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.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
SuperGlobalCalculator is independently built and maintained. See how we build and verify our calculators.