What is the probability of having a stalemate with 5 players in paper sissors rock?
Obviously 1/1
Obviously 1/3 (1/1 * 1/3)
Now we get two possible stalemate scenarios.
- All players get the same value:
1/1 * 1/3 * 1/3 = 1/9
- All players get a different value:
1/1 * 2/3 * 1/3 = 2/9
Add the two probabilities together and get: 1/9 + 2/9 = 1/3
This is where its gets interesting (or difficult).
If we try the two possible stalemate scenarios:
- All same value:
1/1 * 1/3 * 1/3 * 1/3 = 1/27
or ifn = number of players
then(1/3)^n-1
- All values chosen:
1/1 * 2/3 * (at least one of the remaining players chooses the third value)
hmm Ok, it is easier to calculate that none of the players get a certain value (call thatpNone
) thenpAtLeastOne = 1 - pNone
Try again:1/1 * 2/3 * (1 - 2/3 * 2/3) = ~0.37
or ifn = number of players
then2/3 * (1 - (2/3)^n-2)
This does not actually work, it does not approach 1 as n increases because of the second scenario (2/3 * (1 - (2/3)^n-2)
has that 2/3 * ..
at the start
I was on a boat and a unified rule came to me: It does not matter what the first two players choose, each player after that has a 1/3 chance of picking a value that causes a stalemate.
So using the pAtLeastOne = 1 - pNone
rule: 1 - (2/3)^n-2
Plug in 4: 1 - (2/3)^4-2 = 0.56
use the rule found previously (1 - (2/3)^n-2
): 1 - (2/3)^5-2 = 0.70
unfortunately brute forcing the result (see below) give about 63% stalemates for 5 players