>>I think that this adding up is valid.
>
>Why?

You take one pancake and look at one side of it. You can have two situations: either that side is brown or it is gold. Those situations are mutually exclusive and together they add up to 100%. I thought that in those situations those chances can be added up. I may be mistaken, however, it's been years since I studied maths or statistics other than just as a hobby.

>There's indeed an a priori 2/3 chance of drawing a
>monochromatic pancake simply because there are two of them
>among three. Your addition does not help to comprehend this
>simple fact.

The addition was how I got to the idea of getting the general colour X, for which we can then substitute brown or gold. IMHO my addition shows how the general situation of drawing colour X can be split up in the specific situations of either drawing brown or gold.

The idea of this general situation is also in a way a validation of the method of the repeated experiment, with the three stacks of cards, that I described.

>
>Let's follow Bob's idea of having say three colors and all
>possible combinations by two. There are 9 in all, of which 3
>are monochromatic. Then the a priori probability of having a
>monochromatic pancake is 3/9. But, once an A is drawn, the
>probability that it's a side of AA is 1/2, because now you
>draw from AA, AA, AB, and AC.

Hehe. There is a mistake in your example here. Do you want to find it yourself or do you want me to tell you? I wrote a hint at the bottom of my post, but I guess that you can find out the mistake without it.
If you correct the mistake, you'll find that my addition still works :)

>
>>Oh, Alex, you do have to explain this.
>
>I've no problem with your stack experiment. It should lead
>to a correct estimate.

You did call it "exactly the case when a wrong argument leads to the
right answer", though. Or did I misread your meaning?

Whymme

PS: Alex, please don't take this post the wrong way. As you may have gathered, I love this site and I respect your knowledge of mathematics and your drive in making this all accessible for the general internet public. It's just that in this case I don't agree with your reasoning, but that doesn't diminish my respect in any way.
I did not mention what was wrong (IMHO) with your example of three colours because I have the idea that you like to figure it out for yourself and I didn't want to deny you that pleasure. If I'm wrong here, I'd like to apologise.
You might want to delete this PS; it is meant as a personal message anyway and not as something that is needed to be communicated to anyone who reads these discussions.

Hint for the three colour pancake mistake: look at the situation with two colours: do we have three pancakes B/b, G/b, G/g or do we have four pancakes B/b, G/b, B/g and G/g?