Rock Paper Scissors Probability Calculator

Rock Paper Scissors as a Game

RPS is a classic zero-sum, simultaneous-move game. With three pure strategies (Rock, Paper, Scissors) and the standard rules, no pure strategy dominates: each beats one option and loses to another. The game-theory solution is a mixed strategy equilibrium.

Nash Equilibrium

The unique Nash equilibrium of RPS is the uniform random mix:

(P_rock, P_paper, P_scissors) = (1/3, 1/3, 1/3)

If both players play this mix, neither can improve their expected outcome by changing their own probabilities, and the game’s expected value is exactly 0 (a draw on average).

Counter-Strategy When Opponent Is Predictable

If your opponent’s mix is known to be (p_R, p_P, p_S):

• You picking Rock wins with probability p_S, loses with p_P, draws with p_R
• You picking Paper wins with p_R, loses with p_S, draws with p_P
• You picking Scissors wins with p_P, loses with p_R, draws with p_S

The optimal pure response is to play the option that beats whichever has the highest probability. Your edge with any given move is the chance it wins minus the chance it loses, so the best move is whichever of these three is largest:

edge(Rock) = p_S − p_P edge(Paper) = p_R − p_S edge(Scissors) = p_P − p_R

You will often see this written as max(p) − min(p), and that shortcut is wrong. It happens to give the right answer when the highest and lowest probabilities sit in the correct cyclic order, and the wrong one otherwise. Against a mix of 50% Rock, 10% Paper, 40% Scissors, max − min says 40%; the real best edge is Rock at 0.40 − 0.10 = 30%. The three differences above are always correct because each one compares the move you beat against the move that beats you.

If the opponent plays a uniform third of each, all three differences are zero, which is what makes it an equilibrium.

Worked example: opponent plays 50% Rock, 30% Paper, 20% Scissors

• Pick Paper: wins on opponent’s Rock (0.50), loses on Scissors (0.20), draws on Paper (0.30)
• Net edge: 0.50 − 0.20 = +30% per round

By contrast:

• Pick Rock: 0.20 − 0.30 = −10% (bad choice)
• Pick Scissors: 0.30 − 0.50 = −20% (worst choice)

So Paper is the optimal pure counter, with an expected gain of 0.30 win-probability per round.

Real Human Play

Studies of human RPS players reveal systematic non-uniformity:

• Most players slightly over-pick Rock (~36–40%)
• Players who lose tend to switch
• Players who win tend to repeat
• Long sessions drift toward the equilibrium but never quite reach it

This is why deliberately exploiting the “win-stay, lose-shift” tendency can yield a small but consistent edge over casual opponents.

Variants

Caveats

In a single round, even the optimal strategy can lose. The expected-value framework only matters over many rounds. Against a true random opponent, your best long-run record is a perfectly even 33% W / 33% L / 33% D. Improving on that requires reading patterns, which is a psychology problem, not a math one.