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Your analysis is fine, Trashos, but as before it is incomplete.
A: To-hit: You have assumed a naked, unbuffed Octavia. While an interesting mental image, it is not a reasonable assumption to make. Cat's Grace (or Belt +4 but I don't remember what kind of belt you have access to at that point), Haste and Heroism give her a total of +5 AB, and are all buffs she can cast herself. With just another +1 she has a 95% chance to hit, just like panda said. The source of that final +1 is trivial, and left as an exercise for the reader.
B: SR: Math checks out, though I will not examine the case of just Spell Penetration without Greater.
C: Combining the above: I do not disagree with panda, but I will do it because it will be useful to what comes next
No SP: 0.95*0.4=0.38 or 38%
GSP: 0.95*0.6=0.57 or 57%
Just as expected, and mirroring panda's results, with GSP we see a 50% increase on the spells that land compared to without. But, that number is not the number we actually care about. It is only useful as a proxy for
D is for Damage
An L12 Octavia has three choices of Rays to use: Scorching Ray, Hellfire Ray and Disintegrate. For a proper conclusion, we have to examine each. I am assuming no bonuses to damage and I don't remember how reasonable an assumption that is.
Scorching Ray: Each ray deals 4d6+6d6 fire damage, reduced by the Astradaemon's Fire Resistance 10. This gives us an average of 25 damage per ray, 3 rays per round. We could go deeper into examining the full distribution, but it is more effort than I'm willing to expend on this. The average will have to serve for our purposes. Her true Damage per Round is going to be 25*0.38*3=28.5 or 25*0.57*3=42.75. Spell Penetration increases her damage by exactly 50%, which is exactly what we would expect.
Now, the Astradaemon has 213 hit points. Octavia needs to land 9 rays, on average, to kill him. Without SP this means firing 9/0.38, or 23.7 rays, or 8 spells. With SP, this means 9/0.57, or 15.8 rays, or 6 spells.
Hellfire Ray: Each ray deals 12d6+6d6-10 damage. The average is 53. This is barely better than 2 Scorching Rays, 2/3 of a casting, so I am not going to examine this case further. Scorching Ray is a strictly better use of her slots and actions in this fight.
Disintegrate: 24d6+6d6, for an average of 105. No resistance applies, but it is subject to a fort save. A successful save reduces the damage to 5d6+6d6, or 38,5. Octavia should have a DC of 10+6 Spell Level+7 INT+1 Focus, or 24. The Astradaemon has a fort save of 12. Taking this into consideration, the average is 105*0.55+38.5*0.45=75.08. 28.53 without SP, 42.8 with. Marginally better than Scorching Ray.
Octavia needs 3 rays to kill him, so she has to cast 8 spells without SP, or 6 with it. We got the same result as with Scorching Ray, so we can use the spells interchangeably in this fight. This is somewhat fortunate, as Octavia cannot cast 6 Disintegrates at L12. The cause of us being able to use them interchangeably is a spell's nature as a distinct packet of damage. Disintegrate is simply slightly more overkill on average. And this is the same cause of us seeing less than the expected 50% improvement. In both cases we were close to needing 5 spells with SP, with Disintegrate requiring a little less luck than Scorching Ray to get there.
Conclusion? I have none. Spell Penetration improves Octavia's damage against foes with Spell Resistance. This was blindingly obvious to everyone, and so my theorycrafting, like most theorycrafting, was a waste of time. And I am still unsatisfied with it. I ignored the distribution, which would allow us to measure exactly how lucky you need to be to finish him off with less spells than average (the variance of our distribution), and how this number changes with and without SP and for the different spells. I ignored possible sources of increased damage, like a Rod of Empower. I ignored completely the contributions of the rest of the party, which is never a wise thing to do in a party-based game. And I possibly ignored other things that would affect the outcome and I cannot think of at the moment.
If you want to theorycraft properly, you cannot ignore variables like those. You need to take everything into account, or the numbers you arrive to, and use to derive your conclusions from, diverge more and more from reality. There are few things worse than assuming your spherical cows behave exactly like the real ones.
But, I hear you say, you cannot expect us to take every single variable into account. It's too much effort, requires too much time, and there's still the risk of missing something and coming to the wrong result. I hear ya. And, well, I agree completely. Scientists who are extremely well paid to do this for a living still struggle with it. We cannot expect a rando on a forum to do it for a game, of all things. And this is the reason Desiderius's (and Pink Eye's) experimental approach, and the conclusions derived from it, is vastly superior to theorycrafting.
A: To-hit: You have assumed a naked, unbuffed Octavia. While an interesting mental image, it is not a reasonable assumption to make. Cat's Grace (or Belt +4 but I don't remember what kind of belt you have access to at that point), Haste and Heroism give her a total of +5 AB, and are all buffs she can cast herself. With just another +1 she has a 95% chance to hit, just like panda said. The source of that final +1 is trivial, and left as an exercise for the reader.
B: SR: Math checks out, though I will not examine the case of just Spell Penetration without Greater.
C: Combining the above: I do not disagree with panda, but I will do it because it will be useful to what comes next
No SP: 0.95*0.4=0.38 or 38%
GSP: 0.95*0.6=0.57 or 57%
Just as expected, and mirroring panda's results, with GSP we see a 50% increase on the spells that land compared to without. But, that number is not the number we actually care about. It is only useful as a proxy for
D is for Damage
An L12 Octavia has three choices of Rays to use: Scorching Ray, Hellfire Ray and Disintegrate. For a proper conclusion, we have to examine each. I am assuming no bonuses to damage and I don't remember how reasonable an assumption that is.
Scorching Ray: Each ray deals 4d6+6d6 fire damage, reduced by the Astradaemon's Fire Resistance 10. This gives us an average of 25 damage per ray, 3 rays per round. We could go deeper into examining the full distribution, but it is more effort than I'm willing to expend on this. The average will have to serve for our purposes. Her true Damage per Round is going to be 25*0.38*3=28.5 or 25*0.57*3=42.75. Spell Penetration increases her damage by exactly 50%, which is exactly what we would expect.
Now, the Astradaemon has 213 hit points. Octavia needs to land 9 rays, on average, to kill him. Without SP this means firing 9/0.38, or 23.7 rays, or 8 spells. With SP, this means 9/0.57, or 15.8 rays, or 6 spells.
Hellfire Ray: Each ray deals 12d6+6d6-10 damage. The average is 53. This is barely better than 2 Scorching Rays, 2/3 of a casting, so I am not going to examine this case further. Scorching Ray is a strictly better use of her slots and actions in this fight.
Disintegrate: 24d6+6d6, for an average of 105. No resistance applies, but it is subject to a fort save. A successful save reduces the damage to 5d6+6d6, or 38,5. Octavia should have a DC of 10+6 Spell Level+7 INT+1 Focus, or 24. The Astradaemon has a fort save of 12. Taking this into consideration, the average is 105*0.55+38.5*0.45=75.08. 28.53 without SP, 42.8 with. Marginally better than Scorching Ray.
Octavia needs 3 rays to kill him, so she has to cast 8 spells without SP, or 6 with it. We got the same result as with Scorching Ray, so we can use the spells interchangeably in this fight. This is somewhat fortunate, as Octavia cannot cast 6 Disintegrates at L12. The cause of us being able to use them interchangeably is a spell's nature as a distinct packet of damage. Disintegrate is simply slightly more overkill on average. And this is the same cause of us seeing less than the expected 50% improvement. In both cases we were close to needing 5 spells with SP, with Disintegrate requiring a little less luck than Scorching Ray to get there.
Conclusion? I have none. Spell Penetration improves Octavia's damage against foes with Spell Resistance. This was blindingly obvious to everyone, and so my theorycrafting, like most theorycrafting, was a waste of time. And I am still unsatisfied with it. I ignored the distribution, which would allow us to measure exactly how lucky you need to be to finish him off with less spells than average (the variance of our distribution), and how this number changes with and without SP and for the different spells. I ignored possible sources of increased damage, like a Rod of Empower. I ignored completely the contributions of the rest of the party, which is never a wise thing to do in a party-based game. And I possibly ignored other things that would affect the outcome and I cannot think of at the moment.
If you want to theorycraft properly, you cannot ignore variables like those. You need to take everything into account, or the numbers you arrive to, and use to derive your conclusions from, diverge more and more from reality. There are few things worse than assuming your spherical cows behave exactly like the real ones.
But, I hear you say, you cannot expect us to take every single variable into account. It's too much effort, requires too much time, and there's still the risk of missing something and coming to the wrong result. I hear ya. And, well, I agree completely. Scientists who are extremely well paid to do this for a living still struggle with it. We cannot expect a rando on a forum to do it for a game, of all things. And this is the reason Desiderius's (and Pink Eye's) experimental approach, and the conclusions derived from it, is vastly superior to theorycrafting.
You fail statistics forever. Please repeat the class if you have taken it. Or ask Trashos to explain. He, at least, understands why this is wrong.
