B J
United States Columbus Indiana

First, special thanks to arkayn for inspiring me to write this after reading his Analysis: Warp Spiders thread. If I find the time I may turn this into a two part series, the follow up analyzing the different troop capacity units and discussing their merits.
Recently, I decided to build a Dark Angels deck. The deck is focused around counterattack and rallying and in support of the counterattack I've tried to add cards that would give firepower or armor bonuses to counterattacking units. With only about 10 infantry assault units in my deck I figured I shouldn't use more than 2 troop capacity units in my whole deck, but then the statistician in me asked, "Well how many troop capacity units is actually optimal?” This question has now been nagging at me for 2 weeks, and I'm still not satisfied with my solution, though it is accuratemy solution just doesn't quite answer the question I was most interested in. Here I will try to present my findings.
Goal Given a certain number of infantry assault units (IAs), how many troop capacity units (TCs) should I include in my deck? This was the initial question I started with. I decided this was a matter of balancing a two objectives: You want to have enough TCs that you have a good chance of getting two IAs and one TC in a deployment draw. You want few enough TCs that you don't draw a TC without getting at least two IAs
Parameters Ideally, I wanted to calculate the probability of drawing a single TC and 2 or more IAs in any 7 card deployment, at any point in the deck. Instead I calculated the probability of drawing a single TC and 2 or more IAs in the first 7 cards. The difference is subtle, but it's an important distinction. Before I present the statistics, this is probably a good time to address some of choices I made:  The deck has 60 cards. (The statistics could be easily modified to consider a larger deck.)  I'm only considering the first 7 cards drawn from the deck (First Wave 1 + 6 card deployment), because First Wave:1 is the most common, and I believe (though I could be wrong) that the probability of getting the desired draw in the first 7 cards is proportional to the probability of getting the desired draw in in ANY 7 card deployment.  I also assumed that drawing 1 TC and 0 or 1 IAs is not desirable, since using the troop capacity ability to charge a single infantry assault unit is not useful, and you would be better off having a different unit instead of the TC.
Results and Analysis So let's start with 10 infantry assault units and 1 troop capacity unit in our deck. The probability* of drawing 1 TC and 2 or more IAs is 3.11%. The probability of drawing 1 TC and 0 or 1 IAs is 8.56%. The odds ratio of desired draw to undesired draw is thus 0.363. This means that you are almost three times as likely to get the undesired draw as the desired draw. Now by increasing the number of IAs in the deck we discover something cool: with 16 IAs in a deck and 1 TC the odds ratio becomes 1.10. Thus, you should never put a troop capacity unit in a deck just for the sake of its troop capacity ability if you have less than 16 infantry assault units in your deck. An odds ratio of 1.10 is pretty near having the same probability of an undesirable draw as having a desirable draw. It isn't until you have 20 IAs in the deck (and still 1 TC) that the odds ratio is greater than 2 (2.05).
Now starting with 20 IAs in the deck, how many TCs should be in the deck? With 1 TC, the odds ratio (as already given) is 2.05 and the probability of the desired draw is 7.8%. If we increase to 4 TCs, the odds ratio drops to 1.91 and the probability of the desired draw increases to 23.8%, about three times the probability with only 1 TC in the deck! To have what I would consider to be a truly comfortable odds ratio, you need 24 IAs in the deck with 4 TCs. This gives an odds ratio of 3.57 and a probability of getting the desired draw becomes 27.7%. We could continue evaluating higher quantities of IAs or TCs in the deck, but I think at this point having 1/3 of the deck being infantry assault units and troop capacity units, you are going to want to put other types of units into the deck.
Conclusions In short, you should NEVER use a troop capacity unit for the sake of its troop capacity ability if you have less than 16 infantry assault units. I wouldn't even really consider it with less than 20 IAs. Secondly, having 4 TCs makes you 3 times as likely to get a TC with 2 or more IAs, while it barely decreases your odds ratio of getting the desired deployment. This same principle could be applied to other card combos, so think about that next time you are building your deck!
*I chose not to include how I went about calculating these probabilities here, primarily because my method was pretty inelegant (lots of excel cells calculating "k choose n" and adding and dividing). But if someone is interested then I could probably be convinced to comment on this thread with the process details.
Note: This post has been edited to correct some of the original statistics presented.

Chris Berger
United States Round Lake Illinois

Cool. Nice analysis.
But what I'm thinking is  basically, troop capacity units will support your infantry assault units, so if you don't have enough infantry assault units to support, it's useless to include the transports that bring them into battle. And if you do have a lot of infantry assault units, then make sure you have enough troop capacity units. But obviously, that's just qualitative stuff, so it's good to see numbers to support it  if the numbers didn't support it, I'd wonder where we went wrong  the numbers or the intuitive reasoning.
It's basically a problem of combos that's present in all CCG's. Two cards combo well together... what if I have 1 copy of one and several of the other? What about the reverse? The only real answer is that if you want a combo to play out, you need multiples of both cards. But the troop capacity thing complicates matters in that a) you need more than one infantry assault for troop capacity to be worthwhile, and b) the troop capacity units are mostly nonflag units with no killing power, so if you're going to use them, it HAS to be worth it.
The Space Marines are an interesting subset, as they have a lot of guys that can shoot for 3 or 4, so you don't NEED to use assault, but you will probably have some assault guys in your deck. Plus, their troop capacity units generally have some killing power of their own.
I wonder if you saved your spreadsheet and could run numbers for my Dark Eldar assault deck. I have 43 infantry assault units in that deck. Currently only 6 troop capacity units  and that's despite having 8 units in the deck with special abilities that relate to troop capacity units. Mostly because I wanted the Raiders for a different deck, and also because the Dark Eldar have very limited troop capacity:
Lightning Raider has Troop Capacity: 1, but lets you take another action after you use the TC on a 4+. I used to have some, but they were part of a box of cards I had stolen many years ago. And anyway, the deck in question has Lelith Hesperax, so it doesn't need the Lightning Raider ability.
Raider has Troop Capacity: 2 and lets you move units to another sector (which is the ability that caused me to pillage them for my other deck)
Hellcrone Gunship has Troop Capacity: 3, has 3 Armor* and a very rare 6 speed (actually, I think it's the only unit in the game with 6 speed), and also gives out firepower to your infantry units when locked, with no test. But it's rare and I only have 2 of them, and besides it's a d2, so not sure I'd want more of them in my deck despite the fact that they're awesome.
The Broken Will has Troop Capacity: 2 and an okay ability, but is mostly awesome because of the die value of 6. Also a common, so easy to get at the time.
Also, I might consider doing the analysis with 7 cards or maybe 8 cards at a time, taking into account the First Wave value of 1 or 2. Orks have their First Wave of 3, but don't have a ton of Troop Capacity or Assault units. Interestingly, the Marines have the most Troop Capacity units (I count 12), and it looks like Imperial Guard only have 2. Chaos and Eldar each have 3. Orks have 5, Tyranids and Dark Eldar, 4 each.
* lies  the Hellcrone Gunship only has 2 armor

B J
United States Columbus Indiana

arkayn wrote: I wonder if you saved your spreadsheet and could run numbers for my Dark Eldar assault deck. I have 43 infantry assault units in that deck. Currently only 6 troop capacity units  and that's despite having 8 units in the deck with special abilities that relate to troop capacity units.... Also, I might consider doing the analysis with 7 cards or maybe 8 cards at a time, taking into account the First Wave value of 1 or 2. Orks have their First Wave of 3, but don't have a ton of Troop Capacity or Assault units. Interestingly, the Marines have the most Troop Capacity units (I count 12), and it looks like Imperial Guard only have 2. Chaos and Eldar each have 3. Orks have 5, Tyranids and Dark Eldar, 4 each.
Well it's good you asked me this question, because I discovered that my spreadsheet was not calculating the "undesirable draw" correctly. It was only looking at the probability of drawing a deployment that included 0 IAs and 1 or more TCs. It left out the "undesirable draw" where you draw only 1 IA and 1 or more TCs. I'll have to make some corrections to my original post.
I was able to modify my spreadsheet to look at any deployment size (6+X first wave). Here are the quick numbers for the deck you described (with the corrected spreadsheet): With 43 IAs and 6 TCs in a 60 card deck, First Wave 1: > 0.1% chance that you will draw 1 or more TCs without getting at least 2 IAs > 40.1% chance that you will draw 2 or more IAs and 1 TC
With 43 IAs and 6 TCs in a 60 card deck, First Wave 2: > 0.03% chance that you will draw 1 or more TCs without getting at least 2 IAs > 40.2% chance that you will draw 2 or more IAs and 1 TC
Note: It may also be "undesirable" to get 2 TCs and only 2 or 3 IAs, but this would barely increase the probabilities given above (I'd guess on the magnitude of 0.01%) so I left it out to make my job easier

Chris Berger
United States Round Lake Illinois

TheMav wrote: With 43 IAs and 6 TCs in a 60 card deck, First Wave 1: > 0.1% chance that you will draw 1 or more TCs without getting at least 2 IAs > 40.1% chance that you will draw 2 or more IAs and 1 TC
With 43 IAs and 6 TCs in a 60 card deck, First Wave 2: > 0.03% chance that you will draw 1 or more TCs without getting at least 2 IAs > 40.2% chance that you will draw 2 or more IAs and 1 TC
Thanks. So my question is: did I hurt myself by removing the Raiders? That is, how much better/worse would the numbers be with 8 TC? Or maybe 4 or 10?

B J
United States Columbus Indiana

arkayn wrote: Thanks. So my question is: did I hurt myself by removing the Raiders? That is, how much better/worse would the numbers be with 8 TC? Or maybe 4 or 10?
Excellent question. Here is a table providing the relevant statistics for the different quantities of troop capacity units. (Let me know if the table needs any more explanation.) This is for a 60 card deck, with 43 IAs, and a 7 card draw.
#TCs Good P Bad P Odds Ratio  4 33.6% 0.12% 272 6 40.1% 0.14% 291 8 42.2% 0.14% 297 10 41.1% 0.14% 288
With 43 IAs in the deck going from 4 TCs to 10 TCs means you go from 13 to 7 "other" units. From the numbers, increasing from 6 to 8 to 10 TCs, doesn't change probabilities that much, so I would be of the opinion that 6 TCs is the best choice since it allows you to still have 11 "other" cards in the deck (Probably some really helpful characters).

B J
United States Columbus Indiana

Just as a general note, the original post has been edited after making some corrections in my spreadsheet used to calculate the probabilities.
Quick Summary of Changes:  A 60 card deck should have a minimum of 16 infantry assault units before you consider including any troop capacity units.  To make effective use of troop capacity abilities, you need at least 20 to 24 infantry assault units and 4 Troop capacity units.


