My gut-reaction to this was 1/2, and after reading everyone's answers I had a heck of a time wrapping my head around the correct answer. Here's the really simple, non-technical way I finally convinced myself:
Let's simplify the original experiment and imagine there are only two cards in the hat: a W-W card and a W-R card. Now let's reach in and draw one out, and imagine we see a white face. What are the chances that the other face is also white? Well, you night say, there are only two possible outcomes: I drew the all white card or I drew the white and red card. The answer must be 50%.
In fact, there are three possible outcomes: (1) You may have drawn red and white card, (2) you may have drawn the all white card, or (3) you may have drawn the all white card. No, that's not a typo. Yes, outcomes (2) and (3) are the same. Or are they? The all white card may be drawn in two different ways: either one white side up, or the other white side up. The unfortunate fact that both ways look the same is inconsequential.
In other words, looking at my white card, the possible outcomes are:
- I draw W, the flip side is R.
- I draw W1, the flip side is W2.
- I draw W2, the flip side is W1.
I don't know whether I'm seeing W, W1 or W2, but the chance the flip side is also white (that is, either W2 or W1) is 2/3.