If someone who remembers statistics 101 better than me would be so kind to test if these observations differ from chance (assuming uniform distribution), that would be nice to know.

well this is the reason I asked people here as I am no statistician either But those high and low numbers indicated to me there might be something different than simply chance.

I chose 99 test samples because it is sufficiently enough to see at least smaller trends. Of course these results may be totally affected and only by chances but as the person, I think W(insertrandomnumbershere) said somewhere he thought AI prioritizes colony capital ships.

So additional information that might help solving this issue:

1. capital ships having highest number here for one type of AI were almost built in every game of that type, and several times not just one was built.

2. Lowest numbers, in like half of those 11 games the AI's did not even build one of those, in the other games 1, rarely 2

3. There were some spikes, but only with capital ships already having higher numbers. Aggressor had 5 Raptures in one game, Fortifier had 4, 4, and 5 Halcyons in 3 games, Researcher had no spike, Economist built 4,4,4  Radiance, Discord, Revelation in 3 different games.

I will try to find out how to upload a picture here so you may see these numbers as well, I have these numbers in an XLS file but I think a simple screenshot of it would be the easiest.

To me it always seemed like the AI picked their capships based on a random number generator. It never really seemed like the AI actually tried to counter enemy fleet composition using a specific choice of capital ships. This is more or less confirmed by your tests (and Goa's spreadsheet).

Vasari Rebel tests are finished, updating OP now.

To be honest I don't think the other factions would have a very different outcome than this, but I will make these tests for Advent Loyalist to see if I'm right.

Yeah, they may have weight values to determine build chance, similar to how the fleet beacon abilities can.

I think they only have negative ones based on these numbers, so for some types one or two capital ships are not favoured, and the remaining have the same value.

TBH I would expect every capital ship to be built on average 1 or two times in each game if every capital ship had random chances (of course reaching this tendency after several games), but for the low numbers it was not like that. In many games there were none, remaining games only one, and 1-2 games when 2 or 3. Highest numbers had strange rows as well.

The Vasari Rebel Aggressive one had this:

in consecutive 11 games how many Skirantras were built by the 9 AI's: 0,1,0,3,0,0,0,1,1,1,0

Radiance by fortifier AR: 0,2,0,1,1,1,0,0,1,1,1

Rapture by Economist AR: 0,1,2,1,0,0,1,1,1,1,0

Sova by researcher TL: 1,0,0,2,1,0,3,1,0,0,1

Marza by economist TL: 0,1,0,0,1,0,1,2,3,1,0

These are not really the 1212121212 sequence I would expect if all ships had equal initial chances. 1 player has 16,7% chance to get any capital ship, leading to 222111 in a (perfect) game, so if we think along this line of thoughts each capital ship should have a 11222121211212 line, and the lack of 2's in the lines above is more than suspicious to me, and the presence of several high numbers in the rows below as well.

There were other strange things, look:

Fortifier TL Kol 2,1,2,3,1,2,4,2,3,1,0

Economist TL Corsev 4,4,6,0,2,2,0,0,1,2,3

Economist VR Antorak 3,3,2,2,2,2,2,5,0,0,2

Fortifier VR Vulkoras 1,1,2,0,4,1,4,2,1,2,2

Aggressive AR Rapture 2,2,3,1,3,1,3,2,3,2,2

Fortifier AR Halcyon 2,1,4,4,0,1,2,2,3,3,5

Such a wide variety of numbers, not even near the 12121212 line someone would expect.

From my AI observations, the only thing the AI "sees" whenever it builds a ship or a module is the RoleType. If you give the AI two frigates that both only have one frigateRoleType of Light, it will not be able to differentiate between the two, and will build (roughly) equal amounts of both as a result. Transporting this logic to capital ships, all capital ships except for the colony ones have a roleType of Invalid, so there should be no reason why the AI would be able to differentiate between those capital ships. Go ahead, try changing the stats of a particular capital ship drastically and see if you see similarly drastic changes in distribution. I didn't, and I tried everything from health amount to strike craft count to damage (I paid special notice to strike craft count, since someone, I think Lavo, reported seeing the AI prefer ships with strike craft over ones without in one of his mods).

Colony capital ships are where things might get interesting. Since their roleType is COLONY, which is different from the other capital ships' roleType, it is possible that the AI can differentiate it from the others. The odd thing is that unlike the other colony capital ships, the Jarrasul doesn't have a COLONY roleType in unmodded Sins, it has a roleType of Invalid. However, I'm not seeing any significant variation between the Jarrasul's stats compared to Akkan's and Progenitor's stats (you doing additional tests now armed with this information might result in confirmation bias poisoning your conclusions, possibly even your method), so if the AI can differentiate between the two, it likely still has no effect on the first capital ship built.

Added note on random distribution: you must take into account the possibility of a non-ideal scenario taking place even when tests are repeated ad infinitum. According to my calculations, the possibility of a non-ideal distribution at countably infinite runs is ((n - 1) / n)^n, limit n to total number of states; for this system, it will be (5/6)^6, or about 33.49% (when number of states approaches infinite, the probability approaches 1/e, or about 36.79%). Note that this does not indicate what the non-ideal distribution might be, only that it will be non-ideal. Caveat: I can mathematically prove this fact for systems where all probabilities are rational numbers, but I cannot when at least one probability is an irrational (or complex) number. Since I haven't found any proofs online, I cannot be confident that this theorem holds true for any system that has at least one probability whose value is irrational or complex.

We could always ask a dev.

First of all I don't really know anything about statistics, I just use my common sense here. From the Skirantra line my guess would be that from the next game 0 or 1 Skirantras will be built, and not the supposed one or two.

Uhm, thanks I guess, but these things are horrible even in Hungarian LOL, once our teachers in university tried to teach us these things, I can tell you without any major success

Yes I am aware of something like this that's why I chose a higher number of test samples (99 cases from 11 games for each AI type), and my interpretation of this randon generation dictates that there should be a tendency after so many games that reaches the trend of 121212. And this is not the case in many capital ships, instead they have high fluctuations.

So can the line of Skirantra still be considered acceptable even though it clearly has a different trend with different numbers? Can someone tell me if the Skirantra has the same chance as the other capital ships than how can you make me understand it from the results of it? (if possible without highlevel statistics)

Looks like the Invalid and Colony roletypes has nothing to do here, if there is some weight here it is clearly based on other thing(s).

End game statistics can differentiate capital ships, what is this based upon do you know? Cause I don't.

In unmodded Sins, the Jarrasul (Vasari colony cap) has the same roleType as all other caps ("Invalid"), while the other two races' colony caps have a roleType of "COLONY". Unless the AI deep-analyzes abilities (meaning it would have to identify colonize abilities that ships might have before it even builds those ships, something highly unlikely given its reliance on the Colony, ResourceCapturer, AntiMine, and Scout frigateRoleTypes instead of simply identifying frigates with those types of abilities), the Vasari AI in unmodded Sins effectively plays like an AI without a colony cap.

Hmm, nem gondoltam volna, hogy találok itt egy másik magyart. Mindenesetre a lényeg az, hogy annak ellenére, hogy meglehetősen nagy a variáció a számok közt, valójában statisztikailag teljesen lehetségesek véletlen eloszlásra. Több ezer, esetleg több száz ezer, próbát kéne tenni, hogy kisimuljon, így kb. 100 próbával teljesen elképzelhető az egyenlőtlen eloszlás.

Vigyázat a "confirmation bias"-re (nem tudom a pontos magyar nevét, Google Translate megerősítési torzítást dob ki): mivel egyenlőtlen eloszlást keresel, torzítottan látod a saját adataidat. Az egy darab fura eloszlás mellett ne feletkezz el a maradék 17 eloszlásról, ami mind teljesen elfogadható. Ha további próbák után is szokatlan eredményeket kapsz, elkezdeném vizsgálni, hogy pontosan mi alapján különböztet meg két hajót az AI (fregattokra és épületekre koncentráló vizsgálásaim arra mutatnak, hogy csak roleType alapján különböztet).

statCountType alapján működik az, mint ahogy a fregattoknál és épületeknél. AI vizsgálásaim alapján elég korán megállapítottam, hogy teljesen ignorálja a statCountType-ot (össze-vissza változtattam őket, de AI viselkedése nem változott akkor sem, amikor pl. felderítőknek nehéz fregatt statCountType-ot adtam).

For everyone else, sorry about the Hungarian post; I know it's impolite, but in case of misunderstandings or debates, I find it's often better to talk in a person's mother tongue to avoid mistranslations and misinterpretations. I'm fortunate enough to have two mother tongues, so you needn't worry about me. Since Hungarian machine translations are so poor because of the language's agglutinative nature, I'll roughly summarize: first paragraph states that though the distribution is uneven, it's completely viable considering the small sample size, second paragraph states to watch out for confirmation bias and remember the 17 regular distributions when considering the one odd one, third paragraph states that post-game stats work with the statCountType parameter, which seems to be completely ignored by the AI in-game.

To test whether or not an AI can "see" a certain parameter, I usually record a few "control" cases where nothing is changed, then run a few test cases where I either change a parameter drastically (eg. eliminate it entirely or multiply it by 10) or swap the parameter values of two different ships (eg. swapping the weapons of a utility frigate and a long-range frigate). If the AI notices, its build pattern will shift accordingly (its average total build amount on the same preset, faction, and map at the same time mark with no pirates or random events will differ noticeably; the map needs to be both perfectly symmetrical and fair).

For capital ships, my suggestion would be to do the same: pick a parameter, change it in a large way, and see if the AI's behavior changes compared to the control case (this can be either the theoretical even distribution or the distributions you found through a hundred or so runs, just remember to give more leeway when comparing stats derived from smaller amounts of runs). Remember to do this in a loaded mod though: it's better to copy over the game files to a mod folder and make modifications there than to modify the base game files, possibly mess something up, and be forced to reinstall the game as a result. Through process of elimination, you will find all stats that the AI will pick up on when choosing what ships to build. If the AI does not differentiate between any stats, then you've deduced through process of elimination that the first capital ship the AI builds is completely random.

I already put quite a lot of time behind something similar back in January when looking into how I could alter AI fleet composition. I eventually found that the AI relies on only one parameter type: frigateRoleType for frigates, roleType for strike craft, and planetModuleRoleType for buildings (there weren't enough allowed roleType values to properly test capital ships). Everything I did afterwards further confirmed this: I've successfully crippled AIs by making them have no ships with the Scout frigateRoleType, made them produce carriers with bombers instead of antimodule frigates by slapping an AntiModule frigateRoleType onto the carrier (frigates can thankfully have more than one frigateRoleType), made them build and use siege frigates instead of LF's by swapping their frigateRoleTypes, and even cut down on their turrets by changing its planetModuleRoleType to that of a repair platform.