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Let's say I have three bins of red and blue balls. Each bin has exactly 1000 balls in it, but the problem is that I don't know exactly how many balls there are of each color in the three bins. I only have time to sample 100 balls from each bin, and I want to know how many red and blue balls there are in total across all the bins.

So what I do is I take 100 balls from bin A, find the percentage of red and blue balls there (let's say I count 20 red and 80 blue). I do the same for bin b (30 red, 70 blue), and bin C (40 red, 60 blue). From this I infer that each bin contains (to within some margin of error):

Bin A: 200 red, 800 blue

Bin B: 300 red, 700 blue

Bin C: 400 red, 600 blue

And then from that I infer across all three bins: 900 red, 2100 blue.

My question is: is this a reasonable way to proceed? Is there a better way to go? Is there something wrong with this line of inference? What would you do?

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  • $\begingroup$ In order for this question to be answerable, you need to specify some way to quantify the accuracy of the estimate. For instance, suppose there are 800 red and 2200 blue balls total. If you estimate 900 red and I estimate 700 red, who is closer to the truth and by how much? Let the answer seem obvious, note that the true ratio of red to blue is $800/2200\approx 0.3636$, your guess is $900/2100\approx 0.42857$, and my guess is $700/2300\approx 0.30435$. My guess is within $0.0593$ of the true ratio while yours is only within $0.0649$, so it would seem my guess is better! $\endgroup$ – whuber Apr 2 '16 at 21:52
  • $\begingroup$ so is there a way to quantify the accuracy of the estimate? imagine yourself in the situation - for some mysterious reason, you need to know what the total number of balls of each color is, and you only have time to sample 100 balls from each bin - what would you do? $\endgroup$ – Mungy Apr 2 '16 at 22:13
  • $\begingroup$ There are myriad ways to quantify the accuracy. Please tell us how you intend it to be quantified. $\endgroup$ – whuber Apr 2 '16 at 22:14
  • $\begingroup$ I'd love to but I'm not that knowledgeable. That's why I'm seeking the wisdom of others. If you don't have the time or patience, that's ok - I appreciate the assistance you've given so far. $\endgroup$ – Mungy Apr 2 '16 at 22:27
  • $\begingroup$ This sounds very much like a [Polya Urn ](en.m.wikipedia.org/wiki/P%C3%B3lya_urn_model)problem. These notes give an overview of several ways the problem can be thought about. $\endgroup$ – Daniel Johnson Apr 3 '16 at 0:30

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