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Re: The probability of getting zero counts
Posted:
Oct 15, 2010 3:14 AM
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On Oct 14, 10:04 pm, Richard Wright <richwrigREM...@tig.com.au> wrote:
> Consider a series of squares randomly placed over a research area.
>
> The number of stones within each square is counted.
>
> Ten out of 12 squares contain stones, in varying numbers.
>
> Two of the 12 squares contain zero stones.
>
> Person A argues that the fact there are two squares with zero
> stones shows that the distribution is especially patchy in those
> spots. Person B, by contrast, argues that one must expect to get
> some zero counts, given the variability in numbers seen in the
> squares that have stones.
>
> How might one choose between these two claims?
> It seems to me that we must ask the following question.
>
> Given the variability of numbers in the 10 squares that have stones,
> what is the probability of obtaining two squares with zero stones
> if we assume the variability is uniform over the area sampled?
>
> To my mind this question relates to confidence intervals,
> but I can't see how to harness them.
>
> I would welcome any suggestions about how one might approach this
> question of statistical inference.
It depends on how big the squares are relative to the research area,
and how many stones there are in the research area. If we ignore
physical considerations such as the maximum number of stones that
a square can have, here is an analogous problem:
M stones are randomly and independently distributed into N urns.
n of the N urns are then randomly chosen for inspection.
What is the probability that k of the n urns have no stones?
The answer is the sum, m = 0..M, of the product of two probabilities:
Pr(the n urns contain m stones) *
Pr(k of the n urns have no stones | the n urns contain m stones).
Can you take it from there?
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