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
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
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 LoseI 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.