choose a model.
Define a trial.
Define a statistic of interest.
Repeat trials.
Find an average.