mathematical explorations
Cheryl Nelson and Nicole Williams
A Fair Game? The Case of
Rock, Paper, Scissors
r
Rock, paper, scissors is a popular game.
Middle-grades students often use it
to choose the person who will go first
in some activity or larger game. This
game also provides a rich context
to explore the relationship between
experimental and theoretical probability through the concept of a fair game.
Research finds that students learn best
from classroom activities that connect
to real-world problems (NCTM 2000;
Sutton and Krueger 2002).
Using the game of rock, paper,
scissors, we distinguish between two
types of probabilities: experimental
and theoretical. Experimental probability, determined by conducting an experiment repeatedly, uses the frequency of the outcomes and the relative
frequency to estimate probabilities. In
contrast, theoretical probability calculates the probability of each outcome
based on known or theoretical premises and possibly calculations.
Cheryl Nelson, [email protected],
and Nicole Williams, nwilliams@
winona.edu, are colleagues in the Department of Mathematics and Statistics
at Winona State University in Minnesota.
They are interested in strengthening
preservice teachers’ understanding of
mathematics through rich investigative
activities.
The activities described here address the following Standards from
Principles and Standards for School
Mathematics (NCTM 2000, p. 248):
• Develop and evaluate inferences
and predictions that are based on
data
• Understand and apply basic concepts of probability
In this article, we present three activities that we have used to promote
probabilistic thinking with middlegrades students.
Activity 1 uses experimental probability as students play the game of
rock, paper, scissors several times and
record their results. In activity 2, a
table and a tree diagram help students
understand the theoretical probability behind the game. The extension
in activity 3 allows three people to
play the game. The student results
Edited by Denisse R. Thompson, [email protected], Mathematics
Education, University of South Florida in Tampa, and Gwen Johnson, gjohnson@coedu
.usf.edu, Secondary Education, University of South Florida. This department is designed
to provide activities appropriate for students in grades 5–9. The material may be re­produced by classroom teachers for use in their classes. Readers who have developed
successful classroom activities are encouraged to submit manuscripts in a format similar
to this “Mathematics Exploration.” Of particular interest are activities focusing on the
NCTM’s Content and Process Standards and Curriculum Focal Points. Send submissions
by accessing mtms.msubmit.net.
Vol. 14, No. 5, December 2008/January 2009
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Mathematics Teaching in the Middle School
311
Copyright © 2008 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
Activity 1: Playing the
Game—Experimental
Probability
Before starting the activity, the rules
should be discussed. Typically, two
volunteers come to the front of the
class and demonstrate the game.
Before playing, we ask the class if the
game is fair and provide an explanation. Making predictions and testing
conjectures aligns with the NCTM’s
Data Analysis and Probability
Standard for grades 6−8, so this is
an important part of the activity. At
this point, rich classroom discussion
results in comments such as these:
• “I think it is fair because each person
has three shapes to choose from.”
• “It is not fair because some people
cheat. They delay showing their
shape and wait for the other person
to show their shape.”
When we ask students how they
would prove that the game is fair,
none of them have any ideas. In response, we suggest that they play the
game to explore its fairness.
As indicated on the activity sheet,
students should form groups of three,
in which two students play the game
45 times and the third student records
the results. Table 1 offers a convenient
record-keeping device.
After completing the experiment,
students use their results to determine
whether the game seemed fair or not,
which leads to a discussion of what
fair means from a probability perspective. In a fair game, each player has an
equal chance of winning. The fairness
of the game can then be considered
from a theoretical perspective using two formats: a table and a tree
diagram.
312
Table 1 Results from 45 trials of rock, paper, scissors
Outcome
Tally
Frequency
A wins
|||| |||| |||| |
16
B wins
|||| |||| ||||
14
Tie
|||| |||| ||||
15
Table 2 Table showing outcomes of rock, paper, scissors
Player A
Player B
we describe can give middle-grades
mathematics teachers valuable insight
into adapting these activities to fit
their classroom needs.
Rock
Paper
Scissors
Rock
T
B
A
Paper
A
T
B
Scissors
B
A
T
Note: A = A wins; B = B wins; T = Tie
Activity 2: Using
Mathematics—
theoretical probability
Activity 2 begins with students
completing a table to find the
theoretical probability of each hand
choice. It continues as students
create a tree diagram to obtain the
same probability results. Students
compare the two formats and extend
the tree diagram to compute additional probabilities.
As table 2 shows, if two players
randomly choose a hand shape, then
each of the nine outcomes in the table
is equally likely. Hence, it can be seen
that P(player A wins) = 3/9, P(player
B wins) = 3/9, and P(tie) = 3/9.
Rock, paper, scissors can be considered a multistage experiment and analyzed using a tree diagram. We filled
out the first branch to get students
started, and they completed the rest of
the diagram, as shown in figure 1.
Fig. 1 Tree diagram illustrating the probabilities for rock (R), paper (P), and scissors (S)
Mathematics Teaching in the Middle School
●
Vol. 14, No. 5, December 2008/January 2009
Our class discussion included how
to calculate the final probability along
the branches of a tree diagram. We
discussed that generally, player A will
choose rock 1/3 of the time and of that
1/3, player B will choose rock 1/3 of the
time; 1/3 of 1/3 leads to RR (rock rock)
occurring 1/9 of the time. It is important for students to compare the table
and the tree diagram formats to verify
that the probability results are the same.
They can see that the results are simply
being represented differently. When
we asked students to compare the two
formats, responses included these:
Fig. 2 Students try to make rock, paper, scissors fair as a three-player game.
(a)
• “The ties are on the diagonal in the
table. There are no T’s on the tree
diagram.”
• “The ties on the tree diagram are
indicated by RR, PP, and SS.”
• “The table is simpler than the tree
diagram because all you have to do is
count the number of T’s, A’s, or B’s.”
(b)
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Vol. 14, No. 5, December 2008/January 2009
●
Mathematics Teaching in the Middle School
313
Activity 3:
PlayING WITH Three
Students are surprised that this game
can be played with three people.
Including a third player challenges
students to determine how to make the
game fair. There are many ways to accomplish this fairness goal. It is interesting to see how creative students can
be when left to their own resources.
As our students played this game using the rules given on the activity page,
they soon discovered that the game was
not fair and that player B had an overwhelming advantage over the other two.
We compared and contrasted the twoplayer game to the three-player game. At
some point, students realized there was
no possibility for a tie in our version of
the three-player game. Each group then
decided to make new rules when players
A, B, and C win, and some incorporated
the possibility of ties. Figure 2 explores
two different sets of rules that students
created to make the game fair.
Concluding Remarks
As students discuss the fairness of this
game, interesting questions can be
posed.
• Is it a fair game if you play the
game only once?
• When people play this game, do
they tend to choose a certain hand
shape more often than another
shape in the other two?
• Can someone be “better than”
another person at this game? How?
Middle-grades students need tasks
that develop their mathematical
reasoning. Principles and Standards
advocates that students should make
and validate conjectures based on data.
Rock, paper, scissors for two or three
players meets these criteria and piques
students’ curiosity. Throughout these
activities, students were encouraged
to reflect and defend their intuition
regarding the game.
Do You Want to Add
Your Two Cents?
Y
ou may have opinions about
articles or de­partments
mentioned in this issue.
Perhaps you have a classroom
activity to share that is related
to one of them. If so, please
share them with other
teachers by sending
an e-mail to
[email protected],
writing “Readers
Write” in the
message line.
314
Mathematics Teaching in the Middle School
SOLUTIONS
Playing the Game
1. Answers will vary.
2. Answers will vary.
3. Fair means that each player has an
equal probability of winning.
4. Answers will vary.
5. Answers will vary.
Using Mathematics
1. See table 2.
2. Answers will vary.
3. See figure 1.
4. Answers will vary.
5. The ties in the table are along the
diagonals. The ties in the tree diagram are at RR, PP, and SS.
6. a. 1/9; b. 2/9; c. 3/9, or 1/3; d. 3/9,
or 1/3; e. 5/9
7. Explanations will vary, but the
probability that no player shows
rock is 4/9.
Playing with Three
1. Answers will vary.
2. Answers will vary.
3. Player B.
4. Answers will vary.
5. If the tree diagram is continued,
these are the possible outcomes:
RRR, RRP, RRS, RPR, RPP, RPS,
RSR, RSP, RSS, PRR, PRP, PRS,
PPR, PPP, PPS, PSR, PSP, PSS,
SRR, SRP, SRS, SPR, SPP, SPS,
SSR, SSP, SSS.
6. a. 3/27, or 1/9; b. 18/27, or 6/9, or
2/3; c. 6/27, or 2/9
References
National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston,
VA: NCTM, 2000.
Sutton, John, and Alice Krueger.
EDThoughts: What We Know about
Mathematics Teaching and Learning.
Aurora, CO: Mid-Continent Research
for Education and Learning, 2002. l
●
Vol. 14, No. 5, December 2008/January 2009
activity sheet 1
Name ______________________________
Rock, Paper, Scissors: Playing the Game
Rules: Each of two players makes a fist and counts aloud to three, each time raising one hand in a fist and swinging it
down on each count. On the third count, each player displays one of three hand shapes: scissors (two fingers); paper (four
fingers); and rock (a fist).
•
•
•
•
If
If
If
If
scissors and paper are shown, the player showing scissors wins, because scissors cut paper.
scissors and rock are shown, rock wins, because a rock crushes scissors.
paper and rock are shown, paper wins, because paper covers a rock.
both players pick the same shape, the round ends in a tie.
Work in a group of three. In problems 1 and 4, the nonplayer should record the results.
1.Play the game 45 times with a partner, and tally the results in the table.
Outcome
Tally
Frequency
Player A wins
Player B wins
Tie
2.On the basis of your results, do you think the game is fair? Why, or why not?
3.What does fair mean from a probability perspective?
4.Find a different partner. Play the game 45 times with your new partner, and tally the results in the table. In the last
column, compile your results from both trials.
Outcome
Tally
Frequency
Compiled Results from
Both Trials
Player A wins
Player B wins
Tie
5.Did switching partners affect the fairness of the game? Explain.
from the December 2008/January 2009 issue of
activity sheet 2
Name ______________________________
Rock, Paper, Scissors: Using Mathematics
Refer to the rules of the game. Determine if the game rock, paper, scissors is fair using mathematics.
1.If players choose rock, paper, or scissors randomly, each of the nine outcomes in the table below is equally likely.
Fill in the table with A if player A wins, B if player B wins, or T if there is a tie.
Player B
Player A
Rock
Paper
Scissors
Rock
Paper
Scissors
2.According to the table, does the game seem fair? Explain using probabilities.
3.You can think of rock, paper, scissors as a multistage experiment and analyze it using a tree diagram. Let R = rock,
P = paper, and S = scissors. Complete the outcomes for each branch of the tree. Record the probability for each
choice by each player.
Player A
Player B
Outcome
R
RR
R
P
S
4.Find outcome RS in the table and on the tree diagram. Which format, the table or the tree diagram, is easier to interpret?
Why?
Player A
Player B
Player C
Outcome
R
RRR
R
R
P
from the December 2008/January 2009 issue of
activity sheet 2
(continued)
5.Where are the ties in the table? On the tree diagram?
6.Use the table or the tree diagram to find the following theoretical probabilities.
a. P(RP) = _________
(The probability of rock and paper being selected)
b. P(SP or SR) = _________
c. P(A wins) = _________
d. P(B wins) = _________
e. P(at least one player shows scissors) = _________
7.Explain how to calculate the probability that no player shows rock using either the table or the tree diagram.
from the December 2008/January 2009 issue of
activity sheet 3
Name ______________________________
Rock, Paper, Scissors: Playing with Three
What would happen if you wanted to play rock, paper, scissors with three people? Form groups of three, and decide which
student is player A, player B, and player C.
1.Consider the following rules for playing rock, paper, scissors with three players.
• Player A wins if all three make the same hand shape.
• Player B wins if exactly two players make the same hand shape.
• Player C wins if none of the players make the same hand shape.
Do you think this game is fair? Explain your reasoning.
2.Play the game 45 times and tally the results in the table.
Name of Player
Outcome
A− _____________
A wins
B− _____________
B wins
C− ____________
C wins
Tally
Frequency
3.If you wanted to win this game, which player would you choose to be? Explain your answer.
4.Compare and contrast the two-player game to the three-player game.
from the December 2008/January 2009 issue of
activity sheet 3
S
(continued)
Name ______________________________
5.Make a tree diagram of the possible outcomes of the game.
Player A
Player B
Player C
Outcome
R
RRR
R
R
P
S
R
P
P
S
R
S
P
S
6. Find the probability of each player winning using the rules at the beginning of the activity.
a. Player A wins (all three make the same hand shape) = _________
b. Player B wins (exactly two players make the same hand shape) = _________
c. Player C wins (none of the players make the same hand shape) = _________
7. Using the information from the tree diagram, rewrite the rules for rock, paper, scissors so that three people can play this
game and so that it will be fair from a probability perspective.
from the December 2008/January 2009 issue of