Yeah, there are 4 50% chances to have a site, which is why the average is 2.
A brief explanation of what order to search in:
Let us pretend that each province has 2 50% chances to have a site, of which half will be fire sites and half will be earth sites.
So, if we look at 160 provinces:
40 will have no sites.
40 will have one fire site.
40 will have one earth site.
20 will have one earth site and one fire site.
10 will have two fire sites.
10 will have two earth sites.
Now, we've done some earth searching, and we want to know: which provinces should we search for fire?
a) Sites which haven't been searched are expected to have: (40 + 20 + 10 * 2) / 160 = 0.5 fire sites each.
b) Sites which have been searched, and which contained two earth sites, have 0 fire sites each (guaranteed).
c) Sites which have been searched, and which contained one earth site are expected to have: 20 / (20 + 40) = 1/3 fire sites each.
d) Sites which ahve been searched, and which contained zero earth sites, are expected to have: (40 + 10 * 2) / (40 + 40 + 10) = 2/3 of a fire site each!
So category B is not worth searching at all.
Category D > Category A > Category B.
This is a particular case of the well-known "Monty Hall" paradox:
http://en.wikipedia.org/wiki/Monty_Hall_problem