Site Searching Statistics Questions

[lch]

Okay, I think Baalz has trouble with the reverse scenario than I described: If I already found something, how big is the chance I'll continue to find something if I search more?

Baalz said:
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 non-earth sites, which complicates the math and doesn't change the chances too much)

Those things are not independent. For every site searching, there are two things that come into the equation: One, how many sites there are in the province, two, how many you found already. If there are two sites in a province and you already found two, then you can search all you want but that won't make a new site pop up. That's because those sites are rolled for before the game starts. If they'd be rolled for when the site search is being made, you would be correct.

[Evilhomer]

The inaccuracy of indypendents example is that the unknown "head" probability has four positions to be at (so you would have to compensate with a factor of 4 here), while each tail in the example only has one position (while having four "cases" instead).

Probably pretty fuzzy explanation, but if we simplify to Baalz original 4 random coin flips, 3 known the chance for the last unknown is 50% heads and 50% tails (as Baalz wrote).

That said I am not sure about the mechanic behind site distribution so I am not sure how well the coin flipp theory works. It is however always better to search provinces with few sites for the simple reason that 4 sites is the maximum (thus you cannot find 2 new sites, if you have already found 3 for example).

[VedalkenBear]

Ich has it correct, and there's a distribution that is used precisely for this issue; it is the hypergeometric distribution.

Consider the following. A province in ME, by default, has a 40% chance of having a site. Let us assume no terrain modifiers are in place.

The number of sites that are available to find is given by the binomial distribution. The probabilities are:

0 - 0.1296
1 - 0.3456
2 - 0.3456
3 - 0.1536
4 - 0.0256

(Incidentally, this means that the average number of sites in a province with a per-site chance of 40% is 1.6.)

Therefore, if you have an unsearched province, and you cast Gnome Lore on that province, the probability that you find an earth site is the probability that there are X sites, and that there is at least one site among those X. That is:

[P(1 site) AND (Probability that site is Earth)] OR [P(2 sites) AND (Probability that the first site is earth OR Probability that the second site Earth)] OR [P(3 sites) AND (Probability that first site is Earth OR Probability that second site is Earth OR Probability of third site is Earth)] OR [P(4 sites) AND (Probability of first site is Earth OR Probability of second site is Earth OR Probability of third site being Earth OR Probability of fourth site being Earth). Since AND is mathematically multiplication, OR is mathematically addition, and P1 OR P2 OR P3 ... is equal to 1-P1*P2*P3..., we have, assuming equal chances of each type of site in this province:

(0.1296)(0) + (0.3456)(0.111) + (0.3456)(0.210) + (.1536)(0.298) + (0.0256)(0.376)

= 0.1663

or an 16.33% chance of finding an Earth site in that province.

Now, what happens if you have an Earth site in a province, and then cast Astral Probing on the same Province? Then the probability is:

P(2 sites)*(1-P(that site is not Earth) + P(3 sites)*(1-P(neither site remaining is Earth) + P(4 sites)*(1-P(none of the three sites remaining is Earth).

Or

0.3456*0.111 + 0.1536*0.210 + 0.0256*0.298

= 0.0783

Or an 7.83% chance of finding an Astral site in a province that already has another Earth site, _provided you did not search for the Earth site_. If you did, then it is 0.0876, or an 8.76% chance.

I hope this makes sense. If anyone wants other probabilities (especially the way this all changes if you select a different per-site probability), just let me know.

[Evilhomer]

I do belive you are incorrect VedalkenBear. It was a while since i studied statistic (I will give you that) but probabilites change if we get to know something about the result (like knowing that we have 1 earth site). The binomical chances you listed:

0 - 0.1296
1 - 0.3456
2 - 0.3456
3 - 0.1536
4 - 0.0256

are probably correct if you have no prior knowledge (I have not dubble checked this but I will trust them). They will however change once it is known that we have 1 site (from the earth check - obviously now the chance of having 0 sites is 0). This will change the result on the following calculation....

[Edratman]

VedalkenBear - first class analysis.

What I think would be most helpful (and you have provided ample evidence of ability) would be a table showing the average number of sites per province with the settings of 10,20,.....100%. (The values higher than 75 are available using map edit.)

Thank you.

[thejeff]

Ok, without actually checking the numbers I followed you up through the 16.33% chance of an earth site.

For the astral search:
Why (1-P(site is not Earth))? Isn't that the same as the chance that the site is Earth? Where does the chance that the site is astral come in?

Nor do I see where you get the 8.76% if you have searched for an earth site.

If there isn't an earth site, but you've searched for one, how does that change the probability of finding an Astral site?

[VedalkenBear]

Evilhomer: Actually it doesn't. The other calculation used the probability of 2 sites (0.3456) multiplied by the chance that the second site (since the first is already occupied) is the appropriate kind.

The binomial numbers above came right out of Excel's BINOMDIST function. Also, I know they're correct because I've had to use that exact P for Work Measurement several times.

Remember, the number of sites are mutually exclusive. Having 2 sites is not dependent on having 1. Obviously, you can't have 2 without having 1, but in a probabilistic sense, the states are mutually exclusive and independent.

[Panpiper]

VedalkenBear said:
...

Zounds!

[VedalkenBear]

Hey, I'm an Industrial Engineering student, and probability/statistics are used in every single one of my classes. I hope I'd be good at it.

As for the expected number of sites per province, it really is just (4*p). I've verified it for several different values of p in Excel, and it follows exactly.

Now, if someone were to give me the actual terrain differences on the different types of sites, I could do some more advanced analysis.

[Evilhomer]

I am not sure we understand each other here I must say. I will give you an example:

The same hypergeometric distribution GIVEN that we have 4 sites in that province would be:

0 - 0.0
1 - 0.0
2 - 0.0
3 - 0.0
4 - 1.0

In the same way the distribution you gave would change GIVEN that we have atleast 1 site, so using this:

0 - 0.1296
1 - 0.3456
2 - 0.3456
3 - 0.1536
4 - 0.0256

just cannot be accurate for the second calculation (when we know we have atleast 1 site). Maybe I misunderstood you somewhere along the way, a bit to tired to go into detailed math analysis.