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 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.

Ans. If each person chooses different symbols, then the total no. of possible outcome is = (3)^3=27

Total no. of possible outcomes when no one wins which means that no one will win when all the three symbols are same or each person will present a different symbol

When all the three symbols are same, no. of outcomes = 3

When all present a different symbol then no. of outcomes= 3! = 6 ways

Therefore, total outcomes in which no one wins= 3+6 =9

Probability that no one wins in one round = 9/27

= 1/3

For 4rounds, that probability that no one wins =** = 1/81

part a: 1.08

part b: 27.59