TWF or increased threat range: What's more likely to deliver a critical hit?

Question

In Pathfinder, what is more likely to deliver a critical hit:

• Attacking once per round with an Elven Curve Blade (critical threat range 18-20)

or

• Attacking twice per round with Two Weapon Fighting (TWF) and two light blades, such as two short swords (critical threat range 19-20) but at -2 to hit

I am working on an Eldritch Knight (EK) build. I'd like to understand what is more likely to deliver that Spell Critical in a full attack: attacking twice per round at -2 on a threat range of 19-20, or attacking once per round on a threat range of 18-20.

I am looking for a way to graph these two probabilities against a range of Armor Class values.

I understand that the TWF is only applicable in a Full Attack. For the purposes of this question I'm only interested in a Full Attack.

I understand that there are other ways to increase a threat range or attack more often. For the purposes of this question I'm not applying haste nor keen edge nor strength bonuses nor any other bonus to attack. I'm also not going to use a kukri (light weapon, threat range 18-20) in this comparison.

If it makes a difference, the EK is Fighter 1, Wizard 5, EK 10. That would make a Base Attack Bonus of +13 (1 + 2 + 10).

EDIT: The EK has the Two Weapon Fighting feat and both weapons are light. The modifier is -2 for each attack: primary hand and off hand. The EK does not have the Improved Two-Weapon Fighting feat.

I'm assuming that the opponent would have an AC of 29, so the additional attacks at +8 and +3 would only hit on a roll of 20 and therefore not make a difference to the probabilities. I'd take two twenties in a row but I understand (1/20)^2 is only 0.0025 likely to occur.

I'm interested in knowing how the probability changes over a range of Armor Class values below and above 29, but to help narrow the question I've picked one AC.

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Answer 1

Assuming I got the mechanics down correctly, here's an AnyDice script to model them:

function: hit ROLL:n plus BONUS:n ac AC:n {
  if ROLL =  1 { result: 0 }
  if ROLL = 20 { result: 1 }
  result: ROLL + BONUS >= AC
}

function: crit ROLL:n plus BONUS:n ac AC:n range THREAT:s {
  if [hit ROLL plus BONUS ac AC] & ROLL = THREAT {
    result: [hit d20 plus BONUS ac AC]  \ confirm? \
  }
  result: 0
}

BAB: 1 + 2 + 10
loop AC over {20..40} {
  ECB: [crit d20 plus BAB ac AC range {18..20}]
  output ECB named "Elven Curved Blade vs AC [AC]"
}
loop AC over {20..40} {
  TWF: [crit d20 plus BAB-2 ac AC range {19..20}]
  output 2dTWF > 0 named "Two-weapon fighting vs AC [AC]"
}

A few notes about this code:

• The function hit ROLL plus BONUS ac AC returns 1 if the roll hits and 0 if it misses. When called with d20 as ROLL, AnyDice automatically calls it for every possible roll of the d20 and returns a biased die that rolls 1 with the probability of hitting the target and 0 with the probability of missing.

• The function crit ROLL plus BONUS ac AC range THREAT returns either 0 (if the attack misses or fails to be a crit threat) or biased die representing the outcome (0 or 1) of the crit confirmation roll. Again, when called with d20 as ROLL, AnyDice automatically evaluates it for all possible rolls of the d20 and combines the results into a single biased die that rolls 1 with a probability equal to that of getting a confirmed crit against the target.

• In the first loop, ECB is thus a biased die representing the probability of striking a confirmed crit with an Elven Curved Blade.

• In the second loop (which I split off from the first to keep the output a bit more readable), TWF is a biased die rolling 1 with a probability equal to that of getting a confirmed critical with one of the two attacks, and 0 otherwise. Thus, rolling 2dTWF yields either 0, 1 or 2 depending on how many of your two attacks crit, and 2dTWF > 0 just maps the outcomes 1 and 2 both to 1 (since you can't use Spell Critical more than once per turn anyway).

Looking at the numerical results, we can see that:

• For AC ≤ 25, Two Weapon Fighting is slightly better than the Elven Curved Blade, with the difference being more pronounced at lower AC. For 26 ≤ AC ≤ 32, the Elven Curved Blade is slightly better than Two Weapon Fighting.

• For AC = 29, the results from this script indeed match those from your own answer (after correcting the minor probability addition error pointed out by Thanuir). Specifically, like you, I get a 3.75% chance of critting with the Elven Curved Blade, and a 2.9775% ≈ 3% chance of critting at least once with Two-Weapon Fighting.

• For AC ≥ 33, you'll only hit (and crit) on a natural 20 either way. Thus, Two Weapon Fighting is always better in this case, since you have two chances of rolling a nat 20 (and confirming it with another nat 20) as opposed to just one with the Elven Curved Blade. This gives you an overall change of 0.499375% of getting Spell Critical off with TWF, as opposed to just 0.25% with ECB. Of course, neither of those are odds you really should be betting on.

BTW, a quick way to compare the two approaches is to replace the two loops in the script above with this single loop:

loop AC over {20..40} {
  ECB: [crit d20 plus BAB ac AC range {18..20}]
  TWF: [crit d20 plus BAB-2 ac AC range {19..20}]
  output ECB - (2dTWF > 0) named "ECB - TWF vs AC [AC]"
}

and then look at the "mean" graph in the summary view. If the mean difference is positive, then ECB is better than TWF (when it comes to scoring at least one crit per turn, anyway); if it's negative, it's worse.

Answer 2

With a BAB of 13 against an AC of 29, the die has to roll 16 or higher to hit at all. This is also the target to confirm a critical threat without any other modifiers.

The short sword threatens a critical 10% of the time, on 19 or 20.

To confirm the critical at -2 to hit, the dice have to roll 18 or higher, which is 0.15.

So that's a 0.10 chance to threaten a crit followed by a 0.15 chance to confirm the critical. 0.10 * 0.15 = 0.015 with one attack.

There are two attacks per round, so 0.015 + 0.015 = 0.03 or 3% chance each round in TWF Full Attack to confirm a critical.

The Elven Curve Blade threatens a critical 15% of the time, on 18, 19 or 20. To confirm, the dice against AC 29 has to roll 16. That's 0.15 to score a critical threat, followed by 16 or better to confirm, which is 0.25.

So 0.15 * 0.25 = 0.0375 or 3.75% chance.

So, fraction of a percentage point more likely to hit with the single attack threatening on 18-20.

This makes me think that the success of scoring (confirming) a critical hit depends more on the armor class of the opponent than on (a) the rate of attacks or (b) the threat range of the weapon, all else being equal. A lower AC increases the chance of confirming the critical. A higher AC reduces the value of a large threat range.

Answer 3

With the specific criteria you listed, the Elven Curve Blade option is better, at 4.23% to crit, against a 3.46% chance to crit with the TWH option. I assumed you do not have Improved Two-Weapon Fighting.

For each attack you calculate the chance to crit as the product of the chance to threaten a critical and the chance to confirm. Since you only care about a single crit, what you really want to know is the chance you don't crit on any attacks. You find this by subtracting your crit chance from 1.

So for the Elven Curve Blade:

1 Full BaB: 3.75% chance to crit, 96.25% not

1 -5 Iterative: 0.25% crit, 99.75% not

1 -10 Iterative: 0.25% crit, 99.75% not

The product of the chances not to crit is 95.77%, giving you a 4.23% chance to crit one or more times.

For the TWH:

1 Main Hand Full BaB -2: 1.5% crit, 98.5% not

1 Off Hand Full BaB -2: 1.5% crit, 98.5% not

1 -5 Iterative: 0.25% crit, 99.75% not

1 -10 Iterative: 0.25% crit, 99.75% not

The product of the chances not to crit is 96.54%, giving you a 3.46% chance to crit one or more times.