Rock-Paper-Scissors Odds: Why Ties Keep Happening, and How to Decide in Big Groups

The chance of a tie by group size

The calculation assumes everyone throws rock, paper or scissors at random with equal probability. A round only produces a result when exactly two of the three hands appear. If everyone throws the same hand it's a tie, and if all three hands appear it's also a tie.

Two things stand out. First, the odds are the same for 2 and 3 players; the feeling that three is harder to settle is actually a misconception. Second, things get worse fast beyond four players: with 6 you redo it almost 4 times on average, and with 10 it takes about 19 rounds on average. When ten people can't finish rock-paper-scissors, it has nothing to do with how they're throwing.

How likely is a single winner in one round?

When you want to choose one representative or one organiser, what you really want to know is the chance of ending up with exactly one person in a single round. That number is even harsher than the tie rate.

With 6 people trying to pick one, a single round settles it only about once in 40 tries. With 10, it's once in 2,000. Rock-paper-scissors is a superbly designed tool for two people, but it quickly stops working as the group grows. A practical rule is to switch to another method once you have more than five people.

Can you improve your odds at rock-paper-scissors?

If everyone is perfectly random, your chance of winning is the same whatever you throw. People, however, are slightly less than random. These tendencies are often pointed out:

• Rock is said to be the most common first throw. The usual explanation is that your hand starts as a fist, so it's easy to keep it that way.
• People tend to avoid throwing the same hand three times in a row, and someone who has won twice in a row tends to switch.
• Right after losing, people sometimes throw the hand that just beat them.
• When a game starts with a fixed chant (in Japan, “saisho wa guu”, starting from rock), the rhythm can bias the next throw.

All of these give at most a small edge when your opponent has habits; none of them is a sure way to win. If the other person deliberately plays at random, the tendencies disappear. More importantly, the very existence of these biases means rock-paper-scissors can't be called perfectly fair. Using it for decisions where fairness matters, like who pays or who does a chore, favours whoever is better at reading habits.

Other ways to decide when there are many people

Once you have more than five people, switching to a method that produces a result in one go is faster than repeating rock-paper-scissors over and over, and it's easier to show it was fair. The right method depends on what you're deciding.

1. 1

   Say first what you're deciding

   Whether you're picking one person or deciding everyone's order changes the tool. If the purpose changes after you start, you'll end up redoing it and leaving distrust behind.

2. 2

   Put the candidates where everyone can see them

   List the participants or options and check together that nothing is missing or duplicated. Skip this and, after the result, someone will say their name wasn't in it.

3. 3

   Run it once, on the spot

   The key is to leave no room for a redraw. Announce “one go decides it” before running it.

4. 4

   Keep the result as it is

   Share a screenshot or write the result down so it can be checked later. If it's only said out loud, memories will differ.

When assigning a heavy role, some dissatisfaction remains even if the method was fair. What makes the difference is whether everyone agreed beforehand to leave it to a draw. Treat the fairness of the method and how people feel about the result as separate issues. How to guarantee fairness is covered in how to run a fair draw, and how dice rolls come out in dice probability. For working out how many draws it takes to win in lotteries and gacha, see how to calculate gacha odds.

FAQ

What is the chance of a tie in rock-paper-scissors?

About 33.3% for both 2 and 3 players. It rises to about 48.1% with 4, about 63.0% with 5 and about 94.8% with 10. A round only produces a result when exactly two hands appear, so the more players there are, the more likely all three hands show up and the tie continues. With 10 people it takes about 19 rounds on average to get a result.

Is 3-player rock-paper-scissors harder to settle than 2-player?

The chance of a tie is about 33.3% for both; it's actually the same. It feels harder because with three players, a round with a result can leave two winners who then have to play again. The chance that exactly one of three wins is about 33.3%, so picking one person takes more rounds than with two players.

Is there a way to win rock-paper-scissors more often?

If your opponent plays completely at random, your chance of winning is the same whatever you throw. People are said to have habits, such as throwing rock first or avoiding the same hand three times, but these biases vanish if the opponent pays attention. Put the other way, once it becomes a game of reading habits it's no longer perfectly fair, so a draw or other method suits decisions about chores or money better.

What should we use to pick one person from a big group?

Once you have more than five people, switching to amidakuji (ghost leg) or a wheel is the quickest route. Use a wheel to pick just one person, and amidakuji to decide everyone's order at once. Check the candidate list together before starting, announce that one go decides it, then run it and record the result. Following this order means nobody asks for a redraw afterwards.