## Thursday, December 11, 2014

### Math by Request: Screamers vs Landraider

I had a request from reader Jon Glas for the probability related to screamers attacking a landraider.  Landraiders can be tough to kill, especially for daemons.

We are going to look at two cases here: attacking normally, and attacking with prescience.

Attacking Normally:

With 9 screamers there is an overall 20.28% chance of destroying a landraider in one round of combat.  Within that chance there is a 15.48% chance of exploding it, and a 4.8% chance or glancing it out.

Initially this doesn't look good. A one in five chance is not all that great.

**Warning: here is my work**

I started by calculating the chances of an 'explodes' result.
2/3 * 1/6 * 1/6 = 1/54

Using this we find that there is an 84.52% chance of not getting an 'explodes' result which means there is a 15.48% chance of getting at least one 'explodes' result.

and the chances to cause a hull point without causing causing it to explode.
2/3 * (1/9 + 1/6 * 5/6)= 1/6

Rather than finding the chances of getting 4-9 hull points I instead found the chances of 0-3

0: (5/6)^9 = 0.1938

1: 9(1/6)(5/6)^8 = 0.3489

2: 36(1/6)^2 * (5/6)^7 = 0.2791

3: 84(1/6)^3 * (5/6)^6 = 0.1302

This gives a 95.2% chance of doing 0-3 hull points which means there is a 4.8% chance of doing 4+ hull points.  If we combine this with the 15.48% chance to explode this gives an overall chance of 20.28%

**End of math**

Attacking With Prescience:

Now if we look at what happens when the unit has prescience the results are fairly dramatic.  All of a sudden we have a 40% chance of destroying the landraider.  The chance to explode increases to 20.15% and the chance to glance out the vehicle increases to 19.85%

**More math!**

Chance to explode:
8/9 * 1/6 *1/6 = 2/81 = 0.2015

Chance to glance:
8/9 * (1/9 + 1/6 * 5/6) = 2/9

Same style as last time:

0: (7/9)^9 = 0.1042

1: 9(2/9)(7/9)^8 =  0.2678

2: 36(2/9)^2 * (7/9)^6 = 0.3061

3: 84(2/9)^3 * (7/9)^8 = 0.1234

This adds up to an 80.15% chance of 0-3 hull points and a 19.85% to do 4+ hull points.  Combining this with the chance to explode this leads to a 40% overall chance.

**End math**

Wow!  I was completely shocked by this result.  The increase to the chance to explode was about what I expected, but the chance to cause 4+ glances increased by a dramatic amount!  Having a 40% chance to destroy a landraider in one turn doesn't seem incredible, but consider four melta guns in melta range only have a 44.14% chance of destroying that same landraider.  All of a sudden our attacks look much better.

If we manage to knock a single hull point off of the landraider (with all that daemon shooting) our chances of destroying it with the screamers jumps up to 52.34%.

Now if you opponent is letting a squad of 9 screamers run around they are doing things wrong.  What happens if you get knocked down to only 5 screamers?

Without prescience 5 screamers have a 9.25% of wrecking a landraider with 8.92% of that chance being an explodes result.  With prescience our chances increase to 12.24% overall with 9.4% for exploding.

What we see is that when the unit gets smaller prescience becomes less helpful. With a full unit prescience increases our unit's damage to 197% of our normal damage, but when we are at half strength we only do 132% of the normal damage.

So, if you want to wreck a landraider fast make sure your screamers get there in one piece, or be prepared to dedicate more than one squad to the job.  This falls in line from most of what I had learned while playing daemons.  Half strength squads are not going to get the job done, and it is generally better to make sure you kill your target the first time.

1. Neat. Help me figure out your equations. You posted this:

I started by calculating the chances of an 'explodes' result.
2/3 * 1/6 * 1/6 = 1/54 = 0.1548

I would think there is a 1/2 chance of hitting, a 1/6 chance of exploding assuming you scored a penetrating hit. So the hard part is calcing the chance of penetrating because there are multiple dice combo results that net a penetrating hit. For example, you need a 10+ on the armorbane roll to penetrate. That is a 5,5 or a 5,6 or a 6,5 or a 6, 6. That is 4 of potentially 36 results. So to me the calc would be: 1/2 * 1/9 * 1/6 = 0.009 or 0.9%.

So obviously my math is wrong. So help me understand yours :)

1. I skipped a couple of steps in what I was showing. The 1/54 was correct but that lead to the 15.48% chance to explode. With the 1/54 there was an 84.52% chance of causing no 'explodes' result giving us the 15.48%.

I also updated the main post. Thanks for pointing this out.

2. Oh and with the chances to penetrate: You are correct we need a 10+: (6,6); (5,6); (6,5); (5,5); (4,6); and (6,4), so there are a total of 6/36 possibilities that will cause a penetrating hit.

3. Oh, doh Lamprey's Bite is S5 not S4... that totally changes the little bit of math I did in the last post. This is what I get for not playing a 40k game for like 3 months.

4. Also to John, the chance to hit is not 1/2. Vehicles that move gain WS1 until the end of the next turn. So it's a 3+ to hit not 4+.

5. How did you calculate the overall % chance of 9 screamers exploding the LR? We determined a single attack has a 1/54 chance of doing it, but where I get lost is factoring in multiple die rolls, like in this case where 9 screamers are making a single attack each. Do you multiply the chance of a single attack causing an explode by the number of attacks? If so that is a 16.66% chance (1/54 * 9) or (0.0185 * 9).

This is fun :) Should change the name of this thread to mathhammer is awesomesauce :)

6. The chances of not getting an explodes result is 53/54 so the chances of 9 guys getting 0 explodes is (53/54)^9 = 0.8452 So, I did 1-0.8452 and this gave me a 0.1548 chance of getting at least one 'explodes' result.

2. I really like this blog.
:-)

1. Thanks! It may sounds cheesy, but it means a lot to hear.

By sheer coincidence when I posted this update yesterday my students were taking their test over probabilities.

3. Totally quiz them on screamers attacking land raiders :)

1. Unfortunately we didn't have time to go over combinatorics, but finding the probability of getting an explodes result is certainly within the realm of possibility.

As far as I know I don't have any students who play 40k, but I did catch one girl reading up on the history of the Dark Angels in class.

4. I have a somewhat related question then. When a Screamer charges and makes it's special attack it says that all of it's normal attacks are consumed, but what about the attack you generate by the charge itself?
Does this attack feed into the special attack (making it 2 special attacks per Screamer), since it's an attack gained by a special circumstance (charging) and thus can not be classified as a 'normal' attack surely (i.e., the amount under the Attacks column). -Or-
Does the Screamer gain 1 Screamer special attack and one 'normal' charge attack to be done at the Screamers usual stats. -Or-
Does the screamer no longer have this attack at all for some reason?

It's always been a question that I've considered and I've never really come to a firm decision myself

1. It loses the charge bonus. It's the same thing as a MC with smash which replaces all of its normal attacks. You get one and that's it.

2. You can think of it exactly like stat modifiers. If you have a x 2 multiplier, that goes first. Then any + or -. And finally if you are supposed to set a stat to a fixed level (in this case one) that overrides everything else.

So you get your base attacks + charge bonus when you charge. If you want to, you can replace all of those for a single lamprey's bite.

5. I would assume that the attack gained by charging is part of its normal attacks, so even on the charge it would only have one special attack.

The FAQ doesn't have anything on this either.

6. One math question: It could be interesting to calculate ho many "weak" models would be needed to make reasonable to assault a strong unit. Ex: how many gretchin, or witches, or eldar guardian, or AM troops wold be needed to assault some of the nonsense space wolf units to balance the match....

1. This is a mathematically difficult problem to solve. I may go so far as to say that a general solution of how to solve your proposed problem is unsolved.

The reason a general solution is (presumably) unsolved is because it would involve solving for an unknown number of exponents at the same time as solving for an unknown number of bases.

However, that doesn't mean I can't answer your question. Specific solutions can be found by trying different numbers of troops until a desired percent is achieved.

I encountered a particularly nasty Space Marine commander on a bike, and I have been wondering how many nurglings it would take to stop him. These calculations may be my distraction from grading finals.

7. :-) yes, maybe I will try by myself with some different units... Then we could check the process to achieve the result desired and maybe find a general equation....

1. +50% of success will be my obj)

2. If you get a result I would be happy to check and see if I get the same thing.

I don't thing a general equation is possible. A solution to this type of general equation would have some profound impacts to mathematics in general. More specifically it would mean that the encryption of the internet could easily be broken.

http://en.wikipedia.org/wiki/Discrete_logarithm

3. So we will be famous, bro'! :-D :-P

4. If you define what you mean by "reasonable odds" then you could make three pseudo-general equations that you just have to plug numbers into. The three equations would be for the different cases of good unit having higher I, same I, or bad unit having higher I.

Basically what you're left with is a general formula in which you plug in number of attacks per model, chance to hit, chance to wound, and chance to fail saves.

It's only general in the sense that the computations are the exact same every time but you have to know the exact statlines of the units fighting each other and which ones get the charge.

I would think the easiest way of going about it would be to calculate expected values rather than probabilities. Something like "how many gretchin need to assault 5 space marines in order to have an expected value of killing all 5 of them" would be a pretty simple calculation. Something like "how many gretchin need to assault 5 space marines to have a 75% probability of killing them all" would require a few more steps.

8. To win a fight is more complicated than just put "n" wound in one turn. I'm thinking that a good start is to have an expected value to put one wound more than the opponent, calculate the possibility of him running away and ours to catch and destroy him (if possible). Then re-do the calculation for the 2nd phase, without the bonus of the charge and see where the fight is going. [Maybe we have some possibility to win the fight on the first turn, but maybe the bigger probability is that the fight goes on and we begin loosing it from the second turn....]

then the problem of the terrain we are charging (or are charged) trough...

What else?

The purpose of this calculation is not the exercise in himself, but rather is to have an idea of what to do when we find our self in some specific situation. Like the poker player that when he find with J10 in his hand and looking at that 3 card on the table knows if the best move for him is to raise or to check/fold the hand. :-)

1. Back in 6th edition I actually wrote a formula to calculate the probability of winning combat. It was very complicated, but theoretically worked. Practically speaking there was too much going on for it to be very helpful, so I attempted to write a program. I am not very good at programing, and as a result never got it to work. Also with the ever-changing rules this would be a continuous endeavor.