10th Mathhammer: The Basics and Termagants vs. Tac Marines

[synapticinsight]

Greetings Tyranid dice junkies, today I'm posting some numbers I've crunched for termagants and how their weapon options from the index stack up against the tactical space marine.  You can skip to that below, but first I'd like to go over some of the basics of calculating probabilities and how I present the data.  If there is interest, I'll post more numbers and stats for various units and matchups in 10th edition.

--The Basics:  Ok, many of you probably already know what's in this section, but I'd like to make sure we start from the beginning so everyone is on the same page.  First up are some basic statistics from dice rolls.  Whenever we have multiple outcomes with different probabilities, we call that a probability distribution function (PDF).  For a simple d6 roll, we have a flat probability distribution function where any outcome has a 1/6th probability.  So the PDF for a d6 roll is:

1: 1/6
2: 1/6
3: 1/6
4: 1/6
5: 1/6
6: 1/6

And that isn't very illuminating, because most of the time in 40k we don't care about the probability to find a single die-roll probability...  Instead we want to know the probability of finding a particular value OR GREATER.  This type of distribution is known as a cumulative distribution function (CDF).  [For the big math nerds out there, we technically want the (1 - CDF), but the principle is effectively the same].  The CDF or probability of rolling a result or greater (X+) is shown below:

1+:  100%
2+:  83.3%
3+:  66.7%
4+:  50%
5+:  33.3%
6+:  16.7%

OK, cool, I think those numbers are pretty well known by most 40kers, but maybe the terminology is a little new.  The other big CDF to know is for 2d6, because this is used for battleshock tests and charge rolls.  The CDF numbers work out like this:

1+:  100%
2+:  100%
3+:  97.2%
4+:  91.7%
5+:  83.3%
6+:  72.2%
7+:  58.3%
8+:  41.7%
9+:  27.8%
10+ 16.7%
11+:  8.3%
12+:  2.8%

Alright, lets flesh those numbers out a bit:  Typical battleshock test for a space marine (6+) passes 72% of the time.  A charge roll out of deep-strike (9+) will only succeed 27.8% of the time.  A typical not-in-synapse tyranid unit will only pass a battleshock (8+) test 41.7% of the time.  Ok, enough of the basics...

--Termagants vs Tac Marines: Alrighty, here's how I break things down.  For a minimum size unit of termagants (10), I will give the probabilities for A) most likely amount of wounds inflicted, B) The cumulative probability that LESS than the most-likely value will be inflicted, and C) the cumlative probability that MORE than the most-likely value will be inflicted.  For example:  If I say: "2 wounds (35%), < (25%), > (40%)",  then this means the most likely outcome of the attack would be two wounds inflicted (which happens 35% of the time).  There would then be a 25% that one or less wounds are inflicted, and a 40% chance that 3 or more wounds are inflicted.  It is important to note that it is very possible that cumulative sum of probabilities for the "greater-than" or "less-than" number EXCEEDS the most-likely number.  This might seem counter-intuitive, but it is very important to understand to best grasp the amount of damage you may be able to inflict, and which weapons or units outperform other ones.  Ok, on to the numbers.  Lets look a the three weapon choices for termagants: Fleshborers, Devourers, and Spinefists:


Table 1: Termagants (10 models) vs Tac Marines expected wounds delivered in a single shooting round:

	Most Likely	<	>
Fleshborers:	1 wound (38.5%)	30.9%	30.6%
Devourers:	1 wound (32%)	17.5%	50.5%
Spinefists:	1 wound (29.1%)	14.1%	56.5%

Hmmm...  Ok, generally not great, but we can gleam a few interesting details.  Spinefists have the least chance of failure and the best chance of inflicting 2 or more wounds at 56.5% (more often than not you will remove a tac marine model).  The Devourers come in second and the Fleshborers are the clear worst option, damage-wise.  We can also take a look at how the numbers change with swarming instincts and those "sustained hits 1":

Table 2: Termagants (10 models) with Swarming Instincts vs Tac Marines expected wounds delivered in a single shooting round:

	Most Likely	<	>
Fleshborers:	1 wound (33.9%)	22.2%	43.9%
Devourers:	2 wounds (27.1%)	34.8%	38.2%
Spinefists:	2 wounds (25.9%)	28.8%	45.3%

Ok, better, slightly.  Now the most likely outcome for Spinefists and Devourers will remove a tac marine model, and spinefists have a reasonable chance of pushing 3 or more wounds at 45.3%.  For me, this makes spinefists an obvious pick against fleshborers.  Even though fleshborers gain "assault", one round of spinefist fire can outpace 2 rounds of fleshborer shooting...  so I might not mind giving up the shooting for an advance roll if I can get the shots off next round.

Finally, we can look at the best-case scenario where the termagants get Swarming Instincts and the benefit of Symbiotic Targeting from an Exocrine (reroll 1s to hit):

Table 3: Termagants (10 models) with Swarming Instincts and Symbiotic Targeting vs Tac Marines expected wounds delivered in a single shooting round:

	Most Likely	<	>
Fleshborers:	1 (30.7%)	16.7%	52.6%
Devourers:	2 (25.2%)	26.1%	48.7%
Spinefists:	2 (23.4%)	20.9%	55.8%

As expected, best-case, but still not especially great.  At 60pts per squad, no one is expecting killing machines.  Termagants should do what they do best, take objectives, soak fire, and annoyingly respawn.

OK, that's it for today.  Let me know if you want some more Tyranid mathhammer, and what stats/numbers you would be interested in seeing in future posts.

[infornography]

cool! I was aware that fleshborers were the weakest, but I didn't know by how much.

I'd like to see how our anti-tank options fare against light and heavy vehicles.

Say maybe a 6 pack of zoans vs rupture cannon vs haruspex melee?
Include relative point efficiencies.

Yeah I know range is going to be a HUGE factor here.

[piersonsmuppet]

There are some spreadsheets here that can be used to show the effectiveness of any weapon into most defensive profiles, I think doing offensive stats might be beating a dead horse. One thing that might be interesting is to calculate which units termagants can come within 6” of and survive HI to still control an objective to deny opponent VP (and if they can survive the next round of combat to still control a point scoring you VP with/without the use of endless swarm).

[mattblowers]

I really can't decide which one I like the best. They all have their weaknesses. As noted, the ability to score objectives is what they are best used for. I don't love spinefists because terms don't want to get short. I like the strength 5 a lot, but double the shots is better in most situations unless your list needs to run and gun then the assault is important. Non are dramatically enough better or worse if you have a plan to take tactical advantage of differences.

[hivefleetkerrigan]

Im going to be using devourers personally due to 2 shots but better range compared to spine fists. I'll probably bring a tervigon or two for lethal hits so I can guarantee ok shooting into tough targets.

[Iryan]

So far I have been running both squads of fleshborers and squads of devourers. The fleshborer ones are more for actions than anything; being able to cleanse an objective or similar even after advancing is quite useful. Of course, usually you would then be better off just taking gargoyles, but still. Devourer-gants at the main force so they are supported by a hive tyrant, borergants at the outer wings. Working fine for me so far.

[infornography]

Given the breakdown of the math, I am inclined to lean on devourers for the extra range. Barbies can hold the longer range role. For close range it really comes down to neuros for super cheap, spinefists for somewhat cheap, Horms for some speed and quality into melee, or Gargs for speed and flexibility.

I'd probably pick depending on points available and secondary roles.

[goodbrood]

What is the probability of no damage getting through on a squad of 20 termagaunts with sustained fire, crits on 5's, rr1's to hit, into Thunderwolf cavalry. Mainly want to see how bad my luck is.

If you want to be accurate, you could math out no damage before getting to saves...

[Iryan]

Aug 11, 2023 15:04:25 GMT goodbrood said:

What is the probability of no damage getting through on a squad of 20 termagaunts with sustained fire, crits on 5's, rr1's to hit, into Thunderwolf cavalry. Mainly want to see how bad my luck is.

If you want to be accurate, you could math out no damage before getting to saves...

I have no idea how you got that combination of buffs on those gants. Sustained hits from the hivefleet bonus does not work against mounted units, Tervigon gives lethal hits not rerolls, the strat for crits on 5 is only for the fight phase... Buuut:

each termagant has either 1 or 2 shots > 20 or 40 shots total

each individual hits on 4+ and with S4/S5 vs T6 wounds on 5+, cavalry saves on 3+

without all those buffs above, a single shot has a chance of (1/2)*(1/3)*(1/3) = 1/18 to deal 1 damage

with 20 shots, the odds of getting 0 damage is then (1-(17/18))^20 = approximately 32 %
with 40 shots, the odds of getting 0 damage is then (1-(17/18))^40 = approximately 10 %

If you explain me what buffs and effects you managed to get on the gants, then I can do the calculations for that.

[nidsallday]

Aug 12, 2023 17:49:42 GMT Iryan said:

Aug 11, 2023 15:04:25 GMT goodbrood said:

What is the probability of no damage getting through on a squad of 20 termagaunts with sustained fire, crits on 5's, rr1's to hit, into Thunderwolf cavalry. Mainly want to see how bad my luck is.

If you want to be accurate, you could math out no damage before getting to saves...

I have no idea how you got that combination of buffs on those gants. Sustained hits from the hivefleet bonus does not work against mounted units, Tervigon gives lethal hits not rerolls, the strat for crits on 5 is only for the fight phase... Buuut:

each termagant has either 1 or 2 shots > 20 or 40 shots total

each individual hits on 4+ and with S4/S5 vs T6 wounds on 5+, cavalry saves on 3+

without all those buffs above, a single shot has a chance of (1/2)*(1/3)*(1/3) = 1/18 to deal 1 damage

with 20 shots, the odds of getting 0 damage is then (1-(17/18))^20 = approximately 32 %
with 40 shots, the odds of getting 0 damage is then (1-(17/18))^40 = approximately 10 %

If you explain me what buffs and effects you managed to get on the gants, then I can do the calculations for that.

rr1 to hit in shooting could be exo, the others i dont know either xD

[hivefleetkerrigan]

Hazarding a guess here is that there was some rules misinterpretation:

Sustained hits from the adaptation, missing that it's only against infantry

5+ crit from the strat missing that it's only in melee

[piersonsmuppet]

Aug 12, 2023 17:49:42 GMT Iryan said:

each termagant has either 1 or 2 shots > 20 or 40 shots total

each individual hits on 4+ and with S4/S5 vs T6 wounds on 5+, cavalry saves on 3+

without all those buffs above, a single shot has a chance of (1/2)*(1/3)*(1/3) = 1/18 to deal 1 damage

with 20 shots, the odds of getting 0 damage is then (1-(17/18))^20 = approximately 32 %
with 40 shots, the odds of getting 0 damage is then (1-(17/18))^40 = approximately 10 %

You sure about those results (0.06^20 is a very small number)? 32% is the result of using 1/18 in your formulas but that would calculate the chance of 1 damage occurring (still not quite right). Binomial probability shows a 37% probability to get only 1 damage (PX=1) for a 1/18 chance using 20 attempts and a 29% probability for 0 damage (PX=0). I find pure binomial probability to be a little off since the multiple rolls between hit/wound/save throw the math off a bit, so I like using weighted avgs of the binomial probability for saves. I found 0 damage when gants shoot into TWC with fleshborers to be 27%:

Going off of 3.3 avg wounds on TWC (20 x 1/2 x 1/3), so avg 3 saves with a 30% chance for avg 4 saves.

probability of 3/3 successful saves x probability of 3 avg saves + probability of 4/4 successful saves x probability of 4 avg saves

2/3^3 x (1 - 0.3) + 2/3^4 x 0.3 = 27%

Out of curiosity, what is the function you used?

[Iryan]

Aug 14, 2023 4:54:34 GMT piersonsmuppet said:

You sure about those results (0.06^20 is a very small number)?
[...]
Out of curiosity, what is the function you used?

...I do not really get what kind of math you used yourself... ^^;

There is no complicated function I used. All the math I did is visible in my post, aside from rounding off decimal points on the end result.

The question was what are the odds of zero damage to come through. If there is zero damage, that means the result of every single attack has to be the same. So there is no need to muddle around with complicated formulas for permutations or binomials... there is only one possible permutation for this. So the odds for getting the same result with every single shot is simply the odds of getting that result for an individual shot, to the power of the number of shots.
The 4+ chance to hit is 1/2, the 5+ chance to wound is 1/3, the 3+ armor gives a 1/3 chance for the damage to get through.
So the odds for a single shot to deal one damage is, as I said, (1/2)*(1/3)*(1/3) = 1/18
So therefore, the odds for a single shot to deal zero damage is 1-(1/18) = 17/18
So the odds for 20 shots to all get the same zero damage result is (17/18)^20 = 0.318807350104...

If you want to get the odds for all other results, you need to get binomial coefficients involved and multiply by the number of permutations of individual attacks results that will give your target result.

Probability P(k) for getting k damage out of n attacks with an individual success probability of p

P(k) = p^k * (1-p)^(n-k) * (n! / (k! * (n-k)!))

That will give you the following results:

n(shots) p(hit) p(miss)
20 0,055555556 0,944444444

k (hits) P(k) P(k) in %
0 0,31880735 31,88
1 0,375067471 37,51
2 0,209596528 20,96
3 0,073975245 7,40
4 0,018493811 1,85
5 0,003481188 0,35
6 0,000511939 0,05
7 6,02282E-05 0,01
8 5,7571E-06 0,00
9 4,51538E-07 0,00
10 2,92171E-08 0,00
11 1,56241E-09 0,00
12 6,893E-11 0,00
13 2,4952E-12 0,00
14 7,33884E-14 0,00
15 1,72678E-15 0,00
16 3,17424E-17 0,00
17 4,39341E-19 0,00
18 4,30726E-21 0,00
19 2,66704E-23 0,00
20 7,84422E-26 0,00

If you want to adjust for different buffs on the unit, you can just look at how that buff affects the probability for a single shot, then adjust accordingly. For instance, getting rerolls of 1 to hit (or to wound) are a simple multiplier of 7/6 to the overall success rate, so the odds would not be 1/18, it would be 7/108. Stuff like lethal hits and critical wounds are a bit more involved, but so long as each shot is independent of another, you still just need to figure out the odds of an individual shot to succeed, then pop that probability and the number of shots in the above formula. If the individual shots' probability of success are NOT independent, like if you have a rule that says "you may reroll a single wound roll for the entire unit"... then stuff becomes a hell of a lot more complicated.

[piersonsmuppet]

Just asked for a function name... but it looks like your original post has a typo. Previous post has (1-(17/18))^20, this post says you used 1-(17/18)^20; those extra parentheses really change the math. Though, binomial coefficient can't determine accurate probability for successive dependent events as a single event; it causes some bloat in the results. Hence the calculation of the avg wounds caused, and the use of weighted avg of binomial coefficients for just the saves.

[Iryan]

...oh. Welp. Yes that would be a typo... sorry.

Aug 14, 2023 12:51:23 GMT piersonsmuppet said:

Though, binomial coefficient can't determine accurate probability for successive dependent events as a single event; it causes some bloat in the results. Hence the calculation of the avg wounds caused, and the use of weighted avg of binomial coefficients for just the saves.

I am not sure what you are saying here... are you saying that because every single shot is resolved as a series of consecutive rolls, using the method I did is inaccurate? Because I do not see how or why.

There are only two outcomes of a single shot. Either it causes 1 damage or it does not. So multiplying the probabilities for the consecutive rolls will give you the accurate probability for the overall event.

And each individual shot resolution is independent of each other shot resolution among the 20 shots. How many other shots have hit/wounded/pierced the foe already has no impact on the probability of the subsequent shots hitting/wounding/piercing. There are cases where this would not be true, such as with 8th edition Salamanders or when you shooting at unit with mixed armor profiles, and stuff gets a lot more complicated if you shoot multi-damage weapon at a unit with a FnP (especially if the unit is also multiwound and/or if the damage is random), but none of that applies here. I am pretty sure that if you do a monte carlo simulation, you will find results that line up with my numbers.

I do not quite get the reasoning behind the math you used in your own example either. If you want to know the PRECISE odds of getting an exact number of results, you cannot go about probabilistic averages. Doing so is an approximation. You will get rounding errors. Your example basically says that you will ONLY get either 3 or 4 wounds before the saves. But that is not true. You can get more or less. And the odds of getting 0, 1, 2, 5, 6, or 7 wounds are approximately 2%, 10%, 20%, 13%, 6.5%, and 2.5%. In over 50% of cases, your pre-save number of wounds will NOT be 3 or 4. You can see why your approximation won't get accurate results. ^^;