
Dice Games  Redefining Winning
Who wins a game? It depends totally on how winning is
defined. What will happen if players use inconsistent definitions of
winning? This simple experiment will demonstrate ideas that may be
generalized to understanding voting, sports and economics.
Objective: Students will use mathematics to question the
definition of winning. Students will analyze experimental results
using methods of probability.
Readings: Beyond Numeracy  voting systems chapter, Mathematician Reads the Newspaper. chapt.1 voting, power, and
mathematics

Reformatted: January 2010 

Many simple games exist whose outcome is dependent on chance. Here's
two arrangements where both players may win, or both may lose.
Materials: 2 blank dice of different colors  to be
renumbered
Method:
 We will call the dice red and white.
 Number the Red die with 1,2,3,3,6,6.
 Number the white die with 1,1,1,2,8,12.
 Have two players select a die and compete for the best roll.
 The first player should keep score by tallying the winner of
each roll.
 The second player should keep score by tallying the total
points that each player scores in each roll.
 They should play at least 20 rounds.
 At the end, they should compare who won by each method.
Sample Rounds:

Round

Rolls

Scoring First Method

Scoring Second Method

Red Die

White Die

Red Player

White Player

Red Player

White Player

1

3

1

1

0

3

1

2

2

2

1

0

5

3

3

6

8

1

1

11

11

4

3

1

2

1

14

12

5

6

12

2

2

20

24

In our example, by the end of the fifth round the player
using the first method thinks they have tied 22, the player
using the second method thinks white has won 24 point to 20.


Related discussions at this site


Analysis:
One of the preferred methods of analyzing probability
is to make a chart. Both games will be analyzed using two 6x6 chart
showing all of the possible roll combinations. And the possible
scoring combinations.
Scoring Method 1: show who gets the point for the
round
Rd \ Wh 
1 
1 
1 
2 
8 
12 
1 
tie

tie

tie

White 
White 
White 
2 
Red 
Red 
Red 
tie

White 
White 
3 
Red 
Red 
Red 
Red 
White 
White 
3 
Red 
Red 
Red 
Red 
White 
White 
6 
Red 
Red 
Red 
Red 
White 
White 
6 
Red 
Red 
Red 
Red 
White 
White 
From this chart, we can see that in a typical set of
36 rounds red will win about 19 times, and white will win
about 12 time, with 6 ties.

Scoring Method 2: show the difference in scores
for each round
Rd \ Wh 
1 
1 
1 
2 
8 
12 
1 
0

0

0

1 
7 
11 
2 
1 
1 
1 
0

6 
10 
3 
2 
2 
2 
1 
5 
9 
3 
2 
2 
2 
1 
5 
9 
6 
5 
5 
5 
4 
2 
6 
6 
5 
5 
5 
4 
2 
6 
If we add up all the differences in this chart, we
can see that in a typical set of 36 rounds white will beat
red by 24 points.

Extensions:
Now that students have seen the same game have different outcomes
depending on how you score it, they need to generalize this concept
to common life experiences.
 We may have noticed that the die with the higher median wins
more rounds; the die with the higher average wins more points.
What does this tell us about interpreting statistical data?
 Is it possible in basketball, football, volleyball, etc. for
the team with the most points not to make the playoffs?
 Is it possible in the American two party system for the
candidate with the most popular votes not to become president? Has
this happened? If so, how often?
 If there were three presidential candidates in America, under
what circumstances could the candidate with the least popular
votes become president?
 Using the books referenced at the top of this page students
can research how this applies to democratic elections.
 Since top 10 and top 40 countdowns are popular with kids they
should be able to show how different scoring methods give totally
different results. For example a count down that asks people to
name their favorite song will give different results than one that
asks people to name their favorite three songs giving 10 points to
the first 5 to the second and 3 to the third, which will surely
give different results than asking people to list their favorite
song and their most hated song.
 For a research project, how did the computer program called
"titfortat" win an entire competition without ever winning a
single round? What does this imply about cooperation in human
relationships?

