Arrow Probability Chart

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jdixon41
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Arrow Probability Chart

Post by jdixon41 »

Hi everyone,

What follows is an image from an excel sheet that I made in order to compare the different arrows across different situations. I do not know the math that well, but I do love spreadsheets.

This is a snippet of the chart that shows what your glade guard might be up against. (green is good, red is bad)

https://www.dropbox.com/s/oc14ys0kgzxmm ... _chart.jpg

To the lefthand side is the base numbers.
HitRoll - the min die needed to hit
WoundRoll - the min die needed to wound
ArmorRoll - the Armor save of the target.
WardRoll - the wardsave of the opponent (in retrospect I shouldn't have included this but I filtered out the results of the full chart to only show 6+.)

Base Asrai - standard longbow str 3, -1 ap
Base Hagbane - str 3 poison, -1 ap
Base Trueflight - str 3, no toHit modifiers, -1 ap
Base Fire - str 3, +1 wound, -1 ap
Base Bodkin - str 3, -3 ap
Base Swiftshiver - str 3, -1 to hit, 2 shots

From there, as you go to the right of the chart, you'll see (-1) Hagbane, which means you have a -1 to hit modifier for Range, Cover or whatever. I took it out to (-5) modifier. so that would be something like, long range (-1), hard cover (-2), moved (-1), and something else (-1)

Enjoy!

EDITED:: Re-uploaded the chart because Domine Nox pointed out an error with the Hanbane calculations. If anyone sees any other errors I will fix them.

EDIT2:: Re-uploaded after brechttomme corrected my Swiftshiver calculations.
Last edited by jdixon41 on 15 May 2014, 00:02, edited 3 times in total.
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Re: Arrow Probability Chart

Post by Domine Nox »

I'm confused by 1 thing in this graph. I notice that the Hagsbane arrows get better the higher the enemy toughness. How do you figure? The numnbers should always go down as the T gets better. If I have poison shots and I shoot T3 vs poison shots vs T4 the T3 should be better because poison is still a 1 in 6, and so the arrows that are not poison should net a loss as it gets harder to wound, no?
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Re: Arrow Probability Chart

Post by jdixon41 »

Domine Nox wrote:I'm confused by 1 thing in this graph. I notice that the Hagsbane arrows get better the higher the enemy toughness. How do you figure? The numnbers should always go down as the T gets better. If I have poison shots and I shoot T3 vs poison shots vs T4 the T3 should be better because poison is still a 1 in 6, and so the arrows that are not poison should net a loss as it gets harder to wound, no?
This was the one that I had the most trouble with, I'm checking it now. I was up late doing it and I thought I had everything worked out; I changed the code to generate the table a couple of times so I might have messed this up.

EDIT::

Looks like I had left out some parenthesis when doing the Hagbane calculations. I'm getting the image put together again and I'll edit the original post.

EDIT2::

Chart fixed and re-uploaded in original post.

Thanks Domine for pointing out the errors.
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Re: Arrow Probability Chart

Post by Domine Nox »

There we go, that looks much better. When I first looked at it I was like "Wow, those are the numbers? Why take anything other than Hagsbane?" :lol:

Nice work though, much more professional than me calculating on a per unit basis each arrow out to figure out which I think is best.
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Re: Arrow Probability Chart

Post by Wooster Shooster »

Thank you! You have saved me a bunch of time making this myself!
I checked your numbers in a few cases and they were the same as what I got.

Awesome. Just awesome.
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Re: Arrow Probability Chart

Post by brechttomme »

First off, this is pretty cool! I've done a similar thing (http://asrai.org/viewtopic.php?f=43&t=26375), but it's a lot more unwieldy as it's literally just a chart showing everything at the same time, while you've made it filterable/sortable and everything. Well done!
jdixon41 wrote:if someone has a 1+ AS, even though they still fail on a 1, does our Armor piercing attacks only bring it up to 2+ effectively doing nothing?
Yep. That is literally the only advantage to having a 1+ Armour Save as opposed to a 2+. AP and S4 still get you a 2+ ASv.
jdixon41 wrote:wardsave of the opponent (in retrospect I shouldn't have included this but I filtered out the results of the full chart to only show 6+. This only changes the arrow choice in extreme corner case situations so can safely be ignored when deciding on arrows
Forgive me if I'm reading this wrong, but how can a Ward Save change the decision on what type of Enchanted Arrow to take? Other than Regeneration, a Ward Save has the same effect on every type, doesn't it? I.e. a 4+ Ward Save against Hagbane Tips should stop exactly half the Wounds, but it should stop half the Wounds on anything else too. So, now you'd be looking at every value divided by two. To me, it seems like that doesn't affect the relative position of which Arrows are the best.
jdixon41 wrote:Base Swiftshiver - str 3, -1 to hit, 2 shots (i did it something like this: Probability of wounding = 1 - ((1 - (toHit * toWound * toArmorSave * toWardSave)) ^ 2), statisticians, is this correct?)
Now, I'm not exactly a statistician (though I am an engineering student ;)), but this formula can't be right. If you take toHit, toWound, toArmorSave and toWardSave as all being somewhere between 2 and 6, then your probability of wounding would be between -224 (when all the above are 2) and -1677024 (when all the above are 6). Unless I'm reading it wrong that is. Maybe toHit is not the same as HitRoll? Are you using toHit as 2/3 in the case of a 3+ then?
If so - it is more logical; after all you can't go above 1 or below 0 with probability - then I think you are just making it difficult for yourself. I don't think that it needs anything being squared, but that could be me getting confused as well. The way I did it is:
To Hit: 2 shots each, so multiply the To Hit-probability per model by 2. If they are hitting on 3+ (which becomes 4+ because of the -1 penalty for Multiple Shots), they would normally hit 50% (0,5) of the time. Because of the two shots, that is now 100% (1 - okay, because of this multiplying by 2 you can actually go above 1 here, but not above 2).
To Wound: This is just the regular continuing of the chain. If they would wound on 3+ (against T2), that translates to wounding 2/3 of the time. So, the rolling total is 1*(2/3) = 0,67, or 67%.
Unsaved Wounds: Again, just the same thing. a natural 4+ Armour Save (which is modified to a 5+ because of AP) and a 6+ Ward Save, for example, would mean you are getting 2/3 of your Wounds through their Armour, and 5/6 of those through their Ward. So, the final total is now: 1*(2/3)*(2/3)*(5/6) = 0,37, or 37%.
It seems your result comes close to that (33,6%) but is off a little bit. Unless, that is, I'm doing it wrong of course! :ninja:
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Re: Arrow Probability Chart

Post by jdixon41 »

brechttomme wrote:First off, this is pretty cool! I've done a similar thing (http://asrai.org/viewtopic.php?f=43&t=26375), but it's a lot more unwieldy as it's literally just a chart showing everything at the same time, while you've made it filterable/sortable and everything. Well done!
jdixon41 wrote:if someone has a 1+ AS, even though they still fail on a 1, does our Armor piercing attacks only bring it up to 2+ effectively doing nothing?
Yep. That is literally the only advantage to having a 1+ Armour Save as opposed to a 2+. AP and S4 still get you a 2+ ASv.
jdixon41 wrote:wardsave of the opponent (in retrospect I shouldn't have included this but I filtered out the results of the full chart to only show 6+. This only changes the arrow choice in extreme corner case situations so can safely be ignored when deciding on arrows
Forgive me if I'm reading this wrong, but how can a Ward Save change the decision on what type of Enchanted Arrow to take? Other than Regeneration, a Ward Save has the same effect on every type, doesn't it? I.e. a 4+ Ward Save against Hagbane Tips should stop exactly half the Wounds, but it should stop half the Wounds on anything else too. So, now you'd be looking at every value divided by two. To me, it seems like that doesn't affect the relative position of which Arrows are the best.
jdixon41 wrote:Base Swiftshiver - str 3, -1 to hit, 2 shots (i did it something like this: Probability of wounding = 1 - ((1 - (toHit * toWound * toArmorSave * toWardSave)) ^ 2), statisticians, is this correct?)
Now, I'm not exactly a statistician (though I am an engineering student ;)), but this formula can't be right. If you take toHit, toWound, toArmorSave and toWardSave as all being somewhere between 2 and 6, then your probability of wounding would be between -224 (when all the above are 2) and -1677024 (when all the above are 6). Unless I'm reading it wrong that is. Maybe toHit is not the same as HitRoll? Are you using toHit as 2/3 in the case of a 3+ then?
If so - it is more logical; after all you can't go above 1 or below 0 with probability - then I think you are just making it difficult for yourself. I don't think that it needs anything being squared, but that could be me getting confused as well. The way I did it is:
To Hit: 2 shots each, so multiply the To Hit-probability per model by 2. If they are hitting on 3+ (which becomes 4+ because of the -1 penalty for Multiple Shots), they would normally hit 50% (0,5) of the time. Because of the two shots, that is now 100% (1 - okay, because of this multiplying by 2 you can actually go above 1 here, but not above 2).
To Wound: This is just the regular continuing of the chain. If they would wound on 3+ (against T2), that translates to wounding 2/3 of the time. So, the rolling total is 1*(2/3) = 0,67, or 67%.
Unsaved Wounds: Again, just the same thing. a natural 4+ Armour Save (which is modified to a 5+ because of AP) and a 6+ Ward Save, for example, would mean you are getting 2/3 of your Wounds through their Armour, and 5/6 of those through their Ward. So, the final total is now: 1*(2/3)*(2/3)*(5/6) = 0,37, or 37%.
It seems your result comes close to that (33,6%) but is off a little bit. Unless, that is, I'm doing it wrong of course! :ninja:
I'll take a row and walk through how I did the math.

HitRoll: 3+
WoundRoll: 4+
ArmorRoll: 5+
WardRoll: 6+

For the HitRoll of 3+ I modified it to 4+ so there is a 0.5 chance to hit.
For the WoundRoll of 4+ i left it alone so there is a 0.5 chance to wound.
for the ArmorRoll it get's AP so it is modified to 6+ so a (5/6) chance to wound through that
For the WardRoll I have another (5/6) chance to get through that.

Multiply all that out and you get : (0.5 * 0.5 * .833 * .833) = 0.1736

Since there is two shots, and I want to know the probability that given two consecutive trials, that at least one will hit; I looked that up.

My understanding of that is you take the chance for it not to happen, raised to the power of the number of events, and that gives you the probability that it will not happen in any trial. Then when you want to know that it will happen at least one time, you take 1 and minus that result.

So 0.1736 is the probability that one shot will hit, inverse it (1-0.1736), and then raise it by the number of multishots (2), and i get ((1-0.1736))^2. That is the chance that given two shots I'll miss both. Inverse it again and 1-((1-0.1736))^2. That equals 0.317, which is what the chart shows, and should be the chance to wound with the swiftshiver arrows.

The reference I found for the probability of two trials said to figure out the chance for it not to happen. That's why I did it first. I'm not very strong at statistics, so I was doing my best to apply the rule into my code that generated the chart.

Please correct me if this is wrong and I'll fix the chart.


EDIT2::

Removed the nonsense I wrote about swiftshiver and ward saves since I was wrong. :)

The original post is updated with the new chart that fixes the swiftshiver issue.
Last edited by jdixon41 on 15 May 2014, 00:08, edited 1 time in total.
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Re: Arrow Probability Chart

Post by brechttomme »

I actually don't think that's correct. Now, maybe you shouldn't trust me all that much as I did fail the course last year, but here is my explanation:

The first part:
jdixon41 wrote: HitRoll: 3+
WoundRoll: 4+
ArmorRoll: 5+
WardRoll: 6+

For the HitRoll of 3+ I modified it to 4+ so there is a 0.5 chance to hit.
For the WoundRoll of 4+ i left it alone so there is a 0.5 chance to wound.
for the ArmorRoll it get's AP so it is modified to 6+ so a (5/6) chance to wound through that
For the WardRoll I have another (5/6) chance to get through that.

Multiply all that out and you get : (0.5 * 0.5 * .833 * .833) = 0.1736
is most definitely correct. What you have done here is taken the archer as if it had one shot, but with an additional -1 To Hit-modifier. In-game, this would look like you taking a single die and trying to cause an Unsaved Wound. With that single die, you have a 17,36% chance of actually causing the Unsaved Wound. What you actually have to do, though, is take 2 dice instead of one, because each archer gets 2 shots. So, double the dice means double the chances. Applying this to the result nets you: 2*0,1736 = 0,3472 = 34,72%. NOT 0,317.

The reason I think what you are doing is wrong is the following: with two dice there are 4 possibilities when rolling To Hit.
- die 1 hits, die 2 hits
- die 1 hits, die 2 misses
- die 1 misses, die 2 hits
- die 1 misses, die 2 misses
What you have calculated is the chance of any of the first three happening. This means you are seeing all of the first three results as equal, while the first would obviously have a double probability of actually causing an Unsaved Wound.

What I do is look at 1 die, so two possibilities:
- die 1 hits
- die 1 misses
I now calculate the chance of the first one happening, and then double it. Consider the game again. Would it be fair to roll each die separately, one at a time? It would go slower, sure, but your opponent can't challenge you on that. Rolling each die separately is what I have calculated. If Swiftshiver Shards actually gave the Multiple Shots (500) rule (imagine that, huh? :p), I'd just roll 500 dice one at a time. The chance to Hit (and Wound, etc.) is the same on all of them, so I just take the result and multiply it by 500 to get the actual amount of Unsaved Wounds I should do.

I hope this clarifies the issue a little. It's often easy to overcomplicate things when dealing with statistics. I've experienced it more than enough myself!

EDIT: This is also why you think a Ward Save changes the results for Swiftshiver Shards. Your exponentiation (is that a word? I'm not actually a native English speaker...) in the formula means a Ward Save doesn't affect each Arrow type equally. If you think about it in single-die-situations again, each die has the same chance to get through their Ward Save, regardless of how many dice you throw in total. For example - with random numbers - if Hagbane Tips had a 50% chance of causing an Unsaved Wound before Ward Saves, and you were targeting something with a 4+ Ward Save, the chance you'd get your Wound through with the one Hagbane Tips is now 25% = 50%/2. If Swiftshiver Shards had a 40% chance of causing an Unsaved Wound before Ward Saves (both dice), then each die would have a 1/2 chance of getting through the Ward Save and thus the total chance of getting your Wound through with the Swiftshiver Shards is now 20% = 40%/2. So, even though you start off rolling a different amount of dice, the end result before and after Ward Saves is the same: division by 2. That means Ward Saves have absolutely no effect on the choice of what Arrows are the best in a given situation, as any and all results would be divided by 2 against a 4+ Ward Save. If Hagbane Tips were better than Swiftshiver Shards before, they are still better when both results are divided by two.

God, it's so hard explaining statistics to make it seem understandable... :paranoid:
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Re: Arrow Probability Chart

Post by jdixon41 »

brechttomme wrote:I actually don't think that's correct. Now, maybe you shouldn't trust me all that much as I did fail the course last year, but here is my explanation:

The first part:
jdixon41 wrote: HitRoll: 3+
WoundRoll: 4+
ArmorRoll: 5+
WardRoll: 6+

For the HitRoll of 3+ I modified it to 4+ so there is a 0.5 chance to hit.
For the WoundRoll of 4+ i left it alone so there is a 0.5 chance to wound.
for the ArmorRoll it get's AP so it is modified to 6+ so a (5/6) chance to wound through that
For the WardRoll I have another (5/6) chance to get through that.

Multiply all that out and you get : (0.5 * 0.5 * .833 * .833) = 0.1736
is most definitely correct. What you have done here is taken the archer as if it had one shot, but with an additional -1 To Hit-modifier. In-game, this would look like you taking a single die and trying to cause an Unsaved Wound. With that single die, you have a 17,36% chance of actually causing the Unsaved Wound. What you actually have to do, though, is take 2 dice instead of one, because each archer gets 2 shots. So, double the dice means double the chances. Applying this to the result nets you: 2*0,1736 = 0,3472 = 34,72%. NOT 0,317.

The reason I think what you are doing is wrong is the following: with two dice there are 4 possibilities when rolling To Hit.
- die 1 hits, die 2 hits
- die 1 hits, die 2 misses
- die 1 misses, die 2 hits
- die 1 misses, die 2 misses
What you have calculated is the chance of any of the first three happening. This means you are seeing all of the first three results as equal, while the first would obviously have a double probability of actually causing an Unsaved Wound.

What I do is look at 1 die, so two possibilities:
- die 1 hits
- die 1 misses
I now calculate the chance of the first one happening, and then double it. Consider the game again. Would it be fair to roll each die separately, one at a time? It would go slower, sure, but your opponent can't challenge you on that. Rolling each die separately is what I have calculated. If Swiftshiver Shards actually gave the Multiple Shots (500) rule (imagine that, huh? :p), I'd just roll 500 dice one at a time. The chance to Hit (and Wound, etc.) is the same on all of them, so I just take the result and multiply it by 500 to get the actual amount of Unsaved Wounds I should do.

I hope this clarifies the issue a little. It's often easy to overcomplicate things when dealing with statistics. I've experienced it more than enough myself!
I was wrong. I ran my problem by a coworker and it looks like he's saying the same thing you are saying. I'll get that chart updated as soon as I can and edit the original post.

EDIT: original post updated with new chart. I really appreciate the help from the community.
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Re: Arrow Probability Chart

Post by jdixon41 »

brechttomme wrote:I actually don't think that's correct. Now, maybe you shouldn't trust me all that much as I did fail the course last year, but here is my explanation:

The first part:
jdixon41 wrote: HitRoll: 3+
WoundRoll: 4+
ArmorRoll: 5+
WardRoll: 6+

For the HitRoll of 3+ I modified it to 4+ so there is a 0.5 chance to hit.
For the WoundRoll of 4+ i left it alone so there is a 0.5 chance to wound.
for the ArmorRoll it get's AP so it is modified to 6+ so a (5/6) chance to wound through that
For the WardRoll I have another (5/6) chance to get through that.

Multiply all that out and you get : (0.5 * 0.5 * .833 * .833) = 0.1736
is most definitely correct. What you have done here is taken the archer as if it had one shot, but with an additional -1 To Hit-modifier. In-game, this would look like you taking a single die and trying to cause an Unsaved Wound. With that single die, you have a 17,36% chance of actually causing the Unsaved Wound. What you actually have to do, though, is take 2 dice instead of one, because each archer gets 2 shots. So, double the dice means double the chances. Applying this to the result nets you: 2*0,1736 = 0,3472 = 34,72%. NOT 0,317.

The reason I think what you are doing is wrong is the following: with two dice there are 4 possibilities when rolling To Hit.
- die 1 hits, die 2 hits
- die 1 hits, die 2 misses
- die 1 misses, die 2 hits
- die 1 misses, die 2 misses
What you have calculated is the chance of any of the first three happening. This means you are seeing all of the first three results as equal, while the first would obviously have a double probability of actually causing an Unsaved Wound.

What I do is look at 1 die, so two possibilities:
- die 1 hits
- die 1 misses
I now calculate the chance of the first one happening, and then double it. Consider the game again. Would it be fair to roll each die separately, one at a time? It would go slower, sure, but your opponent can't challenge you on that. Rolling each die separately is what I have calculated. If Swiftshiver Shards actually gave the Multiple Shots (500) rule (imagine that, huh? :p), I'd just roll 500 dice one at a time. The chance to Hit (and Wound, etc.) is the same on all of them, so I just take the result and multiply it by 500 to get the actual amount of Unsaved Wounds I should do.

I hope this clarifies the issue a little. It's often easy to overcomplicate things when dealing with statistics. I've experienced it more than enough myself!

EDIT: This is also why you think a Ward Save changes the results for Swiftshiver Shards. Your exponentiation (is that a word? I'm not actually a native English speaker...) in the formula means a Ward Save doesn't affect each Arrow type equally. If you think about it in single-die-situations again, each die has the same chance to get through their Ward Save, regardless of how many dice you throw in total. For example - with random numbers - if Hagbane Tips had a 50% chance of causing an Unsaved Wound before Ward Saves, and you were targeting something with a 4+ Ward Save, the chance you'd get your Wound through with the one Hagbane Tips is now 25% = 50%/2. If Swiftshiver Shards had a 40% chance of causing an Unsaved Wound before Ward Saves (both dice), then each die would have a 1/2 chance of getting through the Ward Save and thus the total chance of getting your Wound through with the Swiftshiver Shards is now 20% = 40%/2. So, even though you start off rolling a different amount of dice, the end result before and after Ward Saves is the same: division by 2. That means Ward Saves have absolutely no effect on the choice of what Arrows are the best in a given situation, as any and all results would be divided by 2 against a 4+ Ward Save. If Hagbane Tips were better than Swiftshiver Shards before, they are still better when both results are divided by two.

God, it's so hard explaining statistics to make it seem understandable... :paranoid:

So I wrote a little simulation really fast with the previous example, (3+ hit, 4+ wound, 5+ armor, 6+ ward) with the swiftshiver.

What I found was consistent; 0.1736 is the average unsaved wound result, and when I doubled the trials it's still the average unsaved wound result, but then it hit me, I get double the amount of wounds. So it's not really a probability chart as much as an unsaved wound chart right? It was bothering me thinking that if the swiftshiver arrows were multiple shot (100), that my "probability" would be over 1. That didn't make sense, but now that I see it's unsaved wounds, that makes sense.

Is there a way though to not show Unsaved Wounds, but the probability that you get at least 1 unsaved wound? (It shouldn't be able to go above 1 even if you have multiple shots (100). Even though the chance is very slim to miss all 100 shots, given enough trials you could miss them all; probabilities can't go over 1 right?)
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Re: Arrow Probability Chart

Post by Balthorn »

I have a question about these arrows that might alter the calculations. From my reading last night, I noticed that each of the arrows listed is given the "Armour Piercing" rule, which the asrai longbows also have. However, the Trueflight arrows do not have AP listed in their rules. Does that mean that the magic arrows override the rules for the asrai longbows, and as such Trueflight arrows DON'T have AP?

I've been looking around the net but I haven't found anyone bring this question up. So I thought I'd mention it here. Thoughts?
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Re: Arrow Probability Chart

Post by brechttomme »

jdixon41 wrote: Is there a way though to not show Unsaved Wounds, but the probability that you get at least 1 unsaved wound? (It shouldn't be able to go above 1 even if you have multiple shots (100). Even though the chance is very slim to miss all 100 shots, given enough trials you could miss them all; probabilities can't go over 1 right?)
You're right that probabilities can't go over one and this is indeed an "Unsaved Wounds chart". I imagine there is also a way to get the probability of causing at least one Unsaved Wound, but I'd have to look up how to do that. It doesn't seem like it would be very useful though. With the "Unsaved Wounds chart", you can predict the amount of damage your unit is going to do when shooting at an enemy, and you can already compare each Arrow Type to see which is the best in a given situation.
Balthorn wrote:However, the Trueflight arrows do not have AP listed in their rules. Does that mean that the magic arrows override the rules for the asrai longbows, and as such Trueflight arrows DON'T have AP?
If the Trueflight Arrows didn't have AP then they would indeed override the rules for the Asrai Longbow and not have AP. I just looked in my book though and it does clearly state Trueflight Arrows have AP. Do you have an English version? Maybe there is just a mistake in your book?
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Re: Arrow Probability Chart

Post by Balthorn »

brechttomme wrote:
Balthorn wrote:However, the Trueflight arrows do not have AP listed in their rules. Does that mean that the magic arrows override the rules for the asrai longbows, and as such Trueflight arrows DON'T have AP?
If the Trueflight Arrows didn't have AP then they would indeed override the rules for the Asrai Longbow and not have AP. I just looked in my book though and it does clearly state Trueflight Arrows have AP. Do you have an English version? Maybe there is just a mistake in your book?
I don't actually have the book with me and I was reading it rather early this morning (having only owned it one day), so it seems I misremembered. I guess that's why no-one has been bringing it up anywhere else :D Thanks for the correction.

Edit: I see what's happened now. Arcane Bodkins don't have AP listed, since they essentially have APx3. I somehow mixed that up later on, hehe
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