#### A Math Forum Project

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

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### Rock, Paper, Scissors II - posted January 11, 2000

Try Rock, Paper, Scissors - Part I before you do this problem. If you are working on this activity right after Part I, please make sure to quit the ESCOT runner and start it again.
Again we will be observing the Rock, Paper, Scissors game to examine probabilities. In this simulation there are two players. Ed generates throws randomly, but Vicky throws a Rock for every trial. How do you think this will affect the outcome?

Note your prediction in the solutions box. The players will throw 100 times. There is a graph to the right of the simulation that will show how many games Ed wins out of the total number of games that Ed and Vicky win.

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Many students predicted that the new version of this game would be unfair. It does seem strange to mix random and nonrandom choices; however, once they tried the simulation a few times, students realized that since Ed was continuing to choose randomly, the game would remain fair. Ed and Vicky could be expected to win the same number of games.

Many students correctly pointed out that this game wasn't realistic, since Ed, if he were a real person, would certainly catch on to Vicky's constant choice of Rock and counteract with a constant choice of Paper. Some students were hesitant to say that this game was as fair as the previous one, but most were convinced by the identical behavior of the graph.

### Highlighted solutions:

 From: Hana B., age Karen F., age 14 School: Issaquah Middle School, Issaquah, WA

```Page Second RPS Activity
Press Reset and try the simulation as many times as you wish.  (remember to
first press Reset then clearGraph between tries)   When you are done, press
RESET one more time to answer the following questions:

2. Do you think this simulation represents a fair game? Why? Ed has a 1in
three chance of winning, but had he realized that Vicky was always doing rock,
he could have altered his strategy,  thus forcing her to alter hers and making
the game again a game of strategy. The reason that the resaults all center
around 33 is because Ed had a 1/3 chance of winning, losing, and tieing, which
is demonstrated by the the results of wins:35, loses: 32, ties: 33

3. How would you compare the fairness of this simulation to one in which both
players choose their throw
at random? If one person always does the same, then the results will be 1/3,
1/3, 1/3, but if it is random then the players have a 1/3 base probability
which is altered because of the strategy involved. If the game was played long
enough, a player could theoretically even read the way the other player holds
their hand when they're about to use a certain move and anticipate it.

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

 From: Scott F., age 13Josh H., age 12 School: Issaquah Middle School, Issaquah, WA

```Page Second RPS Activity
Press Reset and try the simulation as many times as you wish.  (remember to
first press Reset then clearGraph between tries)   When you are done, press
RESET one more time to answer the following questions:
1. What was your prediction? That he would win because he'd catch on.

2. Do you think this simulation represents a fair game? Why? Yes it is fair,
because he doesn't catch on.

3. How would you compare the fairness of this simulation to one in which both
players choose their throw
at random?
It is the same probability to win as there was when they were random throws.

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

 From: Andrea B., age Jackie G., age 13 School: Issaquah Middle School, Issaquah, WA

```Page Second RPS Activity
Press Reset and try the simulation as many times as you wish.  (remember to
first press Reset then clearGraph between tries)   When you are done, press
RESET one more time to answer the following questions:
1. What was your prediction? We prodicted that Vicky would win beacuse she was
throwind just rock.

2. Do you think this simulation represents a fair game?  Yes we do. Why?
Beacuse evev if vichy throws all rack she will win 1/3 of the time and so will
ed.

3. How would you compare the fairness of this simulation to one in which both
players choose their throw
at random? Both are they same. They both have the same amount of ties and wins
for each. 1/3 of the times tried.

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

 From: Derek L., age 13 School: Issaquah Middle School, Issaquah, WA

```Page Second RPS Activity
Press Reset and try the simulation as many times as you wish.  (remember to
first press Reset then clearGraph between tries)   When you are done, press
RESET one more time to answer the following questions:
I predict that the results of the rock paper sciciors will be the same as
before with each person either wins, loses, or ties and has a 33% chance.
2. Do you think this simulation represents a fair game? Why?
Yes. each persons chance of winning is equal, at about 33%, the rock wins
loses or ties, and the random wins loses or ties
3. How would you compare the fairness of this simulation to one in which both
players choose their throw
at random?
They are basically be the same. Each player has a 1/3 chance of winning the
game when it is random, and has 1/3 a chance of winning when one person is the
same all of the time.
------------------------------------
```

### 18 students received credit this week.

Andrea B., age - Issaquah Middle School, Issaquah, WA
Christine B., age 13 - Issaquah Middle School, Issaquah, WA
Eric B., age 13 - Issaquah Middle School, Issaquah, WA
Hana B., age - Issaquah Middle School, Issaquah, WA
Kalee B., age 13 - Issaquah Middle School, Issaquah, WA
Shannon C., age 13 - Issaquah Middle School, Issaquah, WA
Karen F., age 14 - Issaquah Middle School, Issaquah, WA
Scott F., age 13 - Issaquah Middle School, Issaquah, WA
Jackie G., age 13 - Issaquah Middle School, Issaquah, WA
Jason H., age 14 - Issaquah Middle School, Issaquah, WA
Josh H., age 12 - Issaquah Middle School, Issaquah, WA
Phillip H., age 13 - Issaquah Middle School, Issaquah, WA
Derek L., age 13 - Issaquah Middle School, Issaquah, WA
Scott S., age 14 - Issaquah Middle School, Issaquah, WA
Ella W., age 13 - Issaquah Middle School, Issaquah, WA
Kirsten W., age 12 - Issaquah Middle School, Issaquah, WA
Mac W., age 14 - Issaquah Middle School, Issaquah, WA
Matthew W., age 13 - Issaquah Middle School, Issaquah, WA