One-Man Armies from Supers vs. Mob Attacks from Zombies vs. other solutions

[Raekai]

Wow! Well, thanks! I appreciate you thoroughly checking it out, and I'm really glad that it seems (mostly) up to spec.

And I agree on the whole math over tables bit. The trickiest part, really, is probably that pesky Recoil, but I realized that it's probably better written as a fraction so that the margin of success can just be multiplied by the inverse.

But what do you mean by it is a bit less likely to give the average? I just want to make sure I understand. For 4 enemies, using that +3 bonus on 10 for a total of 13, the average roll is 10, which is a margin 3, which gets 1 hit for the success plus another 1 hit for beating the recoil. I figured 2 hits would be the average out of 4 shots (at skill 10, of course). The base 1 hit for the success is the part I always forget about RoF, so I just want to make sure you took that into account too.

And really. Thanks again. It means a lot to hear that I was successful in cobbling together some arcane symbols (ya know, math) into something useful!

[ericthered]

But what do you mean by it is a bit less likely to give the average? I just want to make sure I understand. For 4 enemies, using that +3 bonus on 10 for a total of 13, the average roll is 10, which is a margin 3, which gets 1 hit for the success plus another 1 hit for beating the recoil. I figured 2 hits would be the average out of 4 shots (at skill 10, of course). The base 1 hit for the success is the part I always forget about RoF, so I just want to make sure you took that into account too.

base hit for 1 is included.



the binomial (true method) gives an average of 35.3 hits when 37 opponents with skill 15 attack.


your method gives an average of 32.4 hits, slightly less, because skill is high.


the binomial (true method) gives an average of 6.00 hits when 37 opponents with skill 7 attack.


your method gives an average of 6.74 hits, slightly more, because skill is low.

Originally Posted by Raekai

And I agree on the whole math over tables bit. The trickiest part, really, is probably that pesky Recoil, but I realized that it's probably better written as a fraction so that the margin of success can just be multiplied by the inverse.

I just realized my method and yours are slightly different. You're looking up the recoil from a table, and I'm just dividing 8 by the number of attacks. The division method is a fair deal more accurate away from powers of 2.


I really like the method though! its slick and it mostly just works! Do you mind if I stick it up on my blog? how do you want me to credit you?

[Raekai]

Originally Posted by ericthered

base hit for 1 is included.



the binomial (true method) gives an average of 35.3 hits when 37 opponents with skill 15 attack.


your method gives an average of 32.4 hits, slightly less, because skill is high.


the binomial (true method) gives an average of 6.00 hits when 37 opponents with skill 7 attack.


your method gives an average of 6.74 hits, slightly more, because skill is low.

Oh, okay! That makes sense. I misunderstood what you meant!

Originally Posted by ericthered

I just realized my method and yours are slightly different. You're looking up the recoil from a table, and I'm just dividing 8 by the number of attacks. The division method is a fair deal more accurate away from powers of 2.

Wow! Okay! That makes a ton of sense with dividing the number of attacks by eight. I was halving recoil as the number of enemies doubled, which ends up giving the same numbers anyway, but—you're absolutely right!—dividing by eight makes that so much easier for numbers that aren't powers of 2, which is exactly what you said, but I hadn't made that connection. So, six attacks would have a recoil of 1.33 (or 4/3, of course, which would be easier for me to just multiply the margin of success by 3/4 instead of dividing by 1.33). Wow. Yep. That's awesome.

Originally Posted by ericthered

I really like the method though! its slick and it mostly just works! Do you mind if I stick it up on my blog? how do you want me to credit you?

You are more than welcome to stick it up on your blog, and please feel free to polish it as well. Your insight about dividing by eight was very helpful, and, if there's a simpler way to calculate the bonus for the lower numbers before it just becomes +4 forever, I'd love to know.

As for crediting me, you can credit me by Raekai and/or my name Greyson Yandt, and, if you wouldn't my linking my name to my blog, I would really appreciate it. (And I appreciate you offering to credit me, too!) I was planning on throwing up a post about this on my blog, and I would happily link to yours in return! Plus, that saves me some of the effort of actually writing this up myself...

[Varyon]

I previously started to build a whole system, doing all manner of probability calculations, where you'd make one roll and it would handle a number of attacks, then you make another, and so forth. After testing it a bit, I found that rolling 3d three times (rather than a variable number of times depending on various factors) worked best.

Basically, as noted above, you roll 3d against 10 three times. On a success, add your MoS+1 to your effective skill; on a failure, subtract your MoF. Once you have the final modified skill, check it on the below table. Multiply the success probability by the number of rolls you're resolving to determine how many successes you got (round normally; if you're more concerned about failures, use that column; in either case, round in favor of successes for 50%). If MoS/MoF matters, note that the results above simply state how many successes or failures you got; to determine margins, simply go down the line to the next probability (which tells you how many had MoS 1+), then the next (2+), and so forth.

Skill	Success	Failure
3	0.46%	99.54%
4	1.85%	98.15%
5	4.63%	95.37%
6	9.26%	90.74%
7	16.20%	83.80%
8	25.93%	74.07%
9	37.50%	62.50%
10	50.00%	50.00%
11	62.50%	37.50%
12	74.07%	25.93%
13	83.80%	16.20%
14	90.74%	9.26%
15	95.37%	4.63%
16	98.15%	1.85%
17	99.54%	0.46%
18	100.00%	0.00%

As an example of it in action, consider someone with a RoF 10, Rcl 2, Malf 17 weapon firing it at full RoF for 100 (!) seconds. After accounting for the rapid fire bonus and SM/range, he's at effective skill 12. He rolls 15 (MoF 5), 8 (MoS 2), and 10 (MoS 0). That's -5, +3, +1, for a total of -1. We check the table and see 63 hit, 37 missed. With Rcl 2, we want to check each MoS 2 to see if we got additional hits. 9 shows 38 hits, so 38 of our 63 hits are MoS 2; 7 shows 16 hits (MoS 4); 5 shows 5 hits (MoS 6); 3 shows 0 hits. This means a total of 63+38+16+5=122 bullets (of the 1000 we sent flying through the air) hit the target. Of course, with skill 12 and Malf 17, MoF 5 corresponds to a malfunction; 16 (which is MoF 5 for our effective skill of 11) shows 2 failures, so we suffered 2 malfunctions. Optionally, if either would have prevented future attacks, you may want to randomly set when they happened (which will prevent some number of your attacks); alternatively, just assume they happened at the end.

If you find the above too swingy, you could opt to average the results of the roll (in which case, I suggest getting rid of the +1 on successes; that's there so the average result is +0). This can result in fractional skill values; have another table for that.

Skill	Success	Failure
3	0.46%	99.54%
3.33	0.93%	99.07%
3.67	1.39%	98.61%
4	1.85%	98.15%
4.33	2.78%	97.22%
4.67	3.70%	96.30%
5	4.63%	95.37%
5.33	6.17%	93.83%
5.67	7.72%	92.28%
6	9.26%	90.74%
6.33	11.57%	88.43%
6.67	13.89%	86.11%
7	16.20%	83.80%
7.33	19.44%	80.56%
7.67	22.69%	77.31%
8	25.93%	74.07%
8.33	29.78%	70.22%
8.67	33.64%	66.36%
9	37.50%	62.50%
9.33	41.67%	58.33%
9.67	45.83%	54.17%
10	50.00%	50.00%
10.33	54.17%	45.83%
10.67	58.33%	41.67%
11	62.50%	37.50%
11.33	66.36%	33.64%
11.67	70.22%	29.78%
12	74.07%	25.93%
12.33	77.31%	22.69%
12.67	80.56%	19.44%
13	83.80%	16.20%
13.33	86.11%	13.89%
13.67	88.43%	11.57%
14	90.74%	9.26%
14.33	92.28%	7.72%
14.67	93.83%	6.17%
15	95.37%	4.63%
15.33	96.30%	3.70%
15.67	97.22%	2.78%
16	98.15%	1.85%
16.33	98.61%	1.39%
16.67	99.07%	0.93%
17	99.54%	0.46%
17.33	99.69%	0.31%
17.67	99.85%	0.15%
18	100.00%	0.00%

Note this works equally well for attacks, defenses, point defense fire, HT rolls to avoid losing FP while running, and so forth. If it calls for a lot of Success Rolls, this will have you covered. It's an alternative to what you have; it's not necessarily better or worse, but some people may favor one over the other.

[Raekai]

Originally Posted by Varyon

Note this works equally well for attacks, defenses, point defense fire, HT rolls to avoid losing FP while running, and so forth. If it calls for a lot of Success Rolls, this will have you covered. It's an alternative to what you have; it's not necessarily better or worse, but some people may favor one over the other.

That looks amazing—more complex, but amazing. That's definitely a great alternative, and I appreciate its universality, and it's also really neat that it uses the percentages of 3d6 results. It fits GURPS very well. If I were looking for more crunch, I'd definitely use this!

[Varyon]

Originally Posted by Raekai

That looks amazing—more complex, but amazing. That's definitely a great alternative, and I appreciate its universality, and it's also really neat that it uses the percentages of 3d6 results. It fits GURPS very well. If I were looking for more crunch, I'd definitely use this!

It gets a bit less messy if you're willing to (le gasp!) use non-standard dice. Rolling 3dF (dF is a d6 that has 2 +'s, 2 -'s, and 2 blank faces rather than numbers) would give results comparable to using the average (it has a max of +3 and minimum of -3, but going beyond that with averages likely calls for statistical outliers anyway), while avoiding the need for fractional skill levels. If there's an attack that is important to know when in the sequence it occurred (such as a Critical Failure that invalidated future attacks), you can get an approximation of when it happened with a 1d6 roll - a 1 means it happened on the first attack (tough luck), 2 means it happened 20% of the way in, 3 is 40%, and so forth, out to 6 meaning it happened on the last attack. You can get more fine-grained with another 1d6 roll - 2,1 would mean 20%, 2,2 would mean 24%, 2,3 would mean 28%, and so forth, out to 2,6 meaning 40%; then another 1d6 could break it down further. Of course, a better option would be to just roll d% (2 d10's, with one designated as the 1's place, one as the 10's place, and 00 treated as either 0% or 100%, depending on your inclination).

This does require a calculator at the table, of course.

[ericthered]

Originally Posted by Varyon

This does require a calculator at the table, of course.

You know, I'm surprised I haven't seen any dice rolling websites that are designed to just take N rolls, a target number, and tell you how many of them hit the specified thresholds. This is the computer age after all...

[Celjabba]

Originally Posted by Varyon

I previously started to build a whole system, doing all manner of probability calculations, where you'd make one roll and it would handle a number of attacks, then you make another, and so forth. After testing it a bit, I found that rolling 3d three times (rather than a variable number of times depending on various factors) worked best.

Basically, as noted above, you roll 3d against 10 three times. On a success, add your MoS+1 to your effective skill; on a failure, subtract your MoF. Once you have the final modified skill, check it on the below table. Multiply the success probability by the number of rolls you're resolving to determine how many successes you got (round normally; if you're more concerned about failures, use that column; in either case, round in favor of successes for 50%). If MoS/MoF matters, note that the results above simply state how many successes or failures you got; to determine margins, simply go down the line to the next probability (which tells you how many had MoS 1+), then the next (2+), and so forth.

Skill	Success	Failure
3	0.46%	99.54%
4	1.85%	98.15%
5	4.63%	95.37%
6	9.26%	90.74%
7	16.20%	83.80%
8	25.93%	74.07%
9	37.50%	62.50%
10	50.00%	50.00%
11	62.50%	37.50%
12	74.07%	25.93%
13	83.80%	16.20%
14	90.74%	9.26%
15	95.37%	4.63%
16	98.15%	1.85%
17	99.54%	0.46%
18	100.00%	0.00%

As an example of it in action, consider someone with a RoF 10, Rcl 2, Malf 17 weapon firing it at full RoF for 100 (!) seconds. After accounting for the rapid fire bonus and SM/range, he's at effective skill 12. He rolls 15 (MoF 5), 8 (MoS 2), and 10 (MoS 0). That's -5, +3, +1, for a total of -1. We check the table and see 63 hit, 37 missed. With Rcl 2, we want to check each MoS 2 to see if we got additional hits. 9 shows 38 hits, so 38 of our 63 hits are MoS 2; 7 shows 16 hits (MoS 4); 5 shows 5 hits (MoS 6); 3 shows 0 hits. This means a total of 63+38+16+5=122 bullets (of the 1000 we sent flying through the air) hit the target. Of course, with skill 12 and Malf 17, MoF 5 corresponds to a malfunction; 16 (which is MoF 5 for our effective skill of 11) shows 2 failures, so we suffered 2 malfunctions. Optionally, if either would have prevented future attacks, you may want to randomly set when they happened (which will prevent some number of your attacks); alternatively, just assume they happened at the end.

If you find the above too swingy, you could opt to average the results of the roll (in which case, I suggest getting rid of the +1 on successes; that's there so the average result is +0). This can result in fractional skill values; have another table for that.

Skill	Success	Failure
3	0.46%	99.54%
3.33	0.93%	99.07%
3.67	1.39%	98.61%
4	1.85%	98.15%
4.33	2.78%	97.22%
4.67	3.70%	96.30%
5	4.63%	95.37%
5.33	6.17%	93.83%
5.67	7.72%	92.28%
6	9.26%	90.74%
6.33	11.57%	88.43%
6.67	13.89%	86.11%
7	16.20%	83.80%
7.33	19.44%	80.56%
7.67	22.69%	77.31%
8	25.93%	74.07%
8.33	29.78%	70.22%
8.67	33.64%	66.36%
9	37.50%	62.50%
9.33	41.67%	58.33%
9.67	45.83%	54.17%
10	50.00%	50.00%
10.33	54.17%	45.83%
10.67	58.33%	41.67%
11	62.50%	37.50%
11.33	66.36%	33.64%
11.67	70.22%	29.78%
12	74.07%	25.93%
12.33	77.31%	22.69%
12.67	80.56%	19.44%
13	83.80%	16.20%
13.33	86.11%	13.89%
13.67	88.43%	11.57%
14	90.74%	9.26%
14.33	92.28%	7.72%
14.67	93.83%	6.17%
15	95.37%	4.63%
15.33	96.30%	3.70%
15.67	97.22%	2.78%
16	98.15%	1.85%
16.33	98.61%	1.39%
16.67	99.07%	0.93%
17	99.54%	0.46%
17.33	99.69%	0.31%
17.67	99.85%	0.15%
18	100.00%	0.00%

Note this works equally well for attacks, defenses, point defense fire, HT rolls to avoid losing FP while running, and so forth. If it calls for a lot of Success Rolls, this will have you covered. It's an alternative to what you have; it's not necessarily better or worse, but some people may favor one over the other.

That look great !
Thanks, printed and filled just in case :)

[ericthered]

Originally Posted by Raekai

If there's a simpler way to calculate the bonus for the lower numbers before it just becomes +4 forever, I'd love to know.

There is.

4* (Attacks-1)/(Attacks)

[Raekai]

Originally Posted by ericthered

There is.

4* (Attacks-1)/(Attacks)

Incredible. So, my math (The Bonus is equal to Recoil multiplied by Half#-0.5) just happened to work. Your math is so much easier. Thank you!