The thing about the change of "colour" in this analogy is you don't know in which direction it changes. So let's say you observe you "marble" through a "purple filter", which gives has:
- a 50% chance of being transparent to your marble (corresponding to a red-blue superposition marble collapsing to a purple marble)
- a 50% chance of being opaque to your marble (corresponding to red-blue superposition marble collapsing to a green marble).
The issue is that when you learn your marble is purple, while you know with 100% certainty the marble in australia is green, there is no way you can send information to Australia using that. This is because the other 50% of the time, your marble will be green, and the marble in Australia is purple.
So if I'm sitting in Australia, when I measure the marbles in my envelopes with purple filters, all I see is purple marbles 50% of the time and green marbles 50% of the time no matter what measurements you are performing at your end. So you can't send me messages by performing measurements at your end because you can't change the statistics of those measurements.
But you'll know the answer to every measurement I performed, if you've measured the other marble with a purple filter too.
The problem comes in when the angle between your two measurements is anything else. The chance that the measurements match is based on the cosine of that angle. There's no way for this to happen if the measurements are independent.
If you try to write two equations, where the first equation takes the secret particle state and first angle and gives you 1 or 0, and the second equation takes the secret particle state and second angle and gives you 1 or 0, you won't be able to reproduce the odds you get in the real world. Only equations that know both angles will work.
