[VedalkenBear]

Ich: Actually, there isn't a 20% chance for each scenario. There is a 0.4 probability of a site being placed, with a maximum of 4 sites being able to be placed. This is a pure binomial distribution. Even in the case of 0.5 probability, there isn't a flat chance.

I agree that they are determined when the game starts. However, we cannot look at a province and tell how many sites are in it. Therefore, we must take into account the various probabilities regarding the number of sites.

Once that is done, you have the scenario I described in my initial post. To go over it again:

If there are 0 sites in a province, and you search for an Earth site, the probability that you will find an Earth site is 0.

If there is 1 site in a province, and you search for an Earth site, the probability that you will find an Earth site is 1/9. (Assuming equal probabilities)

If there are 2 sites in a province, and you search for an Earth site, the probability that you will find at least one Earth site is the probability that both sites are not Earth sites, subtracted from 1.

etc.

Now, those are all _conditional_ probabilities. Since the events are independent (the number of sites does not affect the types of sites), the total probability is the product of the two. Therefore, the probability of one site (0.1296) multiplied by the probability that the site is an Earth site (0.1111) will be the probability of their being one site in the province AND that site being Earth.

Since we do not know the number of sites in the province, we must find the probabilities of each case, and then sum them to find the overall probability of finding at least one Earth site in a province when we search it. That is the probability I gave you.