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Let's say we now have two stacks of powders, one is $a$ the other is $b$. The ratio of amount of $a:b$ is $p:(1-p)$.

The two stacks of powders are poured together (without mixing). It forms two compartments $A$ and $B$. Now we draw a sample of $n$ particles. We are interested in the proportion of $a:b$ drawn in sample.

Let $X$ be the number of particles successfully drawn as $a$. The book said $Var(\frac{X}{n})=p(1-p)$ (variance in proportion).

This is reasonable if we are limited in one compartment. If $A$ is picked then all $n$ particles are $a$, and same applies for $B$. But what if we pick samples near the boundary of $A$ and $B$?

My argument is $$ \begin{align*} Pr(X>0) &=Pr(X>0|A)Pr(A)+Pr(X>0|B)Pr(B)+Pr(X>0|boundary)Pr(boundary) \\ &=1+0+Pr(X>0|boundary)Pr(boundary) \end{align*}$$ and $Pr(X=n)\le1$ which forces $Pr(X>0|boundary)Pr(boundary)=0$.

Why is this term zero in physical meaning?

p.s. in the original question, the powders would then be well-mixed and the new variance is compared with the above one.

user2513881
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    Given that "$A$" and "$B$" refer to *compartments*, which presumably are fixed during the drawing of particles, what could the probabilistic expressions "$pr(A)$" and "$pr(B)$" possibly mean? – whuber Oct 20 '15 at 14:24
  • Clarifications: $Pr(A)$ and $Pr(B)$ refers to the probability that samples are "drawn solely from" compartment $A$ and $B$ respectively. $Pr(boundary)$ is the probability that samples are drawn in an area containing both $A$ and $B$. – user2513881 Oct 20 '15 at 15:24
  • You appear to have in mind some physical model of drawing particles in which groups, rather than individual particles, are extracted. Such a model, although useful in this particular application, is an irrelevant distraction because the purpose of this physical example is to motivate a purely theoretical concept. – whuber Oct 20 '15 at 15:39
  • What is the theoretical concept? Can you kindly explain more? I am studying drug manufacturing and the book gives minimal explanations but many formulas. – user2513881 Oct 20 '15 at 16:21
  • The theoretical concepts that seem to be involved here are random sampling, expectation, and variance, with a focus on a Binomial model. For a different physical analogy that has been effectively used to motivate and explain these things, see http://stats.stackexchange.com/a/54894 *inter alia*. However, I'm purely guessing, because you haven't told us even what book you are reading, much less what it's actually about! – whuber Oct 20 '15 at 16:26
  • Please see https://books.google.com/books?id=LxdFnHQcXhgC&pg=PA28&dq=completely+segregated. (not the book I am using but the content is similar). I tried to traced back to the original paper quoted but in vain. – user2513881 Oct 21 '15 at 07:05
  • Ah, so the book really is about physical mixing processes! That's an interesting subject. In reading through this section, I notice references to "limiting" behavior and "ideal" mixtures. That might help you interpret what it says. Incidentally, the statistical material in this section appears to be quite old and outdated, especially the statements that suggest it's not possible to construct CIs when the sampling distributions are non-normal. – whuber Oct 21 '15 at 15:13
  • @whuber is there any good news in your reading? – user2513881 Oct 31 '15 at 07:32

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