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bananabandana
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Homework Statement
We have a 1-D lattice [a line] of ##L## sites. Sites are occupied with probability ##p##. Find the probability that a given site is a member of a cluster of size ##s##. (A cluster is a set of adjacent occupied sites. The cluster size is the number of occupied sites in the cluster)
Homework Equations
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The Attempt at a Solution
For some site ##x##, I'd say:
$$ Pr( x \in s) = \begin{pmatrix} L \\ 1 \end{pmatrix} \ \frac{ s \times Pr( \text{Make cluster s}) }{L} = sPr(\text{Make Cluster s})$$
##Pr(\text{Make Cluster s})## is the probability that a cluster of size s exists. This is given by (supposing we are not near the ends of the lattice) :
$$Pr(\text{Make Cluster s}) = (1-p)^{2}p^{s}$$
Is this reasoning correct? My textbook also gets to the same answer, but simply states the result, so I am curious [being very rusty on anything to do with statistics] if I have actually done this correctly. (Apologies if this is the wrong forum - I'm aware it's elementary probability theory, but the rest of the text isn't)
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