Suppose we have an urn with $2$ white and $2$ black balls then probability of drawing a black ball is $1/2$ .
In text books solutions to more difficult versions of this kind of ball problems are explained by naming black balls as $ b_1 b_2 ..$ and whites as $ w_1 w_2 .. $ etc..
My question:
These balls are identical, i.e blacks are identical in them and so does whites. Why do we name them as $b_1 b_2 … w_1 w_2…$ is it a correct way to deal with problem how can an identical ball be numbered and thought that way ? How do you know which one you picked ? This confuses me.
Best Answer
The labeling here is unimportant. The random variables for each color possess the property of being exchangeable, roughly meaning that permuting their labels will not change the result. If you permute $b_1$ and $b_2$, $w_1$ and $w_2$ (call $b_1$ by the label $b_2$ etc..., You will still end up with th same answer.
So basically, the labels here are only for mathematical convenience.