Rock Paper Scissors Probability Calculator
Calculate win, loss, and draw probabilities in Rock Paper Scissors against an opponent with any chosen mix.
Find the optimal Nash counter-strategy.
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 — 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. Net win probability for the best pure response is:
P_win = max(p_R, p_P, p_S) − min(p_R, p_P, p_S)
If the opponent plays uniform 1/3, this difference is zero — confirming the 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
| Variant | Strategies | Equilibrium |
|---|---|---|
| Standard RPS | 3 | (1/3, 1/3, 1/3) |
| RPS-Lizard-Spock | 5 | (1/5, 1/5, 1/5, 1/5, 1/5) |
| Weighted (e.g. tax win) | 3 | Adjusted by payoffs |
| Sequential RPS | 3 | First mover indifferent (still uniform) |
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 above that requires reading patterns, which is a psychology problem, not a math one.