A lot of people provided solutions on fb. I'm posting some of them. They are quite elaborate. Save me the time to write it up. :)
Black, the only way the last one can be sure of the hat on her head is if both the one's in front of her are white. So we can conclude that they are not. Thus, if the hat of the first one was white, the second one could safely have said hers was black, but since she doesn't do that, the first one can only be black.
Assumption: Each woman can hear the reaction of the others when they answer the question.
Now, Looking from the last woman's position. there is only one case when she can definitely tell which cap is on her head. The situation will be WWB ( from first to third). Ruling out this situation, now we get to the second woman, she knows that the third one behind her does not know the color of her cap. She also knows that had the color of the cap of the woman in front of her ( the first woman) been white, then there is only one case that the third woman would have been unaware of the color of her cap. Meaning, the second woman would have known the color in that case. But she does not too. This rejects the second case of WB(W/B). The last two cases in which both the second and the third woman do not know the colors of their respective caps, have the first woman wearing a black cap only. Thus hearing from the second woman that she also cannot determine the color of her cap tells the first woman for sure that her cap is Black.
There you go, BLACK is the answer.
Since there are only two white caps and three women, at-least one of them is wearing black. If the last women sees two white caps in front of her, she knows, she herself is wearing black, and declares.
If she is keeping quite, that means, the two women in front are either wearing both black or one white and one black. Now if the middle woman sees a black cap in front of her, she doesnt know if she herself is wearing black or white. If the middle woman sees a white cap in front of her, she knows she herself is wearing black.
Since second lady is keeping quite, the cap on the third lady is black and the third lady comes to know.