Welcome to part two of the MathHammer series. This time we will be looking at different ways of dealing with flyers. When I looked at the new Chaos codex I wondered if the new flakk missiles were actually worth an extra 10 points. I also wanted to compare the quad gun and the icarus lascannon. All this was done in my new
MathHammer Calculator which you can download at that link. More below...
To figure out if flakk missiles are really worth the price, and if there are other things that could deal with flyers more efficiently, I looked at a few anti-air units that I thought may be some good choices. First 4 Chaos havocs with krak missiles, havocs with the new flakk missiles, havocs with autocannons, and a helldrake with the Hades autocannon. in the example above you can see that the helldrake has the highest chances of damaging an enemy Stormraven, and despite being the most expensive of the 4, it still has the highest kill point ratio (damage/point cost).
I summarized the probability of getting a penetrating hit in the table above. 4 Flakk missiles are just as effective as the helldrake against AV10 flyers, but the helldrake's S8 cannon means that it's lower ballistic skill is less of a factor and is better against AV11 and 12 flyers(it also can vector strike which adds to it's effectiveness). The average penetration over all AVs per 100 points shows that the helldrake is slightly better than flakk missiles than 4 havocs with flakk missiles.
An important thing to keep in mind about all MathHammer is to make sure you keep the context in mind. Just because something is better at destroying a unit and has more potential damage/points does not mean it is the better choice. For example, taking a helldrake with the autocannon may be better against flyers, but if the enemy has one flyer and a bunch of infantry then it would be much better to take the baleflamer. Also, it may not be best to take any dedicated anti-air at all because it would take away points for something that is more important. Just remember that figuring out probabilities can be very helpful, but you need to test out your lists and determine the weaknesses and strengths.
Aegis Defense Weapons
This one is a little more straightforward. Which is best, Quad-gun or Icarus lascannon? According to this, the quad gun is much better in terms of getting glancing hits to strip hull points and surprisingly, it is almost twice as likely to get a lucky roll on the damage chart to explode the flyer! The Icarus would come out on top if you were using the weapon against a land raider since the quad gun can't even dent it, but other than that the quad gun wins!
Every once and awhile I catch the bug to really math out my warhammer stuff, so it's always pleasing to see others doing it as well. I would, however, like to offer a friendly critique. There is a subtle but important difference between what your calculating and a "probability." What you've actually calculated is an average number of successes (putting percents on this is misleading). You've probably already noticed this in your hades cannon calculation listed as 100% glance. We all know you don't have a 100% chance to glance. What you've calculated is, on average, you will have 1.00 glances; maybe more, maybe less. Although still a useful quantity, better to see it for what it is. If you want to take your mathhammering to the next level, investigate excel's binomial distribution function. Although more complex, it will allow you to calculate actual probabilities of different outcomes which, in my opinion, is a much better comparison.
ReplyDeleteI'll use the heldrake as a quick example. Each shot has 1/4 chance to glance (4+ to hit, 4+ to wound). So the chance for ALL 4 shots to glance is tiny: (1/4)^4 = 0.3%. The chance ALL of them do NOT clance is (3/4)^4 = 32%. It gets much more complex to calculate by hand the other outcomes but thats whats so great about spread sheets; it's automatic!
You are absolutely right about the whole percent vs. probability. I have actually been changing them over as I remember (that's why the percentages are mixed in with the probabilities). I am actually in a statistics class right now (Biostatistics). I had started making this before the class started so i'm adding new things as we learn them and fixing errors as I see them, so it's a work in progress for sure. And I had no idea excel had a binomial distribution function. I will look into adding that. And a quick question if you don't mind. I was trying to think of a good way to add standard deviation or confidence intervals but I can't think of any way of doing it while still keeping the worksheet simple. Do you have any ideas? And thanks for the input!
DeleteSorry I didn't get back to you sooner. Busy busy busy!
DeleteBy using a the binomial you get your answer in terms of probability for number of successes a specific condition. This can be made as complex as you want. For discussion purposes I'll make a simple one. Say you have 9 bolter shots on your marines and you define a success as just hitting (to keep it simple). A success then has a 2/3 chance of happening for each shot. You already know the average is 9*2/3= 6 hits. Using the binomial distribution will give you a break down of the probability that each number of hits has. In our example:
0 hits: 0.0%
1 hit: 0.1%
2 hits: 0.7%
3 hits: 3.4%
4 hits: 10.2%
5 hits: 20.5%
6 hits: 27.3%
7 hits: 23.4%
8 hits: 11.7%
9 hits: 2.6%
The binomial distribution is not a gaussian or "normal" distribution. As such its my understanding that standard deviation isn't strictly applicable. But the idea of such a thing could be approximated for the binomial distribution. In essence its just a measure of the "width" of the bump you see if you were to plot probability vs # of hits. For a normal distribution, one standard deviation (sigma) away from the mean is ~34% above and below that mean. You'll notice our bump is asymmetric showing why this is an approximation at best. knowing the average and the above probabilities you can give a standard deviation like quantity by simply finding the range of hits that +/- 34% lands in. Another option would to be just to quote the average and perhaps the probability of a certain range to give some idea of the width. For example the above would average 6 hits with about 70% chance to get 5 to 7 hits.
Happy mathing!
I have already played around with binomial distributions in excel a bit. Thanks for the idea. We actually talked about them in class just two weeks before so it's cool to be able to apply it. I suppose you could approximate a normal distribution with this, but just graphing out the binomial distribution really makes that unnecessary since you can easily see the probability of each number of hits. I'll probably update this post when I have some time. And thanks again for commenting!
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