[area51]

This doesn't make sense to me.

See below chart -- The probability that my hat is M/O is the same (50%) regardless of what the others are wearing.

Others Me

MM M

MM O

MO M

MO O

OM M

OM O

OO M

OO O

[bazzargh]

Yes, and when each player says a colour, they're still only right 50% of the time. But the strategy means the cases where you'd be wrong are the same cases that others would also vote incorrectly, but the cases where you're right, the others pass, so you win.

    MMM O    O     O     Lose
    MMO Pass Pass  O     Win
    MOM Pass O     Pass  Win
    MOO M    Pass  Pass  Win
    OMM O    Pass  Pass  Win
    OMO Pass M     Pass  Win
    OOM Pass Pass  M     Win
    OOO M    M     M     Lose

[joosters]

That's a great way of explaining it, thanks!

I had read the solution but it still didn't make sense in my head how seeing the colour of unrelated hats could somehow improve the chance of you predicting your own hat colour. But as you explain, no one individual guess is better than 50%, it's just that the wrong guesses get lumped together.