Fun with Probabilities

This post by Stephen Landsburg on his Big Questions blog has got me thinking about probabilites.

I'd like to post my own (well, plagarised from here) problem to readers:

Three poker chips are in a cup. One is marked with a BLUE dot on each side, another with a RED dot on each side, and the third has a BLUE dot on one side and a RED dot on the other. So there is one blue/blue chip, one red/red chip, and one blue/red chip.

Without looking, you take out one chip, and lay it on the table.

1. Suppose the up-side turns out to be BLUE? What is the chance that the down-side will also be BLUE?

2. What if the up-side is RED? What is the chance that the down-side will also be RED?

3. Before you see how the chip has fallen, what is the chance that it has the same color dot on both sides?

4. Suppose you answered 1/2 in response to Questions 1 & 2. That would mean that whichever the up color of the chip, the chance is 50/50 that the color on the down side is the same. But if at Question 3 you said that chance is 2/3, aren't you contradicting yourself?

If you follow the link to the paper the question comes from you'll get an explanation of the correct answer.  This is essentially a rephrasing of the more familiar Monty Hall problem.

UPDATE: The original link to this problems source no longer appears to be active.  You can try this one instead.