I have a machine that separates lots of sand (called feed) very quickly into two groups: group A (which is larger particles) and group B (which are smaller particles). Because it's so quick and the volume is high, the machine isn't 100% perfect. Thus, we take sample proportions for group A and group B. Given these sample proportions, is there a specific test I should use to determine the overall proportion of the original feed?
Here's an example. Let's say that we want the frequency distribution of particles ranging in size from 0 micro-units up to 100 micro-units. Let's break them up into four ranges: 1-25 micro-units, 26-50 micro-units, 51-75 micro-units, 76-100 micro-units.
Now the proportions for group A are left-skewed (because the particles are larger). Let's say they are 5% 1-25, 10% 26-50, 35% 51-75 and 50% 76-100.
The proportions for group B are right-skewed (because the particles are smaller). Let's say they are 45% 1-25, 25% 26-50, 15% 51-75 and 15% 76-100.
How can I use these two sample distributions to determine the proportion of the original feed?
My first thought is to simply combine the two samples and take a proportion of them both together. If each sample was, say, 10g, then we could combine them to get a 20g sample and our new distributions would be:
25% of the 1-25,
17.5% of the 26-50,
25% of the 51-75,
32.5% of the 76-100
For some reason, this approach seems naive to me. Is this really the only (or best) way to calculate the proportions of the original feed?
Corollary: What if I know the proportions of the feed and one of the output groups? For example, let's say I have the feed proportions and the proportions for group A. Could I use this to calculate the proportions for group B? (I assume I'd use the same process I used above...)