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WhosDaDungeonMaster
Guest
Well Ovinomancer, I finally have time to address your concerns, so here we go:
Thanks for introduction, but I think you have some misunderstandings. You do take the initial fumble attack roll into account (sorry about that, but as I wrote I glanced through your post before, and missed it). Since I am not certain what bonus you are using to the check (you seem to jump around a bit), I can't check your probabilities (they seem off..., which I'll explain later). Also, instead of arriving at the cumulative probability by adding up the probabilities of 1 success to N successes (depending on sample size of N), it is much easier to use the probability of zero successes and subtract that from 1 to find the cumulative probability of 1 or more successes.
Is that really how you are doing your calculations? Multiplying 64.2% by 20%? Hmm... That isn't really how it works. You can't look at the binomial distribution "at least" result and then multiple that by the straight probability of failing the DC 10 check with the +5 bonus.
You need to look at the probability of success (droping the weapon) as a conditional probability since the fumble check has to be a given.
P(A) = 0.05 (natural 1 on attack roll)
P(B|A) = 0.20 (fails DC 10 check to drop weapon)
P(A and B) = P(A) * P(B|A) = 0.05 * 0.20 = 0.01 is the probability of a dropped weapon. Now, use the binomial distrubtion with that as the probability of success (again, dropped weapon). So let's go with a binomial distribution with success probability of 0.01 and a sample size of 20 trials (attacks). Using the Excel binomial distribution function, BINOM.DIST(0,20,0.01,FALSE) with parameters of 0 successes (no dropped weapon), 20 trials (attacks), probability of success (dropped weapon), and non-cumulative because we want the probability of exactly 0 successes (the mass function), not the cumulative distribution function, we arrive with 81.8%. So, there is an 81.8% chance the character will never drop their weapon in 20 attacks. Thus, if we subtract that from 1, there is about an 18.2% chance they will drop their weapon once or more in 20 attacks (most likely, only once, btw). The probability of falling prone on any one attack is 0.002 (0.01 * another 4/20), and the chance of falling prone over 20 attacks at least once is roughly 3.9%. Higher than your numbers, but still, as you say, at lower levels that holds okay. I mean, this is over 20 attacks and 18% and 4% are hardly close to garuantees. So, what of the higher levels?
Using your example at 11th level (only increase in prof, so +6 total, but with 60 attacks instead of 20). The probabilities are actually: natural 1: 0.9539 (95.4%), drop weapon: 0.3635 (36.4%), fall down: 0.653 (6.5%). So, with only a +1 boost to the check but 40 more attacks, obviously the probabilities will go up significantly.
The dual wielder with 80 attacks actually has numbers of natural 1: 0.9835 (98.4%), drop: 45.2%, fall 8.6%. Again, adding more attacks without increasing the check bonus will always result in more drops and falls since you are attacking more. So, if that is the case, things will never get better.
Once again, I am not following your numbers as the 11th level fighter with +6 to the check has a 28.9% chance to drop their weapon once in 60 attacks, and less than 7.5% chance of dropping their weapon two or more times over 60 attacks. Increasing the number of attacks to 80, changes the numbers to 33.1% for exactly one drop and about 12.1% for two or more drops. But, as I said earlier, when you increase the number of trials, the likelihood of dropping increases since you have more attacks, so no surprise there. The chances of them falling down at least twice is less than 0.22% (well under 1%) with 60 attacks, and less than 0.38% (again, well under 1%) even with 80 attacks. Your numbers seem way off here. In both cases, however, the peaks of the distributions are still at zero drops and zero falls.
The shifting effect depends on probability of success (if it is greater than 0.5, it happens more quickly, but our probabilities are definitely less than 0.5). If the example above with the +6 check, the number of trials has to equal 133 (more than twice the base 60 attacks), before the peak shifts and you are more likely (36.9%) to drop once, than not at all (36.7%). The trials has to double (266 attacks) before two drops are the most likely outcome. So, with lower probabilities, you need a lot more trials before what you describe happens.
Now, as I have already said, and I will repeat for clarity: when you increase the number of trials, you increase the likelihood of 1 or more successes (i.e. drops and falls), as expected, but the peaks of the distributions remain at 0 successes unless your trial size increases drastically.
Next, it was your jump to DC 15 that I missed also when glancing through your post initially, that was why I thought maybe your numbers didn't include the initial fumble. I apologize for that oversight, but why would you assume that? My groups have never used them because they make the game "harder" (after all, it starts with only a 1 in 20 occurrence), but because it offers an element of suspense when it happens. Maybe the reasoning is to make it harder in your experience, but not in mine.
Let's examine the case for the Level 20 fighter making 80 attacks a day (using DC 10, thank you) with a +8 bonus (Dex +2, prof +6), failing only with a 1 on the drop weapon roll and the fall prone roll (from your numbers it looks like you might have still been using DC 15?).
Probabilities are natural 1: 0.05, drop weapon: 0.0025 (1/20 * 1/20), fall prone: 0.000125 (1/20 * 1/20 * 1/20).
Drop weapon on one attack or more (assuming this does NOT interrupt further attacks that round, which would decrease the chances slightly but is more complicated and does not have enough impact to seriously change the results) on 80 attack rolls: 18.15% (FYI, the probability of exactly 1 drop is most of this, 16.41%, two drops is only 1.62%, and it decreases drastically after that, so don't let the wording of "one or more" scare anyone.)
Fall prone after dropping weapon (same assumption): 1.00%
They will drop their weapon most likely once about 1 in 4 or 5 days and fall down most likely once about 1 in 100 days. Given how many attacks are being made, those odds don't seem overly harsh IMO. True, this is the best case scenario, but having played with such rules for decades, I have never seem them happen so frequently that myself or my players wanted to alter them or remove them completely. If the concept doesn't suit your sensibilities, then don't use them. The OP wanted to add tables (also fine), but this system works well for my group.
The game mechanic in use is "drop weapon" and "fall prone", which on the surface might sound ridiculous to some, but the "in story" effect can be brought about in a myriad of ways. Your attack was overextended and your foe's parry knocked your weapon loose, resulting in your dropping it. You were not able to recover in time from your off balance maneuver, and slipped on loose gravel, sending you to the ground. It doesn't literally have to mean "Oops, I was so inept and/or clumsy with my Dex 16 that my weapon just fell out of my hand and I landed on my butt! I am so ashamed!"
Anyway, for critical fumbles for spell casters and other ranged attacks? Well, I rule the ranged attack (usually a cantrip, arrow, etc.) might hit another player if they are in melee against the target since friendly fire into a melee can definitely have harmful consequences! The bowstring might snap instead of the player falling down, requiring them to either take the time to restring it or switch weapons and join the melee. A caster's spell component pouch might be lose in the dead leaves at their feet, etc. depending on the situation. I can rule the effect however matches the current scenario--but something negative happens.
For anyone who doesn't like the Dex check (Dex is overly used anyway), a simple replacement is you make a check using whatever bonuses you had to the attack roll in the first place. Suppose you had a magic weapon? Well, the bonus means you are less likely to drop it. A spell giving you an attack bonus? That spell will help with your drop and fall checks as well. You can adjust the DC like Ovinomancer if you want it harder to make. Rule 1's on the following checks always fail, no matter what bonuses you get to the roll. So on and so on, there are any number of ways to adjust critical fumbles rules if you want to use them. If not, then don't. Simple.
But for us, critical fumbles can be a fun part of the game when they happen. Other players laugh when a characters drops their weapon (not NEARLY the punishment it was in earlier additions as picking up a dropped weapon is part of a normal move or action now) or when the monsters do (they laugh even harder and love they might get advantage with the target prone!). Dropping the weapon has, at worse, normally the effect of ending any possible additional attacks with that weapon that round (on the 80 attacks above, this results in losing about 6 attacks throughout the 80). Falling prone can be more dangerous, since close attacks have advantage against you, but it only costs half your move to get up, so hardly the end of the world since if you haven't moved before attacking, you can get up the round you fell down.
Okay, my essay is done LOL. I think I have covered everything in your post. I know this is a lot to disgest, so it is time for lunch.
P.S. Attached is an Excel spreadsheet for anyone who wants to see how this work.
Thanks for introduction, but I think you have some misunderstandings. You do take the initial fumble attack roll into account (sorry about that, but as I wrote I glanced through your post before, and missed it). Since I am not certain what bonus you are using to the check (you seem to jump around a bit), I can't check your probabilities (they seem off..., which I'll explain later). Also, instead of arriving at the cumulative probability by adding up the probabilities of 1 success to N successes (depending on sample size of N), it is much easier to use the probability of zero successes and subtract that from 1 to find the cumulative probability of 1 or more successes.
Is that really how you are doing your calculations? Multiplying 64.2% by 20%? Hmm... That isn't really how it works. You can't look at the binomial distribution "at least" result and then multiple that by the straight probability of failing the DC 10 check with the +5 bonus.
You need to look at the probability of success (droping the weapon) as a conditional probability since the fumble check has to be a given.
P(A) = 0.05 (natural 1 on attack roll)
P(B|A) = 0.20 (fails DC 10 check to drop weapon)
P(A and B) = P(A) * P(B|A) = 0.05 * 0.20 = 0.01 is the probability of a dropped weapon. Now, use the binomial distrubtion with that as the probability of success (again, dropped weapon). So let's go with a binomial distribution with success probability of 0.01 and a sample size of 20 trials (attacks). Using the Excel binomial distribution function, BINOM.DIST(0,20,0.01,FALSE) with parameters of 0 successes (no dropped weapon), 20 trials (attacks), probability of success (dropped weapon), and non-cumulative because we want the probability of exactly 0 successes (the mass function), not the cumulative distribution function, we arrive with 81.8%. So, there is an 81.8% chance the character will never drop their weapon in 20 attacks. Thus, if we subtract that from 1, there is about an 18.2% chance they will drop their weapon once or more in 20 attacks (most likely, only once, btw). The probability of falling prone on any one attack is 0.002 (0.01 * another 4/20), and the chance of falling prone over 20 attacks at least once is roughly 3.9%. Higher than your numbers, but still, as you say, at lower levels that holds okay. I mean, this is over 20 attacks and 18% and 4% are hardly close to garuantees. So, what of the higher levels?
Using your example at 11th level (only increase in prof, so +6 total, but with 60 attacks instead of 20). The probabilities are actually: natural 1: 0.9539 (95.4%), drop weapon: 0.3635 (36.4%), fall down: 0.653 (6.5%). So, with only a +1 boost to the check but 40 more attacks, obviously the probabilities will go up significantly.
The dual wielder with 80 attacks actually has numbers of natural 1: 0.9835 (98.4%), drop: 45.2%, fall 8.6%. Again, adding more attacks without increasing the check bonus will always result in more drops and falls since you are attacking more. So, if that is the case, things will never get better.
Once again, I am not following your numbers as the 11th level fighter with +6 to the check has a 28.9% chance to drop their weapon once in 60 attacks, and less than 7.5% chance of dropping their weapon two or more times over 60 attacks. Increasing the number of attacks to 80, changes the numbers to 33.1% for exactly one drop and about 12.1% for two or more drops. But, as I said earlier, when you increase the number of trials, the likelihood of dropping increases since you have more attacks, so no surprise there. The chances of them falling down at least twice is less than 0.22% (well under 1%) with 60 attacks, and less than 0.38% (again, well under 1%) even with 80 attacks. Your numbers seem way off here. In both cases, however, the peaks of the distributions are still at zero drops and zero falls.
The shifting effect depends on probability of success (if it is greater than 0.5, it happens more quickly, but our probabilities are definitely less than 0.5). If the example above with the +6 check, the number of trials has to equal 133 (more than twice the base 60 attacks), before the peak shifts and you are more likely (36.9%) to drop once, than not at all (36.7%). The trials has to double (266 attacks) before two drops are the most likely outcome. So, with lower probabilities, you need a lot more trials before what you describe happens.
Now, as I have already said, and I will repeat for clarity: when you increase the number of trials, you increase the likelihood of 1 or more successes (i.e. drops and falls), as expected, but the peaks of the distributions remain at 0 successes unless your trial size increases drastically.
Next, it was your jump to DC 15 that I missed also when glancing through your post initially, that was why I thought maybe your numbers didn't include the initial fumble. I apologize for that oversight, but why would you assume that? My groups have never used them because they make the game "harder" (after all, it starts with only a 1 in 20 occurrence), but because it offers an element of suspense when it happens. Maybe the reasoning is to make it harder in your experience, but not in mine.
Let's examine the case for the Level 20 fighter making 80 attacks a day (using DC 10, thank you) with a +8 bonus (Dex +2, prof +6), failing only with a 1 on the drop weapon roll and the fall prone roll (from your numbers it looks like you might have still been using DC 15?).
Probabilities are natural 1: 0.05, drop weapon: 0.0025 (1/20 * 1/20), fall prone: 0.000125 (1/20 * 1/20 * 1/20).
Drop weapon on one attack or more (assuming this does NOT interrupt further attacks that round, which would decrease the chances slightly but is more complicated and does not have enough impact to seriously change the results) on 80 attack rolls: 18.15% (FYI, the probability of exactly 1 drop is most of this, 16.41%, two drops is only 1.62%, and it decreases drastically after that, so don't let the wording of "one or more" scare anyone.)
Fall prone after dropping weapon (same assumption): 1.00%
They will drop their weapon most likely once about 1 in 4 or 5 days and fall down most likely once about 1 in 100 days. Given how many attacks are being made, those odds don't seem overly harsh IMO. True, this is the best case scenario, but having played with such rules for decades, I have never seem them happen so frequently that myself or my players wanted to alter them or remove them completely. If the concept doesn't suit your sensibilities, then don't use them. The OP wanted to add tables (also fine), but this system works well for my group.
The game mechanic in use is "drop weapon" and "fall prone", which on the surface might sound ridiculous to some, but the "in story" effect can be brought about in a myriad of ways. Your attack was overextended and your foe's parry knocked your weapon loose, resulting in your dropping it. You were not able to recover in time from your off balance maneuver, and slipped on loose gravel, sending you to the ground. It doesn't literally have to mean "Oops, I was so inept and/or clumsy with my Dex 16 that my weapon just fell out of my hand and I landed on my butt! I am so ashamed!"
Anyway, for critical fumbles for spell casters and other ranged attacks? Well, I rule the ranged attack (usually a cantrip, arrow, etc.) might hit another player if they are in melee against the target since friendly fire into a melee can definitely have harmful consequences! The bowstring might snap instead of the player falling down, requiring them to either take the time to restring it or switch weapons and join the melee. A caster's spell component pouch might be lose in the dead leaves at their feet, etc. depending on the situation. I can rule the effect however matches the current scenario--but something negative happens.
For anyone who doesn't like the Dex check (Dex is overly used anyway), a simple replacement is you make a check using whatever bonuses you had to the attack roll in the first place. Suppose you had a magic weapon? Well, the bonus means you are less likely to drop it. A spell giving you an attack bonus? That spell will help with your drop and fall checks as well. You can adjust the DC like Ovinomancer if you want it harder to make. Rule 1's on the following checks always fail, no matter what bonuses you get to the roll. So on and so on, there are any number of ways to adjust critical fumbles rules if you want to use them. If not, then don't. Simple.
But for us, critical fumbles can be a fun part of the game when they happen. Other players laugh when a characters drops their weapon (not NEARLY the punishment it was in earlier additions as picking up a dropped weapon is part of a normal move or action now) or when the monsters do (they laugh even harder and love they might get advantage with the target prone!). Dropping the weapon has, at worse, normally the effect of ending any possible additional attacks with that weapon that round (on the 80 attacks above, this results in losing about 6 attacks throughout the 80). Falling prone can be more dangerous, since close attacks have advantage against you, but it only costs half your move to get up, so hardly the end of the world since if you haven't moved before attacking, you can get up the round you fell down.
Okay, my essay is done LOL. I think I have covered everything in your post. I know this is a lot to disgest, so it is time for lunch.
P.S. Attached is an Excel spreadsheet for anyone who wants to see how this work.
