Exact statistical probabilities and expected returns for rolls

[Spider-Dan]

warning: math or GTFO

disclaimer: I am not a mathematician. I couldn't figure out any formula to calculate this easily, so I did most of this through a combination of noticing and verifying patterns, and sheer brute force. I've double-checked my math to the best of my ability, but I am certainly open to correction.

--Introduction--

It's been a long standing desire of mine to quantify the exact expected bonuses (statistically) from the rolls for which we know the exact returns. Unfortunately, the lazy method of simply taking the (normally 7) possible results, adding them, and dividing by 7 isn't truly statistically valid, since those 7 outcomes are far from equally likely. In order to find out the real expected average, we need to take into account the exact probability of each possible result.

In order to do this, it is necessary to go beyond simply calculating which possible numbers can add up to a given total; you must actually determine every possible roll sequence that leads to a given total. The reason why is simple: rolling "rules" that tell you to stop rolling at a certain total, even when you have room left to roll before busting. The most obvious example of such a rule is, "don't Double-Up if you land on a Lucky Number."

For example: suppose we are using Corsair's Roll. Given the constraints of a 6-sided die and a maximum number of 8 rolls (the maximum amount of rolls you can do before the Double-Up window expires), there are 482 possible ways to wind up at a total of X; however, 458 of them would require you to pass through a total of V, VII, VIII, or IX and keep rolling. Since the standard rule for Corsair's is "stay on Lucky # or VII+," those 458 combinations must be eliminated, leaving you with 24 possible ways to end up at a total of X.

So basically, what I've done is break down the number of possible ways to arrive at a given total, using a specific rolling rule that has been set forth. Here's the format for the results :

--rule--
total probability percentage
...
total probability percentage
For most rolls, the two rules I've calculated are "stay on VII+" (i.e. normal rolling) and "stay on VI+" (i.e. you have 1 bust). The two exceptions are L3/U7 rolls (e.g. Healer's) where I've also provided a "stay at VIII+" rule (i.e. Double-Up if you land at VII), and Evoker's where I've provided the same rule. For the record, all rules assume staying on Lucky #.

And now, time for the show:

Lucky # 2 / Unlucky # 6
--stay at VII+--
total probability %
II 2/161 1.24%
VII 31/161 19.25%
VIII 30/161 18.63%
IX 30/161 18.63%
X 28/161 17.39%
XI 24/161 14.91%
Bust 16/161 9.94%

--stay at VI+ (1 bust)--
total probability %
II 2/81 2.47%
VI 16/81 19.75%
VII 15/81 18.52%
VIII 14/81 17.28%
IX 14/81 17.28%
X 12/81 14.81%
XI 8/81 9.88%

Lucky # 3 / Unlucky # 7
--stay at VII+--
total probability %
III 4/161 2.48%
VII 31/161 19.25%
VIII 30/161 18.63%
IX 28/161 17.39%
X 28/161 17.39%
XI 24/161 14.91%
Bust 16/161 9.94%

--stay at VIII+--
total probability %
III 4/316 1.27%
VIII 61/316 19.30%
IX 59/316 18.67%
X 59/316 18.67%
XI 55/316 17.41%
Bust 78/316 24.68%

--stay at VI+ (1 bust)--
total probability %
III 4/81 4.94%
VI 16/81 19.75%
VII 15/81 18.52%
VIII 14/81 17.28%
IX 12/81 14.81%
X 12/81 14.81%
XI 8/81 9.88%

Lucky # 4 / Unlucky # 8
--stay at VII+--
total probability %
IV 8/161 4.97%
VII 31/161 19.25%
VIII 30/161 18.63%
IX 28/161 17.39%
X 24/161 14.91%
XI 24/161 14.91%
Bust 16/161 9.94%

--stay at VI+ (1 bust)--
total probability %
IV 8/81 9.88%
VI 16/81 19.75%
VII 15/81 18.52%
VIII 14/81 17.28%
IX 12/81 14.81%
X 8/81 9.88%
XI 8/81 9.88%

Lucky # 5 / Unlucky # 9
--stay at VII+--
total probability %
V 16/161 9.94%
VII 31/161 19.25%
VIII 30/161 18.63%
IX 28/161 17.39%
X 24/161 14.91%
XI 16/161 9.94%
Bust 16/161 9.94%

--stay at VIII+--
total probability %
V 16/316 5.06%
VIII 61/316 19.30%
IX 59/316 18.67%
X 55/316 17.41%
XI 47/316 14.87%
Bust 78/316 24.68%

--stay at VI+ (1 bust)--
total probability %
V 16/81 19.75%
VI 16/81 19.75%
VII 15/81 18.52%
VIII 14/81 17.28%
IX 12/81 14.81%
X 8/81 9.88%

--

OK, so we're halfway there. The next step is to apply known values to the stated probabilities for each rule (there's an Excel spreadsheet here (http://tinyurl.com/hlbng/roll_calculator_v3.xls) that details all the values). However, in calculating these values, I had a bit of a dilemma: give Bust value (0) full weight and it'll artificially lower the average return (as busts can be rerolled in 1 minute instead of 5). Give busts lesser weight and it'll artificially increase the return of high-risk rules ("stay on XI" looks like a fantastic strategy).

What I decided to do is to take the average "stay on VI+" return, and use 5/6th of that number as the bust value in the other roll calculations (1 min of no buff + 5 min of reduced-risk buff). For example, if you bust on Chaos, you would have 1 minute with a buff value of 0, followed by a reroll of Chaos (under "stay at VI+" rule) and a 5 minute buff of that average return.

Healer's Roll (without/with WHM)
stay on VII+: hMP+4.35/7.30
stay on VIII+: hMP+4.94/7.30
stay on VI+ (1 bust): hMP+4.04/7.04

Chaos Roll (without/with DRK)
stay on VII+: ATK+8.36%/14.60%
stay on VI+ (1 bust): ATK+7.99%/14.23%

Corsair's Roll (with COR)
stay on VII+: EXP+12.25%
stay on VI+ (1 bust): EXP+12.14%

Wizard's Roll (without/with BLM):
stay on VII+: MAB+5.47/9.41
stay on VI+ (1 bust): MAB+5.10/9.10

Evoker's Roll (without/with SMN):
stay on VII+: 1.87/2.85 MP/tick
stay on VIII+: 2.01/2.97 MP/tick
stay on VI+ (1 bust): 1.67/2.67 MP/tick

--

One thing I didn't anticipate: this entire experiment seems to make a fairly strong argument for staying on VI+ (except on Evoker's). The tiny increase from staying on VI+ to staying on VII+ may not be worth it if you are also cycling Evoker's (which is pretty much crippled if you have a bust and are staying at VI+).

Comments are welcome.

P.S. On a related note, when is the [code] tag going to be fixed?

[Taskmage]

P.S. On a related note, when is the [code] tag going to be fixed?I dunno. =/ Weak as it was, that was pretty much our only formatting option. If you make a post in the Comments/Suggestions forum, PiNG usually goes through that stuff whenever he stops by the site.

[Omgwtfbbqkitten]

Stay on +7 for Evoker's doesn't make much sense as 1,2,3,4,6 and 7 all give us the same low returns on Evoker's as 9 does, 1 MP a tick. 8 and 10 produce 2 MP a tick while 11 grants 4 and the Lucky #5 gives us 3. That's seven numbers we don't want and four we do want.

So the goal is pretty obvious, if we can't get 5, we should at least try to get 8 or 10. Staying on 7 is no better than rolling 1-4, 6 and 9. Meanwhile 5, 8, 10 and 11 produce more desirable results. Our target should be the most desirable number, yet we should weight that against PT peformance or if we already have a bust on ourselves.

If you have a Bust on you and a mage's MP is very low, then the chance of rolling 8 or 10 when you've already hit 6 of 7 could be detrimental to your PT. I believe this is where your favorite word "situational" comes into play. Going beyond 6 or 7 here can be a higher risk than before because two Busts will sideline us for one to four minutes depending on when the bust took place.

Let's look at the state of Busts for a second and not factor in the Bust Duration merits (which are utterly worthless anyway).

Let's say you roll two consecutive Busts in a minute's time. That puts COR in the worst spot possible, as now the COR can't buff anyone for a full four minutes. Since we don't want that, if we already have one bust, we definately should be more conservative on rolls from there, so if we land on a 6 or double up to 7, then its not unreasonable to stop there.

But what about three minutes later? We've moved from a high risk situation to one where that high risk may actually be acceptable because at worst, you're going to have two busts on you for one minute, dropping to one bust for the next three right after.

I don't think we should be "averaging" our performace to 6 or 7 and just stop there each time we hit them, that only limits us. It should be dealt with situationally.

And the risk really varies with the roll-type. Samurai Roll, Evoker's and Healer's Roll all provide fixed bonuses while the rest are percentage-based bonuses. And the fixed bonus rolls are quite a common sight in EXP and Merit PTs alike, I've rarely strayed from rolling them, whether the job that compliments them is present or not, It usually my fourth buff that changes, the fixed ones seldom go out of style.

Samurai Roll and Healer's roll are rather low-risk rolls. Even if you do get a number that results in a low +TP or +hMP return, the result is still desirable and, more importantly, a cumalative bonus. Any TP return is going to lead to more opportunities to skillchain/WS, while more +hMP will lead to higher MP recovery (and considering all the +hMP options out there, that's pretty insane MP recovery, even RDMs at the very least should be scarfing cookies to capitalize on this one).

So, really, Samurai and Healer's Roll are your best rolls to be conservative with. But for SAM Roll I would double up on 6 since that is an unlucky there, Healer's roll Unlucky #7 is the Unlucky I'm least worried about and I've never really felt pressed to double up on this one anyway.

And all of that said, the basic breakdown of any roll is still this:

With the exception of Lucky and Unlucky numbers, the resulting buffs get stronger from 1 to 11, 1 being the weakest and 11 being the best result

So when we're not landing the Lucky and Unlucky, we should at least be trying to get near 11 (trying to get 11 outright all the time is just plain stupid).

This simpler rule of thumb concedes that doubling up from 5 to 7 results in a minor boost to something like Chaos roll, but also accepts that the possibility of a 5 increasing to 9, 10 or 11 would be best the best route to take, resulting in a stronger buff. That annoying unlucky 8 is still a chance, but considering the possible rewards, I think that risk (or probability :P ) is worth it.

Or we can just look at what Kenny Rogers said: "Know when to hold 'em, know when to fold 'em."

[Lmnop]

However, in calculating these values, I had a bit of a dilemma: give Bust value (0) full weight and it'll artificially lower the average return (as busts can be rerolled in 1 minute instead of 5). Give busts lesser weight and it'll artificially increase the return of high-risk rules ("stay on XI" looks like a fantastic strategy).

What I decided to do is to take the average "stay on VI+" return, and use 5/6th of that number as the bust value in the other roll calculations (1 min of no buff + 5 min of reduced-risk buff). For example, if you bust on Chaos, you would have 1 minute with a buff value of 0, followed by a reroll of Chaos (under "stay at VI+" rule) and a 5 minute buff of that average return.

I chuckled at your fantastic strategy.

I skimmed your math, looks good and stuff (I'm retarded >.>). Something about taking your 5/6th value doesn't want to sit right with me... I'm probably thinking wrong but... like rolling 2 6s to get a 12 is a 1/36 shot. 3d6 gets that number much more frequently, but it's still not 1/18 (the linear amount of adding all dice) but of course you know that. I guess I'm not quite sure how your measure of time (5 minutes of reduced risk) can affect your chances of gaining a particular number. If this is my problem with math and not yours, don't bother to correct me - I'll get it figured out eventually.

Interesting note to me:
Healer's Roll (without/with WHM)
stay on VII+: hMP+4.35/7.30
stay on VIII+: hMP+4.94/7.30

I find it interesting that a difference in probability exists when WHM is not present, but they become the same with a WHM. This means you can use WHMs in your party as a reason to err on the side of caution? Of course, that also ties into what Bbq said about never really caring if Healer's lands unlucky since it's better than nothing.

[Omgwtfbbqkitten]

I find the math on the returns be a little fuzzy since the fixed bonuses of Healer's and Evoker's are given lower or higher decimal values. Its just a flat, static bonus. No give or take to them at all.

The percentage bonuses of the other rolls can feel more varied since other bonuses can come into play on individual players. But the increase in power is still there from 1 to 11 before we consider lucky, unlucky and job presence.

We can number crunch all we like about the probability of where a roll will land, but I think the end returns are the only thing to be concerned about. As it is we have far too many CORs overthinking their buffs (as though it was some form of meditation) and that hinders them from being able to move to functioning as a puller or even playing pure support with /WHM effectively (though DD is usually the underlying excuse). There's no need to get too anal about the rolling aspect.

Know the values of the best possible buffs. Rolling will still always boil down to intuition and situational application. When you hit the number you're happy with, you stand and cease doubling up.

In other words, its still luck. But if you're constantly stopping on 6 or 7 when you can give a better buff, you're just better off playing BRD where all the buffs are a fixed bonus based on singing skill.

[Lmnop]

I find the math on the returns be a little fuzzy since the fixed bonuses of Healer's and Evoker's are given lower or higher decimal values. Its just a flat, static bonus. No give or take to them at all.
The percentage bonuses of the other rolls can feel more varied since other bonuses can come into play on individual players. But the increase in power is still there from 1 to 11 before we consider lucky, unlucky and job presence.
We can number crunch all we like about the probability of where a roll will land, but I think the end returns are the only thing to be concerned about. As it is we have far too many CORs overthinking their buffs (as though it was some form of meditation) and that hinders them from being able to move to functioning as a puller or even playing pure support with /WHM effectively (though DD is usually the underlying excuse). There's no need to get too anal about the rolling aspect.
Know the values of the best possible buffs. Rolling will still always boil down to intuition and situational application. When you hit the number you're happy with, you stand and cease doubling up.
In other words, its still luck. But if you're constantly stopping on 6 or 7 when you can give a better buff, you're just better off playing BRD where all the buffs are a fixed bonus based on singing skill.


Errr I think the overanalyzing is the point of this. Take a quick look around the internet. Black Jack's the obvious example, but Craps especially. Look around and it's very easy to find charts and tables of how to best play the game. All sorts of calculated probability on how to best win the game. There are 2 kinds of people: those who play to have fun, and those who play to win. Corsair isn't any different. I don't see why you're trying to talk down number crunching so much.

Someone who has a general gist of what will always be most effective can incorporate it into their routine, making it second nature. Just like when I see a monk ready Raging Fists right after my Raging Rush. Even though I didn't expect him to WS right then, I know that an Impaction SC is about to hit. Some of us don't have your wonderful gambling intuition. For some of us, study does the trick.

[Omgwtfbbqkitten]

Errr I think the overanalyzing is the point of this. Take a quick look around the internet. Black Jack's the obvious example, but Craps especially. Look around and it's very easy to find charts and tables of how to best play the game. All sorts of calculated probability on how to best win the game. There are 2 kinds of people: those who play to have fun, and those who play to win. Corsair isn't any different. I don't see why you're trying to talk down number crunching so much..

What I'm saying is you can study all you like and I can go on intuition and most of the time, we'll still be following the same means to the same end. The difference is I'm just not looking for a hidden meaning or math game behind Phantom Roll.

All that matters is knowing the end result of the buffs themselves and knowing your target numbers.

There's also the conservative player and their fear of going beyond 6 or 7 can limit them not only in potential but advanced buffing strategies. I'll actually make that another topic since it is actually another discussion, but the conservative will not touch these techniques while the intuitive and number-crunchers can.

I just feel the number-crunchers help create the conservative CORs in a way. Probabilities can create counterproductive strategy just as much it can create the positive kind.

[Spider-Dan]

I just feel the number-crunchers help create the conservative CORs in a way. Probabilities can create counterproductive strategy just as much it can create the positive kind.
Probability is the primary component in defining what makes a strategy productive. To say that probabilities can create a counterproductive strategy is essentially an oxymoron. Ignoring probability in favor of unscientific intuition is what makes strategies counterproductive.

That all being said, I've been made aware of serious flaws in my methodology by a real mathematician, and am currently working to correct them.

[Hantz]

Math, shmath. Without idiots like me running around, Doubling-up willy-nilly, nobody would be able to say "That Corsair has such bad luck!" And if that happens, then the terrorists win.

Edit: Sorry for that bologna. I just wanted to see that great new sig Krilldog cranked out for me. Oh, and to say that last night I dreamt that Sage Sundi posted a message saying that he was done with FFXI. Weird dream.

[Omgwtfbbqkitten]

Probability is the primary component in defining what makes a strategy productive. To say that probabilities can create a counterproductive strategy is essentially an oxymoron. Ignoring probability in favor of unscientific intuition is what makes strategies counterproductive.

You're missing the point. Some people will take scientific data and obsess about it instead of processing it in a calm fashion, leading to them being too conservatve. Meanwhile others will take data as a means to an end and never utilize it fully. I'm not entirely opposed to the study of probability, I'm just accustomed to acting and learning from action as opposed to constant study. FFXI is an environment without real consequences, so I don't have concerns about taking various risks to get my experience with things.

I never consulted one forum about how to pull as a BRD, in fact, there wasn't any real source material at the time. Yet I took all I had learned from one Onzozo PT and applied that to future PTs with great success. The whole process came naturally to me within one Onzoz PT and with no prior pulling experience.

Pulling with Elegy, Lullabying the mob to stage a fight, keeping buffs up and leaving early to pull - all of it just came naturally. I just regarded it as a strategy I threw together, but the PT thought it was something amazing. Its commonplace now because people have had it explained to them on forums by others, but it was just common sense to me. i just looked at what I had, what I could do with it and did it.

I'd say I was the Kirk to your Spock, but I'm more of a McCoy.