Transcript The Cereal Box Problem - University of California, Los Angeles

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
Shopping
Orange Yellow Blue
Trip
1
///
/
//
Pink
Red
////
///////
# of
Green
Boxes
////
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
Shopping
Orange Yellow Blue
Trip
Pink
Red
# of
Green
Boxes
1
///
/
//
////
///////
////
21
2
//
///
///
/
/////
//
16
3
/
//
////
//
//
///
14
4
//
////
///
//
/
/
13
5
////////////
/////
/
/
//
///////
28
The Statistic of Interest

Our experiment gives us an average of
21  16  14  13  28
 18.4boxes
5

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.
Graphs
Can use graphs to organize the data.
Red
Pink
Green
Trip1 Trip2 Trip3 Trip4 Trip5
# of Pens
Blue
# of Boxes
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
109
87
65
43
21
0
Yellow
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
Histogram #2
Orange
Histogram #1