Guess the color of the cap

Question

I have 3 persons which either wear a white or a black cap. They can only see the color of the other caps, but not their own. White and black caps are eqally likely. As a team, they play a game of guessing their own cap color. If they will win, all of them have to guess correctly their own cap color. Once the game begins, they cannot communicate the color of the other two caps.

Now my interesting question: What is a good strategy for the 3 persons such that with 75% probability all answer correctly? (if the strategy should be made before the caps are donned)

The hard thing is that the players cannot communicate anything with each other once the have the caps.

Answer 1

Six of the eight possible cases involve one cap being of the opposite colour to the other two. Therefore, the strategy with a 75% chance of success goes like this (numbering the persons 1, 2, 3):

• Person 1, then person 2, then person 3 in that order will see the other two person's caps. If they see two caps of the same colour, they will say that their own cap is of the opposite colour. Either one or three people will say this; if all three speak out, they all have caps of the same colour and so lose.
• If exactly one person spoke out, the other two people then say their cap is of the opposite colour to the person who spoke out first, winning the game for the group.

Answer 2

Hint: Thinking about the different possibilities of hat distribution, there is one thing which happens $75\%$ of the time. Find a strategy they can follow in that case, and ignore the remaining $25\%$ of cases.