I'm trying to figure out the impact of a reduced threat range with more attacks in respect of a larger one but with less attacks, but I'm
horribly bad at math.
Case 1: rapier with a 15-20 threat range and two iterative attacks.
Case 2: thrown dagger with a 17-20 threat range and three attacks, with a -2 penalty, via Rapid shot.
What are the odds of critting in these cases?
It depends on your attack bonus vs. the enemies' AC.
If you have a +5 ability mod (probably Dex), +1 enhancement, +6 BAB and the enemy has 20 AC then in the first case you have a 30% chance of threatening a crit with each attack and 65% chance of confirming with the first, 40% chance of confirming with the second. Expected crits = 0.3*0.65 + 0.3*0.4 = 0.315
In the second (same assumptions) you have a 20% chance of threatening a crit with each, 55% confirm with the first 2 and 30% confirm with the last. 2*0.2*0.55 + 0.2*0.3 = 0.28
Do the math with 16 AC & the same bonuses and it comes to .435 vs. .4, with 24 AC it comes to .215 vs. .15
Of course complicating factors would be that the thrown daggers will probably encounter range penalties and OTOH point blank shot will likely add +1 attack, and enchanting multiple daggers gets prohibitively expensive.
enchanting multiple daggers gets prohibitively expensive.
Ricochet toss should make up for that.
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Avr, that's not how multiple probabilities combine. If you simply add the probability of critting on each attack over multiple attacks, then with enough attacks you will get a probability higher than 100%, which is impossible. Probability needs to saturate with the increase in number of attacks instead. In your case, you also have to divide by the number of attacks, essentially averaging the probabilities.
Assuming 15 on the d20 is enough to hit, here is the general formula for the probability of an attack being a crit:
p = tr/20 * (21 - AC + ab)/20
where
tr = threat range (1 for a weapon that crits only on 20, 2 for a 19-20 crit weapon, 6 for a 15-20 weapon)
ab = total attack bonus (including BAB, ability modifiers, other bonuses and maluses).
In your case, ab changes between attacks (one attack has a -5 penalty due to being an iterative attack), so you need to calculate p independently for each attack (or incorporate the penalty into the formula, but I think this is easier).
After you have calculated p for each attack (for example, if you have 3 attacks, you will have p1, p2 and p3), you can calculate the probability of critting per roll:
P_roll = average( p1, p2, p3, ...)
If all your attacks have the same attack bonus, then the probability you'll crit each time you roll is the same and does not depend on the number of attacks.
However, this does not tell you how multiple attacks combine to increase your crit chance over a full-attack. In fact, the more attacks you have, the higher the probability you will crit at least once will become. The probability of critting at lest once per round instead is given by:
P_round = 1 - ( (1 - p1) * (1 - p2) * (1 - p3) * ... )
In your case, since you are at least at level 6, let's choose AC = 20 as our target AC, and ab = 6 (BAB) + 5 (Dex) + 1 (masterwork weapon) = 12.
For the rapier, tr = 6, and we have 2 attacks, one made at ab1 = 12, and the other made at ab2 = 12 - 5 = 7
p1 = 6/20 * (21 - 20 + 12)/20 = 0.195 = 19.5%
p2 = 6/20 * (21 - 20 + 7)/20 = 0.12 = 12%
On average, over 100 rounds, you will have 19.5 + 12 = 31.5 crits, with an average P_roll = ( 19.5% + 12% )/2 = 15.75% of critting per roll.
Your overall probability of critting at least once per round with a rapier is:
P_round = 1 - ( (1 - 0.195) * (1 - 0.12) ) = 0.292 = 29.2%
This means that, over 100 rounds, you will have critted at least once during 29 of them.
~ ~ ~
For daggers, tr = 4, and we have 3 attacks, two made at ab1 = ab2 = 12 - 2 = 10, and the other made at ab3 = 12 - 5 -2 = 5
p1 = 4/20 * (21 - 20 + 10)/20 = 0.11 = 11%
p2 = p1 = 0.11 = 11%
p3 = 4/20 * (21 - 20 + 5)/20 = 0.06 = 6%
The average per roll crit probability is P_roll = ( 11% + 11% + 6% )/3 = 9.3%
The overall probability of critting at least once per round with daggers is:
P_round = 1 - ( (1 - 0.11) * (1 - 0.11) * (1 - 0.06) ) = 0.289 = 25.5%
~ ~ ~
In both cases, under these assumptions of AC and ab, rapier is better. If you are interested in recovering as much panache as possible, the rapier gives you a higher crit chance per roll (of course, since its base crit chance is higher). Same if you look at recovering at least one panache point per round, as the extra attack is not enough to compensate for the lower threat-range and attack bonus.
Let's say now that you also had Two Weapon Fighting and Point Blank Shot. You would now do 4 dagger attacks with a net attack bonus of 9/9/9/4. In this case:
P_roll = 8.75%
P_round = 30.75%
So it's even less probable for you to crit on any given roll (again, this is obvious, given the lower attack bonus, and so the lower probability of confirming the crit), but the sheer number of attacks compensates for the lower per-roll crit chance, giving you an overall chance of critting at least once per full-attack higher than the rapier.
That being said, as Avr suggested, you should, however, include also all specific bonuses and maluses coming from weapon-specific feats and effects applying to your character to have a more sensible estimate.
Also, the formula breaks when rolling within the threat range is not enough to hit, but I think for now this is good enough.
EDIT. Some copy-paste mistakes.