math but also probability
posted by homeworkgenius .
Three people play a game of "nonconformity": They each choose rock, paper, or scissors. If two of the three people choose the same symbol, and the third person chooses a different symbol, then the one who chose the different symbol wins. Otherwise, no one wins.
If they play 4 rounds of this game, all choosing their symbols at random, what's the probability that nobody wins any of the 4 games? Express your answer as a common fraction.

math but also probability 
K
1/81

math but also probability 
Michael
First, we compute the total number of possible outcomes for each round. Each person can choose any of three different symbols, so the total number of possible outcomes is $3^3 = 27$.
Now, we compute the number of outcomes in which no one wins. No one wins if all three symbols are the same (3 ways, one for each symbol), or each person presents a different symbol ($3! = 6$ ways), so there are $3 + 6 = 9$ outcomes in which no one wins.
Therefore, the probability that no one wins a given round is $9/27 = 1/3$. This means the probability that no one wins any of four rounds is $(1/3)^4 = \boxed{1/81}$
Respond to this Question
Similar Questions

math
one evening, three people play three rounds of a betting game with the undrestanding that the person who comes in last each round must double the money of the other two. After three rounds, each person has been last just once and each … 
math
susan, joanne, and diane are triplets. susan has red, blue, green, and yellow sweaters. janne has green, red, purple, and white sweaters. diane's sweaters are red, blue, purple, and mauve. each girl has only one sweater of each color, … 
PROBABILITY QUESTION PLEASE HELP!!
Three people play a game of "nonconformity": They each choose rock, paper, or scissors. If two of the three people choose the same symbol, and the third person chooses a different symbol, then the one who chose the different symbol … 
Statistics/Probability
1. Two roommates, roommate A and roommate B, are about to go cruising with a mutual friend and are arguing over who gets to sit in the front seat. Roommate A suggests a game of rockpaperscissors to settle the dispute. Consider the … 
Statistics/Probability
2. Two roommates, roommate X and roommate Y, are expecting company and are arguing over who should have to wash the dishes before the company arrives. Roommate X suggests a game of rockpaperscissors to settle the dispute. Consider … 
Statistics/Probability
3. On a given evening, two roommates, roommate A and roommate B, are arguing over who should control the television remote. Roommate A suggests a game of rockpaperscissors to settle the dispute. It's the first time that the roommates … 
Math
A game is played by throwing darts at a target. A player can choose to throw two or three darts. Darcy plays two games. In Game 1, he chooses to throw two darts, and wins if he hits the target at least once. In Game 2, he chooses to … 
Math
A game is played by throwing darts at a target. A player can choose to throw two or three darts. Darcy plays two games. In Game 1, he chooses to throw two darts, and wins if he hits the target at least once. In Game 2, he chooses to … 
Math Probability
Mario, Yoshi, and Toadette play a game of "nonconformity": They each choose rock, paper, or scissors. If two of the three people choose the same symbol, and the third person chooses a different symbol, then the one who chose the different … 
Math
A game show has three doors labelled A,B, and C behind which there may or may not be a prize. A group of contestants are asked behind which doors they think there is a prize. The results are that 18 people choose A, 19 people choose …