[thejeff]

So what's the generic equation?

I can't quite get there. Even assuming equal chances and that paths are fully searched or not at all.

Having found one or more sites, just shifts the P(1+sites) over in the equation. From:
P(1)*1/8 + P(2)*15/64 + ...
to:
P(2)*1/8 +P(3)*15/64 + ...
for 1 known site.

Having searched a path increases the P(1+sites)?
From:
1/8, (1/8*7/8)+(7/8*1/8)+(1/8*1/8)
To:
1/7, (1/7*6/7)+(6/7*1/7)+(1/7*1/7)
for searching one path?