Percolation Problem Homework: Probability of Cluster Size s

[bananabandana]

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)

 

[mfb]

The probability that a cluster of size s exists in the range depends on L. I guess your Pr(Make Cluster s) is the probability to have a cluster at a given place. In that case the derivation works, as the events are all mutual exclusive you can add the probabilities. This assumes that you are at least s steps away from the border.

 

[Ray Vickson]

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

[/B]

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)

The probability that site $x$ is the first member of an $s$-cluster is $(1-p)^2 p^s$, because the site immediately before $x$ must be unoccupied, then sites $x, x+1, \ldots, x+s$ must be occupied and finally site $x+s+1$ must be unoccupied. You get the same probability if site $x$ is the second or third or fourth or ... or last site in the $s$-cluster. So, indeed, the probability you want is $s (1-p)^2 p^s$.