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