Hat puzzle
You, and I, and your friends, are going to play a game:
- I put a black-or-white hat on each of you.
- Each of you gets to see everybody else's hat-colors, but not your own.
- Each of you writes down "white," "black," or "pass."
- Y'all win if [at least one person wrote down their own hat-color] and [nobody guessed the wrong hat-color].
You can strategize however you want before the game starts.
How often can y'all win? How?
Raw numeric answer for 1 person:
1/2
Raw numeric answer for 3 people:
3/4
Raw numeric answer for one specific larger number of people:
7 people:
7/8. Isn't it distressing how your probability of winning goes _up_ as the number of people does?
Raw numeric answer for one specific yet-larger number of people:
15 people:
15/16