From what I've seen, the amount of money paid depends on the size of the side doing the paying. I'm not sure if it's linked to number of titles or size of economy, but larger empires have to pay more reparations. (Similar to how mighty kings must pay an enormous ransom to buy an indulgence from the Pope to gain piety.)
Here's how it works: If you are their liege, the gains from the war are part of your kingdom. It doesn't matter if they're your king, duke, earl, or even mayor, bishop, or count.
To claim a duchy from somebody else through war (when you don't have a casus belli that lets you claim it directly):
Find out who* has a claim on it by opening the duchy information screen and clicking on "Claimants" to see who has a claim on that kingdom.
Invite one of the male claimants over to your court. (If nobody will come over, or there are no male heirs, you'll have to find another way to get that duchy. Note that if one of your courtiers marries a female claimant that you can't legally give a title to, that female claimant is not your vassal and the newly-conquered duchy will not become part of your kingdom, even if you give your male courtier a title.)
Give that claimant a landed title. Any landed title will do, even a city, church, or castle.
Note that if you land the claimant a city, gaining him a Duchy will found a vassal merchant republic instead of gaining you a new feudal vassal if you are playing version 1.9 of the game or later. Merchant republics are generally considered awesome to have as vassals due to their revenue and tax bonuses, but this might disrupt an existing merchant republic that is already earning you revenue.
Go to war with the ruler of your choice, using the "Claim on duchy X" as your casus belli.
Once you win the war, that claimant will be your duke.
I'm guessing that this will also work with kingdoms if you're an emperor. So far, I've claimed two duchies this way: once by giving away a county to the claimant, and once by giving away a bishopric.
*As revealed in this other answer, you can't actually press the claim in war unless the claim is strong, or the claimant is a pretender (2nd or 3rd in line), or there's already another war going on over the title, or the current holder of the title is a woman (and you have a male claimant) or a child (so there's a regency). You can only use "any claimant with a pulse" if the title is disputed or the ruler is a child; you can only use "any male claimant with a pulse" if the ruler is female; otherwise, you'll need to pick up one of the pretenders or someone with a strong claim: you'll need to do slightly more research to find who they are, and they're often harder to invite to your court.
(Note: if there are three or fewer male claimants, then it's almost certain that they're the heir and pretenders. Just grab one and go on your way. Figuring out the exact claims is more important when there are four or more male claimants.)
Best Answer
Internally, Crusader Kings 2 bases event triggers1 upon mean time to happen (abbreviated MTTH). The weighting formulas for different events are in the game files, often in plain text. One of the simplest ways of modding the game works by editing these text files to provide different values. The game UI will translate that value after all modifiers into the rough percentage chance you see2.
Behind the scenes, the actual engine calculations are something different. Paradox Game developer Johan shared this snippet of source code:
Which basically says that every event has a percentage chance of happening that day. The inclusion of the
noOfDays
is a bit misleading, as the same post says it is "basically, how many days are between checks. (currently 1)". In other words, it's a tuning factor for the game engine to perform checks less often (improving performance) and arrive at the same odds.exp()
is apparently a C/C++ library function for natural exponential function with Euler's number, ex.A few posts down in that thread, user Hakkapeliitta explains the code better than I could:
I bolded the last bit of his explanation, as it is the most pertinent to your question. The last thing to figure out is how the game translates the MTTH in the code. Considering the example of MTTH = 3 months again, let's try the simple conversion of 50% per 3 months to 16.7% per month, which would mean the game would give you a 16.7%/30 days =~ 0.55% chance per day of firing that event - that would be an exceptionally high chance of a councillor accomplishing anything.
So let's go back to your example of 8.29% yearly success. That would equate to a MTTH of 50%/8.29% = 6.031 years, or 72.4 months. Translating that to a daily chance means the code would give you a roughly 0.046% chance of getting the 'Claim Fabricated' event on any given day3. Presumably, the code would then roll a random number and compare it agains this daily chance value.
You are never guaranteed success in this game. All you can do is improve your chances. For your case, it sounds like your ambassador has a diplomacy value of 10 or so. It would (apparently) be very much to your advantage to find someone with a 13+ score.
1: This is true for most long-term events. There are other event types that are "triggered only", which means they have to be chosen through a dialog box/AI decision as a direct and exclusive result to another event. For example, the dialog that pops up upon a successful claim fabrication gives you an option to use the claim or let it go; each of these corresponds to another event in the game engine, and each options' subsequent event is responsible for granting you the claim or the piety from letting it go.
There are other events that trigger from weighting factors in the short term. Apparently events with short timelines place more load on the game engine, so the game does a cheaper calculation to decide whether the event occurs or not. The example the wiki gives is some weighting calculation for whether or not a failed assassination will result in a wound or a maiming.
2: This is true for the events that you can see the odds for, which generally include councilor missions, intrigue plots and faction rebellion chances. Many more events, such as fertility, child rearing and tutoring, disease and death, are either completely hidden or only show the barest hints from traits and opinion modifiers (e.g. strong gives +1 health and +10% fertility, decreasing your chance for death and increasing your chance to have a child; a -100 opinion is more likely that someone will attempt to assassinate you).
3: This is how it really works for true random events: assuming that future probabilities are more in your favor due to past failures is called the Gambler's Fallacy. Imagine your Ambassador trying to fabricate a claim; it's easy to think that it wouldn't be a true-random event, but rather a goal he would progress towards, making it more likely to succeed with each passing day. He doesn't wake up, attempt forging documents all day long until he has a reasonable facsimile, throwing away failed attempt after failed attempt. Instead, he would be bribing people to get looks at official documents or seals to improve his forgery, spending time carving or finding a carver for a wax mould for the seal, aging parchment or finding appropriate parchment until he eventually creates a successful document.
Unfortunately, any reasonably accurate testing will show you that accumulating and increasing the probability over time will radically skew your events. This may sound like it'd be more fun to play – who wants to wait 20 years for your first Causus Belli or your first born heir? – but imagine if every assassination or claim plot against your lands were guaranteed eventual success. You would find your rulers even more death-prone and your lands much less likely to remain under your control. After all, for any simulation to be accurate, the chance for real, total and lasting failure must be an option.