You're almost there, Gollard.

We have the three differently coloured pancakes. You see one side of one pancake, and it has colour X (let's leave in the middle whether this is brown or gold). As you say, there is a 2/3 probability that the other side of that pancake is also colour X; two of the three pancakes have the same colour on both sides, after all.

Let's make that a statement:
If I pick a pancake at random, look at one of its sides and see the colour X, then there is a two third possibility that the other side has the same colour.

Now, if we substitute "brown" for X - just substituting, nothing else - then why should the probability change? And likewise for "gold"?

You can test the thing with the help of a pack of cards and a friend. Take three red and three black cards out of the pack. Ask your friend to make three stacks of them; one with two red cards, one with two black cards and one with one red card and one black one. Then draw one card out of one stack at random and look at its colour. Then check the other card out of that stack and see whether the colour is different or equal. Repeat this a large number of times and count how often the colour is the same.

Whymme