Each player on the hiding team picks a spot to hide and the seeking player picks one spot. If a player is hiding in the picked spot, he's out.
If a player is out he doesn't participate in later rounds.
If any players remain on the hiding team after 3 rounds, the hiding team wins.
Is this game 50/50? Working through the math, start with 1 hider:
- round 1, there's a 3/4 chance of not being found
- round 2, there's a 2/3 chance of not being found
- round 3, there's a 1/2 chance of not being found
Thus, for an individual player on the hiding team, there's a (3/4) * (2/3) * (1/2) chance of not being found. That's 1/4, or 25%.
Inverting that, the seeker has a 3/4, or 75% chance of finding a given player after 3 rounds.
Since there are 3 players and their hiding decisions are independent, the chance of the seeker finding all 3 players in 3 rounds is just (3/4)^3, or 27/64. That's only 42%. It isn't balanced at all.
Is there an obvious way to balance it? What if we did 4 rounds with 5 starting hiding spots. For one hider:
- round 1, there's a 4/5 chance of not being found
- round 2, there's a 3/4 chance of not being found
- round 3, there's a 2/3 chance of not being found
- round 4, there's a 1/2 chance of not being found
Multiplied out, that's a 1/5, or 20% chance of not being found so 4/5, or 80% success chance for the seeker. The chance that the seeker finds all 3 players in 4 rounds then is (4/5)^3, or 51%. 4 rounds is way more balanced than 3 rounds.
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