This problem is of my own construction, which I have constructed to understand Symmetry better.
Say that we have three bags, each having blue and red marbles.
In the bag 1, $1 \over 2$ of the marbles are blue; in bag 2, $1 \over 3$ of the marbles are blue; in bag 3, $1 \over 4$ of the marbles are blue.
Say our experiment consists of first randomly choosing a bag, grabbing a marble from it, and then choosing another bag, and grabbing a marble from this new bag. When considering the probability of the first marble we pick being blue, I am trying to understand why this is the same as the probability of the second marble we pick being blue, by a symmetrical argument (not just a brute force calculation of the probabilities to show they are equal).
So far, I have tried to show that every sample point in the event of the first marble being blue corresponds to a sample point of equal probability in the event of the second marble being blue – and I show such correspondence by switching the first marble we have grabbed with the second. And then the bijection is straightforward to prove.
However, I am struggling with arguing that when we swap which marble we have grabbed in the first and second spots, that this necessarily has the same probability of us sampling it – although I certainly feel intuitively that this is so, I can't explain why it is so at all.
So why is there Symmetry between "the first marble being blue" and "the second marble being blue" here?
EDIT: Is it valid to justify why the experiments of picking from the first bag and picking from the second bag after it are the same by saying that because we know nothing of the first bag and marble drawn before we drew the second, we act as if it never happened when drawing from the second bag? As if we say that the second experiment was different than the first, then given this we could possibly say what more likely happened on the first picking of the bag, in such a way that would contradict the probabilities we have for the first picking of the bag and marble?
Best Answer
The Easiest & Most Intuitive way to see the Symmetry is to think about this Experiment :
Choose 2 Bags , then Choose 2 Balls.
What is the Probability that both Balls are blue ?
Let the Bags be named $1,2,3$.
We have no knowledge of the contents , hence our names will be $A,B,C$.
There are 2 Sources to see the Symmetry here :
Lack of knowledge.
Order is irrelevant.
In our Experiment , we can choose $(A,B)$ , $(B,A)$ , $(B,C)$ , $(C,B)$ , $(C,A)$ , $(A,C)$ , though in our view , those are all the Same Choices for us , due to our lack of knowledge.
We can now see that $A$ occurs first 3 times , $B$ occurs first 3 times , $C$ occurs first 3 times . . .
We can then check that $A$ occurs second 3 times , $B$ occurs second 3 times , $C$ occurs second 3 times . . .
It is totally Symmetric !!
That is the Source of Symmetry here !!
Naturally , no matter what calculations we try , we must have "Blue Ball first" == "Blue Ball second" , in terms of Probability.