Traditionally, the game of Rock, Paper, Scissors is played with two people. Each of the two people, on the count of 3, reveals their hand in the shape of a rock, a piece of paper, or a pair of scissors as best they can (usually with a closed fist, a flat hand, or their index and middle fingers outstretched to form the V of an open pair of scissors). As the rules go, rock smashes scissors, paper covers rock, and scissors cuts paper. The game can be extended to an arbitrarily large number of, say, nn people, where nn is greater than or equal to 2. If when all nn people reveal their play and only 1 or all 3 options appear, that round is deemed indecisiveindecisive and they just play again. If when all nn people reveal their play precisely two of the three options appear, that round is decisivedecisive, and those with the winning play continue while those with the losing play are eliminated. Play continues until only one person remains. Assume the game begins with nn players, each of whom independently and randomly choose between the three options with equal probability, and let R, P, and S count the number of players who play rock, paper, and scissors, respectively. Compute the joint PMF of R, P, and S in terms of n, r, p, and s and present that as part of your written solution. Then, assume that n=12 and derive the probability that r, p, and s and are all equal. Compute this probability out to three significant decimal places.
Traditionally, the game of Rock, Paper, Scissors is played with two people. Each of the two people, on the count of 3, reveals their hand in the shape of a rock, a piece of paper, or a pair of scissors as best they can (usually with a closed fist, a flat hand, or their index and middle fingers outstretched to form the V of an open pair of scissors). As the rules go, rock smashes scissors, paper covers rock, and scissors cuts paper.
The game can be extended to an arbitrarily large number of, say, nn people, where nn is greater than or equal to 2. If when all nn people reveal their play and only 1 or all 3 options appear, that round is deemed indecisiveindecisive and they just play again. If when all nn people reveal their play precisely two of the three options appear, that round is decisivedecisive, and those with the winning play continue while those with the losing play are eliminated. Play continues until only one person remains.
Assume the game begins with nn players, each of whom independently and randomly choose between the three options with equal probability, and let R, P, and S count the number of players who play rock, paper, and scissors, respectively. Compute the joint PMF of R, P, and S in terms of n, r, p, and s and present that as part of your written solution. Then, assume that n=12 and derive the probability that r, p, and s and are all equal. Compute this probability out to three significant decimal places.
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