Chance to Cap & Split Promotions

[Garnet]

I was looking at https://serenesforest.net/path-of-radiance/characters/chance-to-cap/

I didn't like how many zeroes there were. Credits were given to Fire Emblem Averages, as far as I can see they only list the average stat and not the chance to cap, so I can't tell what formula was used. The average stat isn't too hard to calculate, it's just the base stat + promotion bonus + level ups * growth rate. However, calculating the chance of hitting the stat cap is a binomial distribution using the equation P(X=k)=(kn)×pk×(1−p)n−k, where each level up is an independent event with a fixed probability of increasing the stat. Those are complex calculations with varying results depending on the number of decimal places used. Comparing spreadsheet of results I created which calculates up to 10 decimal places, it looks like the table currently listed only calculates with 2 decimal places and then rounds the result. In the table below, I listed the results for Ike calculated with ten decimals and then rounded to two decimal places compared to the current chance to cap listed on the webpage.

For Ike:
HP	Str	Mag	Skl	Spd	Lck	Def	Res
0.02%	70.39%	0.05%	58.81%	79.07%	0%	35.21%	2.08%
vs
0%	59%	0%	56%	78%	0%	46%	8%

That's not too different, and after calculating all characters myself there's only a margin of error of about 10% on any character for any stat, which is honestly kind of nitpicking and I'm glad the results on the site are accurate enough. Also, given that Path of Radiance doesn't have split promotion lines, the chance to cap a stat doesn't really matter as much because the characters classes are always the same.

So all in all, this was a pretty pointless post which I was inspired to make when I somehow max out Ike's HP and thought that was impossible. It turns out there's a less than 1% chance, and either way it's very likely I forgot I used a Seraph Robe in an earlier chapter. I wonder what each characters chance to cap would be if the available stat boosting items were included in the results?

 

Well anyway, I think Path of Radiance would be pretty fun if split promotions were a thing. I mean first of all, you have to remove the whole sage dagger skill thing. What a mistake. Ike as a great lord? He's always been OP anyway, maybe one of the easiest lords to solo a game with. Sothe would make a good Rogue though. Mia as an Assassin doesn't really match her personality but I would probably use her more often. I would totally turn Boyd into a Pirate, and I'm sure that would lead to some funny added support conversations considering all the bandits they used to fight, but him as a Hero makes a lot of sense too. I would probably make Rolf a Ranger, after all Oscar is also a mounted unit. I think it would be a really fun mod, not that people are putting much effort into making mods for PoR. Maybe a De-make of Path of Radiance will get finished someday and then split promotion lines could be added to a version of that.

 

[Eltosian Kadath]

2 hours ago, Garnet said:

Also, given that Path of Radiance doesn't have split promotion lines, the chance to cap a stat doesn't really matter as much because the characters classes are always the same.

Chance to cap matters a fair bit in Path of Radiance due to how the data transfer to Radiant Dawn mechanics work. One of the important things that occurs when you transfer data from Path of Radiance to Radiant Dawn is that any unit that reached their level cap in Path of Radiance, gets a +2 (or +5 for HP) to the base stat in Radiant Dawn, of any stat they capped in the Path of Radiance, with a handful of additional quirks added in there (most notably Sothe's transfer stats work very differently, but that is a bit off topic at this point).

 

[ping]

I think I'm getting different values than both OP and the main site for the chance to cap:

(function used: https://wiki.openoffice.org/wiki/Documentation/How_Tos/Calc:_B_function )

Well, actually, it's only different from the main site for Str, which is odd. I don't know where that comes from - changing the growth by 5% or procs needed by 1, to see if either of those inputs are faulty, results in a chance even lower than the 59% the main site suggests.

I use the cumulative function because I think that should cover cases where a character hits the cap early and then gets "hidden procs" in level-ups after that.

Where things get a bit fucky is unpromoted caps - if Ike gets 15 Spd procs before promotion, two of them "don't count", which means that he will still need 6 after promo, or 21 in total instead of 19. I don't know if there's a way to account for these without the formulas becoming silly, but I'm not a statistican, so I might just not know of a convenient trick.

In any case, accounting for these edge cases should lower the chances to cap - with the simple formula that I (and presumably Maverick for the main page) used, we're falsely counting "15 Spd procs before promo, 4 after promo" as a success, since that's 19 procs in total, and we don't notice that the unpromoted cap got in the way.

So I'm not sure why the chance of success that OP got for Str to cap is a bit higher than what I got, but also why their values for Def and Res are significantly lower. Ike's chance to cap Def before promotion is miniscule, and for Res it's nonexistant, so this should hardly (or not at all) come into play for these stats.

[Garnet]

4 hours ago, Eltosian Kadath said:

Chance to cap matters a fair bit in Path of Radiance due to how the data transfer to Radiant Dawn mechanics work. 

I completely didn't think of that, I have not really ever looked into transferring data considering I only have a gamecube irl, but it seems like a cool feature. 

 

23 minutes ago, ping said:

I think I'm getting different values than both OP and the main site for the chance to cap:

The function I used was 
https://wiki.openoffice.org/wiki/Documentation/How_Tos/Calc:_BINOMDIST_function

First, I will say that I'm not very familiar with either function, but I've read that... 

The B function in OpenOffice Calc is used to calculate the binomial coefficient, which represents the number of ways to choose k successes out of n trials. This is also known as a combination or "n choose k." It's typically used in scenarios where you need to know how many different ways a certain number of successes can occur in a given number of trials, without considering the probability of these successes.

The BINOMDIST function calculates the binomial distribution, which is the probability of a specific number of successes in a given number of trials, with a specified constant probability of success in each trial. This function is used when dealing with scenarios where you have a fixed number of independent trials, each with the same probability of success, and you want to know the probability of a certain number of successes.

So I felt like BINOMDIST was the better choice. It does seem like the data for the main site is a rounded version of your data. I also did not consider that stats could cap early though, and I don't know the effect that would have. Either function, almost all of them are within 5% accurate, with only a few outliers like Soren having a 58% chance to cap res instead of 78%, or Boyd having a 92% chance to cap health instead of 66%, so the two functions must be very similar in this scenario.

[Jotari]

11 hours ago, Garnet said:

I would totally turn Boyd into a Pirate, and I'm sure that would lead to some funny added support conversations considering all the bandits they used to fight,

 

There are no Pirates in Path of Radiance. Well, there are some characters who are by trade, but not as a class. They're all just Bandits.

[ping]

1 hour ago, Garnet said:

The function I used was 
https://wiki.openoffice.org/wiki/Documentation/How_Tos/Calc:_BINOMDIST_function

First, I will say that I'm not very familiar with either function, but I've read that... 

The B function in OpenOffice Calc is used to calculate the binomial coefficient, which represents the number of ways to choose k successes out of n trials. This is also known as a combination or "n choose k." It's typically used in scenarios where you need to know how many different ways a certain number of successes can occur in a given number of trials, without considering the probability of these successes.

The BINOMDIST function calculates the binomial distribution, which is the probability of a specific number of successes in a given number of trials, with a specified constant probability of success in each trial. This function is used when dealing with scenarios where you have a fixed number of independent trials, each with the same probability of success, and you want to know the probability of a certain number of successes.

So I felt like BINOMDIST was the better choice. It does seem like the data for the main site is a rounded version of your data. I also did not consider that stats could cap early though, and I don't know the effect that would have. Either function, almost all of them are within 5% accurate, with only a few outliers like Soren having a 58% chance to cap res instead of 78%, or Boyd having a 92% chance to cap health instead of 66%, so the two functions must be very similar in this scenario.

No, both functions use the same mathematic formula. If B worked as you describe it, it would only give integers as output. BINOMDIST is what MS Excel has, so that's why it's in OpenOffice, too, but B is a somewhat more convenient version because you can set both a lower and an upper boundary.

I'll say that I think that my numbers are correct, since they confirm to what the main site says about Ike. I'm just kinda leery because of the discrepancy for his chance to cap Str, but since I drag-copied the formula across the stats (and, to be sure, used WolframAlpha to get check the result), I will also say that I think that I'm correct for that one, too.

[Shanty Pete's 1st Mate]

14 hours ago, Garnet said:

So all in all, this was a pretty pointless post which I was inspired to make when I somehow max out Ike's HP and thought that was impossible. It turns out there's a less than 1% chance, and either way it's very likely I forgot I used a Seraph Robe in an earlier chapter. I wonder what each characters chance to cap would be if the available stat boosting items were included in the results?

This shouldn't be too difficult to figure out. Just raise the base stat by 2 (or 7, for HP). An Ike who starts with 19 HP, and uses a Seraph Robe, has an equal chance to cap as an Ike who starts with 26 HP, and doesn't use any boosters.

14 hours ago, Garnet said:

Well anyway, I think Path of Radiance would be pretty fun if split promotions were a thing. I mean first of all, you have to remove the whole sage dagger skill thing. What a mistake.

On the contrary, by PoR's design, Mages essentially already have a "split promotion". As you can pick whether they promote into Knives or Staves. It's horribly imbalanced, granted, and there was a thread about what would make Knive Sages "worth it" in PoR. There were some interesting ideas, although it mostly boiled down to "make Knives better in the first place". Although, a formally split promotion could kinda solve this. Obviously a "Knife Sage" should get a larger Strength boost, but what if they got a bigger boost in Magic as well? Or saw their Tome ranks increase by a larger margin? Then there'd be a reason to go "Knife Sage", even if Knives still sucked.

Likewise, Cavaliers enjoy a "split promotion" into Paladin, since they can choose their secondary weapon type. In practice, most will take up Axes, since it's the best weapon type in the game. But here, too, some associated stat changes - i.e. a larger Speed boost for taking up Swords - could sway the logic one way or another.

14 hours ago, Garnet said:

Ike as a great lord? He's always been OP anyway, maybe one of the easiest lords to solo a game with.

Even if PoR had branching promotions, I doubt Ike would get one. In most games with branching promotions, the Lord class still doesn't get a choice. In practice, an option for Ike to get Lances (or Axes, like his father) would be a big deal, in a game with ranged forges. But perhaps they didn't want Ike to become too strong (they were saving that for RD, haha).

14 hours ago, Garnet said:

Maybe a De-make of Path of Radiance will get finished someday and then split promotion lines could be added to a version of that.

Bad idea time: A remake of both Tellius games, but in the Genealogy engine. PoR is part I, and RD is part II. Have fun siezing castles across Tellius and setting up your "inheritance"!

(In truth, RD's structure would not be particularly amenable to this at all, outside of a massive rewrite.)

Edited January 23 by Shanty Pete's 1st Mate
Added link.