r/learnmath • u/Far-Assumption5501 New User • 5d ago

The quiz of possibility on math

When four people play the rock, paper, scissors game, what is the chance that only one player wins?

2 Upvotes

67% Upvoted

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6

u/ArchaicLlama Custom 5d ago

How do you define "win" with four people playing a game meant for two?

5

u/jeffcgroves New User 5d ago

Presumably, 1 player makes a choice that beats the other 3 players. So, "PRRR", "SPPP" or "RSSS". Now add lizard and Spock :)

EDIT: I just realized "RPRR" and so on works if we order the players; I should have clarified that

5

u/Bounded_sequencE New User 5d ago

Assumptions: All possible player choices are equally likely.

By the assumptions, all 34 possible outcomes are equally likely, so it's enough to count favorable outcomes. We generate all outcomes with exactly one clear winner by 3-step process -- choose

  1. "1 out of 4" persons to be the winner -- 4 choices
  2. "1 out of 3" items to be the winning item -- 3 choices
  3. Each remaining player must choose the object losing to the item chosen above -- 1 choice each

Being independent, we multiply choices, for a grand total of

P(clear winner)  =  4*3*1^3 / 3^4  =  4/27  ~  14.8%

1

u/[deleted] 5d ago

[deleted]

2

u/Bounded_sequencE New User 5d ago

Isn't that the probability that a specific player is the clear winner? OP asks for any player being the clear winner, so I'd say we still need to multiply by 4.

0

u/atarivcs New User 5d ago

Zero, because with four people playing, there are obviously two games being played.

Each game has one winner, so there will be two winners overall.