Rock-Paper-Scissors—Art?

In 2005, Takashi Hashiyama, president of a successful Japanese electronics company, wanted to sell his company’s $20 million art collection. He decided to use one of two famous auction companies—Christie’s or Sotheby’s—to offer the collection for sale. Since he felt that the two companies were equally qualified, Mr. Hashiyama invited representatives from Christie’s and Sotheby’s to compete in a game of rock-paper-scissors. (Using games of chance to make such decisions is common in Japanese business.) The winner would get to sell the collection.

In this activity, you will learn more about the game of rock-paper-scissors via simulation and the rules of probability. Let’s review the basic rules of the game before you begin. A “round” of the game consists of a “throw” by each of the two opposing players. There are three possible throws that a player may choose: rock, paper, or scissors. If both players make the same throw, (e.g. rock and rock), then the round is a tie. Otherwise, the winner of the round is determined as follows: rock beats scissors, scissors beats paper, and paper beats rock.

1)Play 10 rounds of rock-paper-scissors against your opponent. Record the result of each round in the table below:

Round / My throw / Opponent’s throw / Outcome
1
2
3
4
5
6
7
8
9
10

2)Count the number of rounds won by each player. Is there a winner?

3)Based on your results, does it appear that both players were selecting among the three possible throws at random in each round? Or do you think one or both players were using some more sophisticated “strategy”? Justify your answer.

4)Work with your partner to design and carry out a simulation to help answer the question posed in Step 4. Use the random digits provided. Let 1 = win, 2 = lose, 3 = tie, and ignore/skip over values 4 thru 9 and 0. Describe your simulation below. Perform 10 trials. What conclusion would you draw? Why?

1387381598950529090873512751828713695761967461214936089253301154214873

5)Suppose that both players in a 10-round game of rock-paper-scissors really did select their throws at random in each round. Work with your partner to calculate each of the following probabilities. Be sure you can explain your method.

(a)P(Player A wins the first round)

(b)P(Player A wins the first round and the second round)

(c)P(Player A wins all 10 rounds)

(d)P(the same player wins all 10 rounds)

(e)P(Player A wins exactly 9 rounds)

6)Let’s consider a different type of question. If both players were selecting throws at random in each round of a game of rock-paper-scissors, how many rounds would you expect them to have to play before someone wins? With your partner, design and carry out a simulation to help answer this question. Use the table of random digits below. Let 1 = win, 2 = lose, 3 = tie, and ignore/skip over values 4 thru 9 and 0. Perform 10 trials.

1922395034057562871396409125314254482853736574115099400019272775442648

7)Suppose that both players in a game of rock-paper-scissors really did select their throws at random in each round. Work with your partner to calculate each of the following probabilities. Be sure you can explain your method.

(a)P(there is a winner in the first round)

(b)P(there is no winner in the first round, but there is a winner in the second round)

(c)P(there is no winner until the third round)

(d)P(there is no winner in the first 10 rounds)