The Cereal Box Problem

By Narineh Movsessian

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?

“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?

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.

Results of One Shopping Trip

ShoppingTrip

Orange Yellow Pink Red Green# of

Boxes

1 /// /

Blue

// //// /////// //// 21

We see that we bought 21 boxes of cereal before we had a complete set.

Method Two

Use a six sided die as a physical model for buying cereal boxes.

Randomly let

1=orange 2=yellow 3=blue

4=pink 5=red 6=green One toss of the die will correspond to the

purchase of one box.

Results of Shopping Trips

ShoppingTrip

Orange Yellow Blue Pink Red Green# of

Boxes

1 /// / // //// /////// //// 21

2 // /// /// / /////// 16

3

4

5

/

//

////////////

//

////

/////

////

///

/

//

//

/

//

/

//

///

/

///////

14

13

28

Our experiment gives us an average of

boxes4.185

2813141621

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.

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.

Can use graphs to organize the data.

Histogram #1 Histogram #2

Graphs

02468

1012141618202224262830

Trip1 Trip2 Trip3 Trip4 Trip5

# of Boxes

0123456789

10111213141516171819202122232425

Ora

nge

Yel

low

Blu

e

Pink

Red

Gre

en

# of Pens