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Note: This question is a follow-up of Estimating population size of a subgroup based on independent samples without replacement

Let a bag have 1000 balls of arbitrary colors and unknowns sizes ($r$). Suppose we also known the total volume occupied by the balls ($t_v$). We want to estimate the total volume occupied by red balls ($t_{v}^{red}$).

We attempt to do this by taking $x$ independent samples (without replacement within each sample) of size $s$ of the original population. Let $s$ be 10% of the original population as an example.

Proceed to count the number of red balls in each sample, e.g., for $x = 10$:

(9, 10, 10, 11, 13, 8, 5, 15, 12, 9)

To estimate the number of red balls in the original population, one can naïvely calculate the average of the above list and divide by the sample. In this case:

(9 + 10 + 10 + 11 + 13 + 8 + 5 + 15 + 12 + 9) / 10 / 0.1 = 102

What about $t_{v}^{red}$? Each sample will give us a set of sizes for the observed red balls. So:

sample1 = [3, 2, 1, 10, 2, 8, 1, 4, 1, 1] => 33 => V = 5098
sample2 = [3, 6, 1, 3, 4, 1, 1, 1, 8, 1] => 29 => V = 2673
sample3 = [1, 4, 3, 10, 1, 1, 5, 1, 1, 2] => 29 => V = 3861
sample4 = [4, 1, 1, 1, 2, 6, 2, 6, 1, 1] => 25 => V = 1624
sample5 = [1, 6, 1, 2, 4, 3, 7, 10, 10, 2] => 46 => V = 8381
sample6 = [6, 6, 1, 2, 1, 10, 3, 2, 3, 1] => 35 => V = 4728
sample7 = [1, 4, 2, 1, 4, 1, 2, 4, 10, 1] => 30 => V = 3807
sample8 = [1, 8, 3, 3, 4, 3, 1, 8, 1, 5] => 37 => V = 4074
sample9 = [7, 9, 2, 4, 4, 6, 6, 1, 8, 1] => 48 => V = 6766
sample10 = [3, 8, 1, 2, 10, 7, 3, 7, 3, 1] => 45 => V = 7191

As you can see, the estimated volume on each sample can vary considerably (from 1624 to 8381). And here lies my problem.

My questions are thus:

  1. How to estimate the total volume occupied by balls of a certain color?
  2. We could hypothetically assume that the ball size follows a known distribution (in this case, zipfian). Would this help?
  3. Instead of a number, how can I obtain a probability distribution of the estimated population size and volume?

Some clarifications over the original problem:

  1. The population size is known à priori.
  2. The population volume is also known à priori.
  3. But the number of colors is unknown. Or the total volume of each color. Or both ratios.
  4. Each sample take balls without replacement.
  5. Samples are independent (after a sample is taken, all balls are replaced).
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