Problem Solving

Jim and Renee will play one game of Rock, Paper, Scissors. In this game, each will select and show a hand sign for one of the three items. Rock beats Scissors, Scissors beat Paper, and Paper beats Rock. Assuming that both Jim and Renee have an equal chance of choosing any one of the hand signs, what is the probability that Jim will win?
5/6
2/3
1/2
5/12
1/3















OA = 1/3; but, why is it not 1/2?? hasn't got anyone the same chance to win? Isn't it a fair bet?? (Source is Manhatten Prep)

Your forgetting that they can tie.

Rock vs. Rock (draw)
Rock vs. Paper (Lose)
Rock vs. Scissors (win)

Paper vs. Rock (Win)
Paper vs. Paper (Tie)
Paper vs. Scissors (Lose)

Scissors vs. Rock (Lose)
Scissors vs. Paper (Win)
Scissors vs. Scissors (Tie)

3/9 chance of winning = 1/3