Quote:
Micah said:
Failed searches will generally mean that there is a higher liklihood that searching other paths will pay off. For example, with a 50% site frequency the chance of having 0 sites in a province (disregarding terrain mods) is approx. 6%, so empty provinces are somewhat of an exception, meaning it's proportionally more likely that sites are hiding in the unsearched paths. Likewise, only 6% of provs will have 4 sites, so a province that already has 3 sites in it is unlikely to merit further searches. (there are 4 potential slots for sites in each province, and .5^4 is how I'm getting the percentages here, the .5 will change based on site frequency). The higher the site % the more this will hold true.

I don't think this is quite right. From what I remember about statistics the logic is like this. If you flip a coin 10 times and it comes up heads every time, what is the chance it'll come up heads the next time you toss it? Answer: 50% because the coin has no memory. Likewise, a province that has generated 3 magic sites has exactly the same chance of having a 4th as an unsearched province has of having at least one site if I understand how the sites are generated. You're right that a completely virgin province has only a 6% chance of having 4 sites, but a province that already has 3 sites revealed has a 50% chance of having a 4th with a 50% site frequency*.
That * is because there is another factor effecting your chances, which is again a bit counterintuitive. The more paths you've searched, the less likely you are to find more sites, regardless of how many you've found so far. As an example, consider a province that you've already searched in 7/8 paths. Assuming for a second that all paths had an equal frequency of sites, with a 50% site frequency the chance of there being at least one site there is (site frequency) * (chance of a given site being the path you haven't searched) or 0.5 / 8. Think about it this way, if you're casting aschric record you'll get the highest payout targeting provinces that have not been searched at all.
This is a little misleading though, as you usually don't care what your chances are that there are undiscovered sites, you care what your chances are of finding a site if you for example cast gnome lore. The thing is, from a statistical standpoint the paths are completely independent, if you search a province for astral it does not effect your chances of finding an earth site there regardless of whether you uncover an astral one. This is because what you really care about is "what is the chances a given province has at least one earth site?" which is not effected by the existence of astral sites  its 0.5 / 8. (disregarding the small chance that there are 4 nonearth sites, which complicates the math and doesn't change the chances too much)