well if there are 3 white hats and 2 princes wear white hats... that means there is 1 possible white hat or two possible black hats on his head.
Just with probability there is a 33% his hat is white, and 66 % chance his hat is black. However...
After some time you realize that the other prince's are unable to deduce the color of their hat, or are unwilling to guess. What color is your hat?
While this would make sense even in our prince's world...Lets imagine it from the other prince's perspective.
3 white - 1 on one prince head = 2 white hats remaining 66% chance
2 black hats - 1 on one prince's head 33% chance
or...
3 white - 2 on two prince's heads = 1 hat remaining = 33% chance
2 black - 0 on princes head... = 2 hats remaining = 66% chance
Or lets go with this 3rd batch I neglected... until now
The king tells them that the first prince to deduce the color of his hat without removing it or looking at it will marry his daughter.
I could just
ask the other two princes' what the color of my hat was, and thus deduce my answer from there. He didnt say the prince's couldn't communicate with each other. Which because they were equally wanting to marry the princess. it is very likely they'd lie to you, so you would take the opposite of their reply. Or perhaps since the king is a man of his word. you ask him what color your hat is. =) yeah going with that... Ask the king to tell you your hat color and recite that back to him and take your prize.
Solely based on probability the answer would be black. However there is always still a chance it could be white. Which is why
asking the color of one's own hat would resolve the uncertainty