>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't do that. Say, you are picking a fruit out of a bag that contains a peach and a pear. The probability of getting a fruit is 1. The probability of getting a peach is 1/2 as is the probability of getting a pear.

>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.

This is exactly the case when a wrong argument leads to the right answer. For a multitude of other examples see Mathematical Fallacies, Flaws and Flimflam by Edward Barbeau.