#### A Math Forum Project

 ESCOT Problem of the Week: Archive of Problems, Submissions, & Commentary

Student Version

### Rock, Paper, Scissors I - posted January 4, 2000

Welcome to the first ESCOT Problem of the Week for the new millennium! We will be playing the Rock, Paper, Scissors game to explore some concepts of probability. For those of you who are not familiar with Rock, Paper, Scissors, it is a choosing game in which two or more people make "throws" with their hands. The throws resemble the shapes of a rock, scissors, and paper. The game is scored as follows: Rock smashes Scissors (rock wins); Paper covers Rock (paper wins); Scissor cuts paper (scissors win). If both players make the same throw, there is a tie.

In this activity you will observe a simulation of the Rock, Paper, Scissors game and tell us what you think about its fairness. In the simulation there are two players, Ed and Vicky. The scorekeeper, Mr. Garrison, is watching over the game. The game is played 100 times. The throws for each player are generated randomly.

There is a graph next to the simulation that helps us think about Ed's winning percentage. It plots, over the course of the 100 throws, the ratio of Ed's wins to the total number of wins, ignoring ties for now. The vertical (up) axis represents this ratio, and it goes from a value of 0, where Ed isn't winning any, to 1, where Ed is winning all. The number of wins for Ed and Vicky are shown below the graph.

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After running the simulation several times, it was clear to most students that the total numbers of Ed's wins, Vicky's wins, and ties were about the same. This means that the game is fair, since we can expect -- in the long run -- Ed and Vicky to win the same number of games.

The graph provided, plotting Ed's wins against the total number of wins, is a visual interpretation of this. Though it jumped around at the start of the simulation, it eventually settled into a straight line at the halfway mark, meaning that Ed was winning about half of the games and showing that the game was fair.

Most students saw that the game was fair, and were able to run the simulation successfully. They found it more difficult to interpret the graph and to explain how its long-term behavior and placement related to the fairness of the game.

### Highlighted solutions:

 From: Kaite T., age 13Tani O., age 12 School: Issaquah Middle School, Issaquah, WA

```Page First RPS Activity
Press Reset and try the simulation as many times as you wish.  (remember to
press Reset then clearGraph between tries)   When you are done, press RESET
one more time to answer the following two questions:
that this game is fair because in it both Vicky and Ed have an equal chance of
winning the game.  Its all luck.  The graph showed us that Ed won 31 games,
Vicky won 35 games, and they tied 34 times.  Then when we ran it again we
noticed that Ed won 43 games, Vicky won 32 games, and they tied 25 games.  All
of these numbers are pretty close to each other and they both had a fair
chance at winning the game.

2. Explain what happens to the graph as the game goes on.  What does this tell
you about the "fairness" of the game?  As the graph goes on, the line starts
to get more level in the middle.  This tells us that as the game goes on, the
more chances of either of them winning increases.  They get a 50/50 chance.
The game gets more fair.  This also means that their chances of tieing also
increases.

------------------------------------
```

 From: Richa K., age 13 School: Issaquah Middle School, Issaquah, WA

```Page First RPS Activity
Press Reset and try the simulation as many times as you wish.  (remember to
press Reset then clearGraph between tries)   When you are done, press RESET
one more time to answer the following two questions:
Yes, I do think that this game is fair. That's because, when you try the
simulation several times and compare the number of times they win, it is about
the same. I tried the simulation about 6 times, and found that Ed won three
times and Vicky won three times, if you exclude the number of ties. The
percentage of both of them winning out of hundred is about the same (there is
not a huge difference).
2. Explain what happens to the graph as the game goes on.  What does this tell
you about the "fairness" of the game?
As the gme goes on, the graph comes toward the middle and levels out. The
thing it tells me about the fairness is that the number of times each person
wins becomes a very close number (meaning that the number of times each person
wins out of hundred excluding the ties is about the same).

------------------------------------
```

 From: Sean R., age 13 School: Issaquah Middle School, Issaquah, WA

```Page First RPS Activity
Press Reset and try the simulation as many times as you wish.  (remember to
press Reset then clearGraph between tries)   When you are done, press RESET
one more time to answer the following two questions:
Yes, because it has the same probability to win as a normal rock, paper,
scissors game does.  In each game there is a one third chance of you winning,
a one third chance of your opponate winning, and a one third chance of tieing.

2. Explain what happens to the graph as the game goes on.  What does this tell
you about the "fairness" of the game?
The graph is usually unfair towards the begining, and then it starts to even
out as the line progresses.  This is because there are more games played, so
there are more chance to win.  This means the graph will be very high or low
towards the begining, and in the middle towards the end.

------------------------------------
```

 From: Allen C., age 12Nelson A., age 12 School: Issaquah Middle School, Issaquah, WA

```Page First RPS Activity

Yes, this game is fair becuase, as the graph shows, the wins for each person
is quite even and constant throughout the whole 100 trials.

2. Explain what happens to the graph as the game goes on.  What does this tell
you about the "fairness" of the game?

The first person that won caused the graph to jump. Then, after that, the wins
for each person began to even out. This tells you that the game is pretty fair
because both people won about the same amount of trials.

------------------------------------
```

### 34 students received credit this week.

Nelson A., age 12 - Issaquah Middle School, Issaquah, WA
Amber B., age 13 - Issaquah Middle School, Issaquah, WA
Christine B., age 13 - Issaquah Middle School, Issaquah, WA
Keturah B., age 13 - Frisbie Middle School, Rialto, CA
Allen C., age 12 - Issaquah Middle School, Issaquah, WA
Jamecia C., age 13 - Frisbie Middle School, Rialto, CA
Katie C., age 13 - Issaquah Middle School, Issaquah, WA
Marshall C., age - Issaquah Middle School, Issaquah, WA
Andrew D., age 14 - Issaquah Middle School, Issaquah, WA
Andy E., age 13 - Issaquah Middle School, Issaquah, WA
Jackie G., age 13 - Issaquah Middle School, Issaquah, WA
La Shanette K., age 13 - Frisbie Middle School, Rialto, CA
Richa K., age 13 - Issaquah Middle School, Issaquah, WA
Bryan L., age 13 - Issaquah Middle School, Issaquah, WA
Derek L., age 13 - Issaquah Middle School, Issaquah, WA
Akira M., age 13 - Frisbie Middle School, Rialto, CA
Ashlie M., age - Issaquah Middle School, Issaquah, WA
Robert M., age 13 - Issaquah Middle School, Issaquah, WA
Tyler M., age 14 - Issaquah Middle School, Issaquah, WA
Brett N., age 12 - Issaquah Middle School, Issaquah, WA
Tani O., age 12 - Issaquah Middle School, Issaquah, WA
Christina R., age 13 - Issaquah Middle School, Issaquah, WA
Sarah R., age 14 - Issaquah Middle School, Issaquah, WA
Sean R., age 13 - Issaquah Middle School, Issaquah, WA
S., average age 12 - Issaquah Middle School, Issaquah, WA
Matthew S., age - Issaquah Middle School, Issaquah, WA
Norma S., age 13 - Frisbie Middle School, Rialto, CA
Vasiliy S., age 13 - Issaquah Middle School, Issaquah, WA
Yusra S., age 13 - Issaquah Middle School, Issaquah, WA
Kaite T., age 13 - Issaquah Middle School, Issaquah, WA
Katy V., age 12 - Issaquah Middle School, Issaquah, WA
Benjamin W., age 14 - Issaquah Middle School, Issaquah, WA
Mac W., age 14 - Issaquah Middle School, Issaquah, WA
David Y., age 12 - Issaquah Middle School, Issaquah, WA