Let's get back a few steps:>>B/B B/G G/G
>>2/3 1/3 0/3 if we see a brown side
>>0/3 1/3 2/3 if we see a golden side
>>-----------
>>2/6 2/6 2/6 when we add the two situations up and then
>>divide by 2.
>>
>>I think that this adding up is valid.
>
>Why?
Well, because of the following:
Let's take the following statements as correct:
1. The probability of having chosen a certain pancake is:
B/B B/G G/G
2/3 1/3 0/3 if we see a brown side
2. The probability of having chosen a certain pancake is:
B/B B/G G/G
0/3 1/3 2/3 if we see a golden side
3. If we randomly grab a pancake and check one of its sides, there is a 50% chance of seeing a golden side and 50% chance of seeing a brown side.
If we assume the above, if we grab just any pancake and show one side of it, the chance that we've taken the B/B pancake is (2/3)*50% + (0/3)*50%. Together this is a chance of 1/3.
The chance that we've taken the G/B pancake is (1/3)*50% + (1/3)*50%. Together this is a chance of 1/3.
The chance that we've taken the B/B pancake is (0/3)*50% + (2/3)*50%. Together this is a chance of 1/3.
Herewith the computation I've shown in a previous post is proven. I probably shouldn't have called it "addition"; that was a bit too simplistic.
We can work from the other side as well. If we start with the following assumptions:
1. There is a 50% chance of seeing a golden side and 50% chance of seeing a brown side.
2. We choose a pancake randomly, with an equal chance of each pancake to be chosen.
There is thus a 2/3 chance that we choose a monochromatic pancake.
Furthermore, the chance that we choose a certain pancake is X(brown)*50% + X(gold)*50%,
where X(colour) is the chance that a certain pancake is taken, and that colour is shown.
We have stated that this chance is 1/3 for each pancake.
In the case of the B/B pancake, the total chance of having grabbed that one is 1/3. We can see that for this pancake, the chance X(gold) is zero. So X(brown)*50% + 0 = 1/3
Therefore, it follows that X(brown) = 2/3
We can make the same computations for the other pancakes. This will produce the scheme above the line in the "addition" above.
And this is why.
Whymme