# Why is there Symmetry (Equality) between "the first marble being blue" and "the second marble being blue" here?

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?

• There are 6 equiprobable orders for the bags: 12, 21, 13, 31, 23, 32. With e.g. 23 I mean: the first chosen bag is bag2 and the second chosen bag is bag3. That confirms that for both the probability to become bag $i$ equals 1/3 for $i=1,2,3$. Commented Nov 3, 2023 at 10:04
• Here is another (equivalent) way to carry out your experiment. (1) Pick a bag at random and throw it away; thus making an unordered selection of two bags. (2) Reach into both bags and draw a marble from each; observe the colors. (3) Flip a coin to decide which bag to call the "first bag" and which the "second bag".
– bof
Commented Nov 5, 2023 at 4:14

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.

In both cases you follow exactly the same recipe: a bag is randomly chosen and from that bag a marble is randomly chosen.

The fact that in the second case on forehand another bag was chosen (and also a marble from that bag) does not make the second experiment an experiment that differs essentially from the first and consequently their probabilities on a success (blue marble) are the same.

The randomness of the second chosen bag is not affected by it and the proportion of blue marbles within the chosen bag is not affected by it.

In both cases the probability on a blue marble is simply:$$\frac13\left(\frac12+\frac13+\frac14\right)$$

If experiments are recognized directly as "the same" (which often is done indirectly by means of symmetry) then symmetry has no added value.

• I'm having trouble seeing why choosing the second bag is essentially the same experiment- after we have chosen our first bag, we have now a constrained set of choices for our second bag. How would you explain how this is irrelevant on a conceptual level? Commented Nov 3, 2023 at 8:33
• The probabilty that the second bag will be bag $i$ equals 1/3 for $i=1,2,3$ (just as it is also for the first bag). Things would be essentially different if we would have any information about the first bag (e.g.: bag3 was chosen as first bag) but we don't have such info. Commented Nov 3, 2023 at 8:39
• Thank you; do you have a way of completing the argument I outlined where for each sample point in the event of the first marble being blue, this is showed to correspond to a unique sample point in the event of the second marbles being blue by swapping which marble was the second and first to be picked? I am having trouble justifying why the 2 sample points we associate have the same probability. Commented Nov 3, 2023 at 8:57
• Maybe, but I don't promise. The fact that the problem at least in my view is solved on a basic way makes me reluctant to investigate more complex efforts to solve it. Commented Nov 3, 2023 at 9:01
• Is it valid to justify why the experiments 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? As if we say that the second experiment was different than the first, then 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? Commented Nov 3, 2023 at 9:13

Clearly, a proportion is present $$\frac{a}{b} = \frac{c}{d}$$. is this symmetry? A ratio (probabilities are ratios?) is being maintained when we go from the "whole" to a "part"? Would you call this fractalish self-similarity? A reflection(?) + dilation?