What color is my hat? 




There are three friends, and each will 
have a hat put on her head.  The hat color is blue 
or red and it is determined separately for each 
of the three by flipping a fair coin. Each friend 
can see the colors of the other hats, but not 
the color of her own hat.  

After the hats are on, simultaneously each friend 
is allowed to guess the color of her own hat, or 
remain silent. The friends win a million dollars 
if at least one makes the correct guess and none 
make the incorrect guess. The friends are allowed 
to collaborate on a strategy before the game begins. 

Can you come up with a strategy which lets the friends 
win more than half the time? 

You can justify your answer with probability computations.  
The sample space for the assignment of hats can be 
thought of as consisting of the eight 
elements BBB,BBR, ... , RRR with each element having 
probability 1/8. You can use this to compute the 
probability of winning with a given strategy -- for 
example if the strategy gives a win in 4 of the 8 cases, 
then the probability of winning is 4/8 = 1/2.