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Wednesday, September 1, 2010

To Frag or not To Frag???

I was doing some thinking about my SW list, and I started to question, when is it better to use Frag Missiles or Krak Missles to kill stock marines? I was somewhat surprised by the analysis, and figured I would share.

Krak Missile:
Chance to Hit: 0.666
Chance to Wound: 0.833
Chance to Kill = 0.666 * 0.833 = 0.5547 or 55.47% chance to kill.
Chance to Kill if in Cover = 55.47 / 2 = 27.7% chance to kill.

Krak is easy to calculate! However, before we can truly analyze Frag's potential, we must first deal with the "deviation" issue. The "deviation" issue is that a Frag missile doesn't have to exactly "hit", to hit. For example, a hit = a deviation of 4 inches or less = a deviation of 5 inches or less (because a centered template won't move off a model with a 1 inch deviation), etc.

So, to begin dealing with this, let's analyze just the flat out odds of a single Frag hit killing a stock marine.

Frag Killing a Marine:
Chance to Wound: 0.5
Chance to Fail Armor Save: 0.333
Chance to Kill = 0.5 * 0.333 = 0.1665 = 16.65% chance to kill a marine if a Frag missile hits it.

So, right at face value, to kill a single Marine, Krak is better than Frag, and we are assuming that the Frag auto hits! It's better even if the Marine is in cover.

Yet, we know that Frag doesn't hit just one marine (in theory anyways). That's the point! It hits multiple targets. So, the fundamental question is, when is it better to use Frag instead of Krak?

Just going off of what it takes to kill a Marine that is hit by Frag, it will take 2 HITS to Equal 1 Krak Missile against a Marine in Cover, and 4 hits to Equal 1 Krak Missile against a Marine in the open. We know that by using my trusty "At Least" calculator. Remember, this is statistics people! We would be so lucky as to just multiply here! Basically, 4 hits = 51% chance. 2 hits = 30% chance. It's close enough (although technically, you would need 5 hits to equal 1 Krak in Open BECAUSE there is no such thing as a fraction of a hit!)

Well! All of this great. The problem is, Frag doesn't auto-hit, does it? Indeed, it does not. That little blast template deviates. In fact, it deviates 44% of the time!

Here's the math:
Chance To Roll a Hit + Chance that 2d6 <= 4 = Total chance to not deviate.
Chance to Roll a Hit = 33%
Chance that a 2d6 <=4 = 16.7%
Total = 33.333% + (16.7777% * 66.66%) = 44.4%

*** In case you were wondering where the 16.7% * 66.6% came from, it's because the chance to roll <=4 on 2d6 is 16.7%, but when the Hit Dice is used, 33% of all outcomes are already a hit. Thus, 16.7% of the remaining 66% of outcomes is all that we need to apply.

So, 44% of the time, the blast marker will go exactly where we put it. This is something we can work with! Okay, let's do some more Mathhammer! Now, let's assume our Blast Template HAS TO HIT for this to work. How many Marines would HAVE TO BE HIT in order for the Frag to be better than the Krak?

Frag to Hit + To Kill:
Chance to Hit: = 0.444
Chance To Wound: = 0.5
Chance to Fail Armor Save: = 0.333
Odds to Exactly Hit and Kill A Marine With Frag = 0.444 * 0.5 * 0.33 = 0.0739 = 7.39%

Not very good at all! So, let's put our handy "at least 1" calculator to work and ask the question: "when will Frag hits EQUAL a Marine in Cover and not in Cover?"

Marine in Cover = 27.7% chance to kill with Krak.
How Many Hits will it take from Frag before we greater than or equal to a 27.7% chance to kill? Survey Says: Approximately 4 hits (26.4%)!

We have to Tag 4 Marines with an exact hit to equal 1 Krak Missile firing into a Marine in cover. For a marine not in cover, we need 11 WHOPPING HITS to equal one Krak Missile. Yikes!


One more thing to consider before we come to our final conclusion. Just because a Frag missile doesn't exactly hit where we want to doesn't mean that it misses!

One could make the argument that slight deviation isn't a big deal. However, now, we are entering into the realm of situational arguments! The next question is, well, how much deviation is acceptable? That's going to depend on how clumped up the target is! Now, I knew you would ask me these questions! So, I'm solving it for you!

Here's your answer to the age old question in RAW form:

How many Marines do I have to be able to HIT with Frag before it is better to use than Krak????

Odds To Kill a Marine where No Deviation is Acceptable: 0.0739
- Need to Hit 4 Marines = 1 Krak in Cover
- Need to Hit 11+ Marines = 1 Krak not in cover
Odds To Kill a Marine where 1 inch Deviation is Acceptable (51.8% to deviate one inch or less): 0.0862
- Need to Hit 4 Marines = 1 Krak in Cover
- Need to Hit 10 Marines = 1 Krak not in cover
Odds To Kill a Marine where 2 inch deviation is Acceptable (61.05% to deviate two inches or less): 0.102
- Need to Hit 3 Marines = 1 Krak in Cover
- Need to Hit 8 Marines = 1 Krak not in cover
Odds to Kill a Marine where 3 inch deviation is Acceptable (72.15% to deviate three inches or less): 0.12
- Need to Hit 3 Marines = 1 Krak in Cover
- Need to Hit 6 Marines = 1 Krak not in cover
Odds to Kill a Marine where 4 inch deviation is Acceptable (81.4% to deviate four inches or less): 0.135
- Need to Hit 2 Marines = 1 Krak in Cover
- Need to Hit 6 Marines = 1 Krak not in Cover

Okay, so, let's wrap it up!

Q) Well, how do I know how much deviation is acceptable?
A) The answer to this is easy! The acceptable deviation = the amount of marines you NEED TO HIT to be better than Krak. So, if Marines are sitting in cover, and you can still hit 3 Marines OR MORE, even if the template deviates 2 inches from where you place it, Frag is better! If not, Krak is better. You see?

Q) So, basically what you are saying is that the likelihood of Frag ever being better than Krak against a Vanilla Marine who is not in cover is slim to none?
A) Yes, that's exactly what I'm saying. If you can't see that here, look again. :) There is a slim possibility that you could tag enough Marines, but they would have to be seriously clumped up, like just having deep struck or something. Remember to pay attention to your deviations before you go "NUH UHZ IT HAPPENZ ALL DUH TIMEZ!"

Q) So, what's the short answer here?????? (You expect me to follow all of this crap???)
A) The short answer is!!!

You need to be able to hit 4 Marines with a direct hit to really make it statistically better than firing a single Krak Missile into cover. It's basically never more efficient to use Frag over Krak if the Marines are not in cover.

Past that, things get more complicated. You have to start "eyeballing" allowed deviation amounts and odds of hitting a certain amount. Refer to the chart to see! Basically though, if you lay that small blast marker down and it touches 4 or more marines, you are better off using Frag (unless they aren't in cover!)


  1. Which is why Lash and Plasma Cannon combo was so powerful before the institution of mechanization. Being able to insure that you are hitting a clump of guys means that you can really negate a lot of the issues coming about because of deviation. However, an astute opponent will usually place his little guys, 2" apart which means a small blast marker will only hit one guy with a hit and it will require a deviation to hit more than one.

  2. WOW neil - nice work. Looks like i was nearly correct with my 3 or more in cover rule with frag missiles vs marines. Then again vs anything its always 3+ for me as a general rule of thumb.

    The applications are fantastic. DS units that need to shoot to be effective = frags

    Units making the most of cover on a gangway (like your terrain and ours) like anti castling!

    When a vehicle is wrecked with an assault unit nearby, meaning the owning player will try to place them on the other side of the wreck - then frag them up nicely (5 isnt so hard in this circumstance) That when your sheer number of missiles in your SW army work brilliantly. you have most angels covered for los

    When a vehicle explodes (marines are in cover and a bit clumped up) also with the threat of a charge he could try to make you travel thru terrain to stop the charge. apply the frags and you could be forcing alot of saves! with 5 X blasts it can be either amazing or a massive anti climax. I really thank my use of Grotzooka's in the past that gave me the knowledge of when to use frag vs certain targets and how to force the use into your advantage.

    Fantastic post mate.

    Well done

  3. Very useful. Confirmed my suspicions.

  4. I'm seeing a flaw with how you do your deviation. When firing at a single target, your math appears correct in assigning 44% chance to hit with frag vs. a 66% chance with krak. However when shooting at a squad the frag's ability to hit is conveyed onto all of the target's you're firing at. Put another way, if I place my marker over 3 marines I won't score a direct hit, a miss, and a slight deviation in the same shot.

    There's also a problem with the 'at least 1' method you're using in that you can't just multiply when dealing with volleys of dice. Consider this, 2 marines take 2 wounds. The odds for each to die is 33%. If you then multiply by the 2 marines you would assume that there exists a 66% chance to kill 'at least one'. That would be incorrect.

    ~The odds of both dying are 11% (4 out 36 possible combinations)
    ~The odds of exactly one dying are 44% (16/36)
    That adds up to 55% (20/36) to kill at least one.

  5. yea but to try to iron out all these details is a bit much - the end result would not deviate much even if you spent hours and hours getting an average. even then it wont be true to 'your' game.

  6. I'm more looking at it as the % chance to do what you want to do, which is hit something. If you want it to hit directly, you have a 44% chance to do that. If you are comfortable with it deviating less than or equal to an inch to hit targets, you have a 51.8% chance to do that. Assuming that it lands where you want it, I'm then calculating how many you need to hit to make it worth your while. (In other words, telling you what it is that you want to do with it). So, if your overall chance of getting a direct hit is 44%, then what you want to do is to tag 4 marines with it because otherwise your odds are not favorable to get what you want, what you really want being having a greater than or equal probability of killing marines than a Krak missile.

    Not sure I'm clear on where you were going with the second question.

    Calculating at least one is done by adding the probabilities of killing 1, or 2, or ....N together, not by multiplying. As is the case here.

    Also, the odds to kill nothing is 0.666 * 0.666 (chance they both make their armor saves) = 0.444, which means that the odds to kill one is about 0.56, like you said.

  7. Now, if you do actually HIT 4 marines, your odds of killing them at that point go up substantially, having an approximate 16% to kill a marine, a 51ish% chance to get at least one.

    The point is though, we can't assume that we hit all 4. To for certain tag all 4, we must either directly hit (44% to happen) or have some acceptable deviation (% to hit shown). So, the odds of killing any marine are effected by the % chance you had to actually hit those marines the way you wanted in the first place.

    That being said, you are right with deviation that you cannot simply, flatly, apply that same % across all models because even some deviation could still be a hit. But then, we are no longer certain of how many marines we are hitting to get a good estimate of the least number of models that need to be hit to make it better than firing Krak. So, to make it easy, we introduce an estimate.

    So, perhaps better titled the estimation of the number you need to tag given an acceptable amount of deviation!

  8. I'm arguing way above my math level :)

    I think I'm getting caught up in the semantics of probability vs. possibility. In practical application the krak missile is probably not going to kill a marine; frag, even more so, is not likely to kill a marine. In practice there's no reasonable expectation of killing a marine so the decision to shoot at him is a pot shot. Pot shots inherently hope for luck, so in their undertaking one wants to maximise they're opportunity for luck.

    As you illustrated, if 1" is acceptable deviation you tip over the 50% threshold making you odds on to hit(ish). A tightly bunched unit will lend better potential to get multiple hits which will in turn offer more opportunity for luck (thus satisfying my stated goals for taking pot shots)

    P.S. Google ID insists on using my real name, but I post as Tathen on your forums

  9. Dunno why it's doing that :)

    You are bringing up a good point about an expectation and maximizing the potential for getting lucky!


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