Question

Rock smashes scissors Almost everyone has played the game rock-paper-scissors at some point. Two players face each other and, at the count of $3$, make a fist (rock), an extended hand, palm side down (paper), or a “V” with the index and middle fingers (scissors). The winner is determined by these rules: rock smashes scissors; paper covers rock; and scissors cut paper. If both players choose the same object, then the game is a tie. Suppose that Player $1$and Player $2$ are both equally likely to choose rock, paper, or scissors. a. Give a probability model for this chance process. b. Find the probability that Player $1$wins the game on the first throw .

Short answer

The probability of all outcomes: (rock, rock), (rock, paper), (rock, scissors), (paper, rock), (paper, paper), (paper, scissors), (scissors, rock), (scissors, paper), (scissors, scissors) The probability of each outcome is $\frac{1}{9}$.

Step-by-step solution

Part (a) - Step 1 : Given Information 

We are given two players that are playing rock-paper-scissors. We need to find the sample space and write the probability for each of this chance process.

Part(a) - Step 2 : Explanation

A player can choose any of three given options: rock, paper and scissors. Assume $(x,y)$ represent the outcomes of the game, where $x$represents the choice of player $1$and $y$represents the choice of player $2$. All possible outcomes of the game are then: (rock, rock), (rock, paper), (rock, scissors), (paper, rock), (paper, paper), (paper, scissors), (scissors, rock), (scissors, paper), (scissors, scissors) . There are $9$possible outcomes, while each of the outcome is equally likely to happen and thus the probability of each outcome is $\frac{1}{9}$.

Part(b)-Step 1 : Given Information

We have been given that two players are playing rock-paper-scissors. We need to find the sample space and write the probability of winning of Player $1$in first throw .

Part (b)-Step 2 : Explanation 

Since sample space comprises of three outcomes for one throw . Player$1$ can choose a single outcome to win the game . So, probability comes out to be$\frac{1}{3}$.