Forum adverts like this one are shown to any user who is not logged in. Join us by filling out a tiny 3 field form and you will get your own, free, dakka user account which gives a good range of benefits to you: No adverts like this in the forums anymore. Times and dates in your local timezone. Full tracking of what you have read so you can skip to your first unread post, easily see what has changed since you last logged in, and easily see what is new at a glance. Email notifications for threads you want to watch closely. Being a part of the oldest wargaming community on the net. If you are already a member then feel free to login now.

My goodness, what had become of Dakka?!?! People saying NOT to use Mathhammer? What the hell is going on around here?

I think the overall message has been "Use Mathhammer, but do it before the game so you don't look like a jackalope."

Fizyx wrote:I ran a numerical analysis and plotted a histogram. There is a sharp drop-off right at the 16-18 mark. The graph is on BOLS forums.

Can't see the link (work comp) but without knowing the minutiae of the calculation all I can say is that it doesn't pass the smell test. The only way you could consistently achieve such a low average is if you're burning out after one spawn in the majority of cases.

Think about it, if your histogram is accurate, then Eldar Runes of Warding would be a marginal psychic defense and LD10 casters would pass their test in the majority of cases, which simply is not true. Same with melta versus AV14.

I think I figured out why the histogram was looking so funky.

The histogram is, by the way, very much correct. I'm just not thinking it through all the way. It explains why there are so many sharp peaks in the histogram.

Here it is: You run the probabilities and on average you should get 2.3 rolls with an average of 10.5 per roll, netting 23.65 gants on average. Obviously, this can't happen, it will be 23 or 24, but we are talking averages.

Now, that is a statistical average, and the numerical average for 100k iterations supports that conclusion. However, what the statistical calculation doesn't take into consideration that the numerical analysis shows is what number you roll changes the odds of if you can roll again. For instance, if you roll a 3, 4, or 5 you will have rolled doubles, so you can not roll again. This creates a deficit in the 13-16 region just because the highest probability 3d6 roll (10.5) can not be added to a initial roll of 3,4,5. Now, a roll of 6 has 10 possible outcomes, but only 4 of them doubles. So even though the chance for an outcome of 5 is 6/216 (or 1/36, same as rolling double 1's or 6's) the probability of rolling a TOTAL of 6 is lower at 4/216. The chance of rolling a 6 and getting to roll again is 1/36. This means that a chance of getting a TOTAL of 6 is lower than a total of 5. Now, this also means that a total of 16-17 should be a little higher than others around it, based solely on the roll of a 6. Now, a roll of a 7 has 15 possible outcomes, but 9 of them are doubles. So if you happen to roll a 7, you are MORE likely to roll a double than not. Which, again, threatens the totals further to the right from subsequent average rolls that will never happen. This is also true on the other end of the range. 16,17,18 will also kill your roll. While they themselves are not very likely, they make rolls of 26-28 even less likely since it caps the amount of number combinations you can roll to get those numbers.

What this ends up doing is creating a large region from 10-17 where the probability of rolling that TOTAL amount is actual very very similar. Then there is a SHARP drop at 18 to a lower plateau from 19-23 which are each about half as likely as 10-17. The median, by the way, is 18 even though the average is 23. The peak of the "probability" curve (which can't be drawn as a simple Gaussian or even double Gaussian) is 14, but a 15 is 2/3 as likely. 16, however, is almost as likely as 14. This is all a artifact of you are more likely to roll doubles with certain numbers. Since there are twice as many numbers in the area twice as likely, you are 4x as likely to roll one of those numbers than you would the larger ones. The curve is a very nice decaying exponential after that point.

Basically, the answer is this. All these probabilities for individual numbers is rather low, but the probability of 10-17 is much more likely than 19-23. What skews the average so far away from the median is the fact that if you do get 3+ rolls, the numbers that are more likely to give you those rolls are in the 8-13 region (24-39 total.) In other words, the more average you roll, the more times you will roll. Honestly, though, looking at the numbers a little more critical, even though the 19-23 is 4 times less likely than the 10-17, the 10-23 region is still well within what you would call one standard deviation from the peak of a Gaussian, if you really could fit one. 10-23 only encompass about about 50% of the area under the "curve" with about 17% below it and 30% above it. It looks really weird to someone who is not used to these kinds of distributions (me).

I hope that makes sense. This is WAY outside of my area of expertise.

GeneralRetreat wrote:I think the overall message has been "Use Mathhammer, but do it before the game so you don't look like a jackalope."

Ah yes, that is right. I got past the first few replies and had a heart attack.

I really liked your post, GernealRetreat on the real worl use of probability in actual warfare.

Also, Phryxis +1 from me as well, thanks for delivering the goods as usual.

GeneralRetreat wrote:I think the overall message has been "Use Mathhammer, but do it before the game so you don't look like a jackalope."

Yup. well put.

Fizyx wrote:
I hope that makes sense. This is WAY outside of my area of expertise.

You are making loads and loads of sense. Thank you for taking the time to not only figure these things out, but also sharing them with us. Your brain makes me horny...

I'd really like to see a repost of the graph, here on dakka for example, or anywhere that doesn't require me to register with BoLS.

Here it is. Honestly, I think the numbers higher than 18 are more likely than I was preaching earlier, but you can see why I was freaking out, lol.

Thanks, Fizyx, this format makes a LOT more sense to me. What this really shows is that probability of spawning less than ~15 models is a little less than 40%. Since spawn burnout will occur after one spawn ~44% of the time or you could roll a low number for two spawns, that seems perfectly reasonable.

The "meat" of the distribution, however, occurs to the right of that threshold. So if you do manage to get past that first critical spawn roll without screwing the pooch, you have greater than a 40% chance of totaling 16 or more total models spawned.

Thank you Fizyx. I really like the seemingly random peaks and lows. Goes to show just how organised a random function can be :-)

And I'll risk stating the obvious: When making educated guesses during the game, always remember that your dice does not have a memory. No matter what you rolled the last time, you always have a ~44% chance to roll a double. As an anecdote: The high-rollers wheel at the monte carlo casino once came up "black" 27 times in a row. A lot of people lost huge amounts of money by thinking "OMG, 25 black in a row, the chance of the wheel coming up black again is less than 1 in 67 millions, let's bet for red." Unfortunately the wheel always has ~49% chance to come up black.

CKO wrote:Why do I see all of these guys pull out their calculators to do mathhammer?

Because it is useful as everyone - including yourself - knows.

Dont you get the same thing from experience?

Sometimes, sometimes not. Everyone can spend some time with a calculator but not everyone can pull out experience from his pocket. Knowing chances on top of experience is even better.

The only problem with mathhammer is that it is like a science experiment under perfect conditions

Rubbish. You're just making up stuff again.

Generally speaking, I'm a rather indecisive player, and being an Imperial Guard player I'm rather often faced with the decision of what to shoot. For example, I've got an autocannon and there's a damaged killa kan squad (lets say 2 guys out of 3, assorted missing weapons and things) coming at me from one angle and a dread on the other. While I don't bring out a calculator I do something much, much worse.

I ask the clubs resident accountants and/or bankers what is the ones I have a higher chance of killing, even if its my opponent!

Shameful, yes, I know :(

I do think experience does count a lot though, if you're either instinctively know, or by experience know that this should do better at killing that, you become that more efficient in your game, with far less uhmming and aaahing like I do!

I do know I've improved in the last 6 months though now that I've gotten more into the swing of the game.

+1 Fizyx. That last analysis lines up a lot better with my memory what I've observed tervigons doing in actual gameplay--often crapping out on the first roll, and I have yet to see a pair of tervies spawn more than about 30-35 gants total.

All the more reason, as Shep pointed out, why nid players should run their tervigons like transports: haul forward at full speed and not spawn until they're in the opponent's grill instead of trying to drop one spawn after another while slogging across the board.

I think another thing that was screwing me up is that I normally run this analysis for a given number of shots. That is, I run 100k iterations of 24 pulse rifle shots that hit on 4, wound on 3 and save on a 3+. That means the maximum number of wounds will be 24 (it is also why the x-axis says "wounds," lol.)

In this case, though, there is no theoretical maximum except for the kill I put in at turn 7. When I ran the simulation for a maximum of three iterations, the real possibility of the 23+ range really shone since basically everything that would normally be after 30 was now added into the final third roll.

In other words, Flavius and others are right. There is really no good reason to spawn anything on turn 1 or 2. Too bad you can't pod it in.

This is a fantastic thread. I've always used probabilities in my games and it has helped tremendously over the years. I apologize before hand if my point has already been covered in this thread, but here goes.

I see one of the major problems with the application of probabilities with players is the misinterpretation of data.

For example, on average you need to cause 3 wounds to expect 1 failed power armor save. So most players expect to kill 1 marine for every 3 wounds they cause. But what this leads to is that people start to believe that this the rule and not the expected average.

Truth of the matter is that when you have caused 3 wounds, there is actually only a 70% chance of actually killing that marine. In other words, there is a 30% chance you can successfully consecutively pass 3, 3+ armor saves. (Odds of pass a single 3+ save is 2/3, odds of passing all 3 3+ save is 2/3x2/3x2/3 = 8/27 or 0.2963).

My point is this, the expected average is just that. An expected AVERAGE. It is not a guarantee that you will get the result you are expecting, and in a lot of cases, the actual chance is a lot lower than you think. What this leads to is a lot of times players either don't apply enough resources to solve a problem, because they have a false sense of security, or they don't take a shot because they think the odds are much lower then in reality.

Expected averages are quick and simple arithmetic problems that everyone can do in the middle of a game. It has its uses, but we all need to think a little deeper to get to the truth.

So if anybody is willing to revisit the infamous lascannon vs. assault cannon against a land raider numbers, I'd be really curious to see that (maybe on a different thread, though?).

It always seemed to me that even though the assault cannon gets higher average penetration, the fact that you only roll that additional die when the rend rule is "triggered" by a 6 on the penetration roll meant that a bunch of relatively infrequent outcomes with big numbers were pulling the mean artificially high, away from the actual probabilities.

Chance of Penetrating hit (any result)

BS 4 Lascannon vs AV 14

Hit on 3, Pen on 6 x1 shot
(66.6% x 16.6%)

11.0556 Chance per round of shooting

BS 4 Assault Cannon vs AV 14

Hit on 3+, Pen on 6, then 5+ x4 shots
(66% x 16.6 x 33.3%)

3.615 Chance per shot
(x4) 14.462X chance per round of shooting


As you can see, the difference is marginal (3%) but distinct, and only results from the higher rate of fire of the Assault Cannon.

Some other factors to consider:
Lascannons have 48" range, so technically, an Assault Cannon has a 0% chance from 25"-48", and 11% is better than 0%.

Twin-linking will raise the statistics of both equally, except for the fact that it allows more chances at a Rending result for the Assault Cannon. Since Rending results are all that matter in this example, this swings the probability even more towards the Assault Cannon (inside 24" of course).

Case in point, I have had my Land Raider popped once by a T-L AC. Never by a Lascannon. But that's just experiential evidence and has no real bearing on the stats above.

Here is the quick and dirty version. I can go into more depth if you think is necessary, but I think the following analysis is sufficient.

Note: The odds reported are the odds of a hit bouncing off (i.e. not causing either a glance or pen), to calculate the odd of getting a glance or pen is the 1 - the odds. Lower is better.

Odds of a single lascannon hit bouncing off against AV10 is 0 (0%), AV11 is 1/6 (17%), AV12 is 2/6 (33%), AV13 is 3/6 (50%), AV14 is 4/6 (67%).

Odds of a single assault cannon hit bouncing off against AV10 is 3/6 (50%), AV11 is 2/3 (67%), AV12 is 5/6 (83%), AV13 is 5/6 (83%), AV14 is 8/9 (89%).

The inverse of these odds are the the odds of either a glance or pen, each additional hit you put on will lower the percentage of a bounce by the percentage of a not bounce.

For example, a single assault cannon hit on AV 14 has 8/9 (88.89%) chance of bouncing, meaning it has a 1/9 (11.11%) chance for a glance or pen. 2 hits will lower the odds of bouncing to 64/81 (79.01%). The difference in the odds of bouncing (8/9 - 64/81 or 88.89%-79.01%) between 1 hit and 2 hit is 8/81 (9.88%), which is a 11.11% improvement (8/81x9/8 or 9.88%/88.89%).

So what does this all mean? Well if you hit with 1 lascannon and all 4 assault cannon shots AV14, your have a 4.24% better chance of getting either a glance or pen. Assuming you have BS4, you have a 66% chance to hit with 1 lascannon, but only 19.75% chance of hitting with all 4 assault cannon shots. 3 hits from the assault cannon hits give you a 70.23% chance of bouncing vs a 66.67% chance of bouncing a lascannon hit.

Bottom line, the expected average of an assault cannon vs AV14 is better than a lascannon. However assuming BS4 shooting, you will only ever see that about 20% of of the time, and during that time it is only better by 4.24%.

*Edited for clarification.

Will wrote:

Odds of a single lascannon hit bouncing off against AV10 is 0 (0%), AV11 is 1/6 (17%), AV12 is 2/6 (33%), AV13 is 3/6 (50%), AV14 is 4/6 (67%).

Wait doesn't a roll of 1 always miss?

Sorry if this was covered in your description but I didn't notice it if it is.

CrazyThang wrote:

Will wrote:

Odds of a single lascannon hit bouncing off against AV10 is 0 (0%), AV11 is 1/6 (17%), AV12 is 2/6 (33%), AV13 is 3/6 (50%), AV14 is 4/6 (67%).

Wait doesn't a roll of 1 always miss?

Sorry if this was covered in your description but I didn't notice it if it is.

I assumed that you have already hit, as the roll to penetrate a tank is independent of how well you hit.

You can also look at it this way. With BS4, 33% of the time you will miss, thus giving you 33% chance of not doing anything to a vehicle right off the bat with a lascannon. An assault cannon firing with BS4 has only a 1.23% chance of missing all four shots, so right off the bat the odds of you not doing anything with an assault cannon is much better than the lascannon. I should have clarified this better.

Ok thanks for the clarification

Will wrote: I apologize before hand if my point has already been covered in this thread, but here goes.

If you truly feel this way you could try READING the thread before posting. It's actually a great way of getting to know what is already covered in it. Much like how real life works actually... Allright, sorry for that sarcasm, you just happened to be the latest one to post that actual sentiment, and it always annoy me. So I took it out on you. Nothing personal.

Apart from that sentence your post is very sound. Data is worthless wihtout interpretation. And yes, that has been stated earlier in this thread, but not nearly enough times. Data != Information. Average != Effect.

Speaking of evaluating the value of various combat/shooting configurations, there's something I'm calling "weighted expected value".

It's basically the expected value of a set of attacks (the product of the total number of attacks multiplied by the likelihood of any particular attack having some effect), multiplied by expected utility if those attacks hit at a 100% rate, or the maximum potential value.

Weighted Expected Value:
EV = (Number of Attacks x Likelihood of Effect)
PV = (Number of Attacks x Likelihood of Effect or 100%)

WEV = EV x PV

The easiest way to explain it is to consider two Bolters and a Twin-Linked Bolter. While they may be equally reliable, having the two Bolters is better because they have a greater potential for damage.

The basic notion is that while thinking only of the expected value is nice, seeing as that's what's most likely to happen, we also need to consider what's less likely to happen: getting lucky and getting unlucky. Getting unlucky, thanks to the Warhammer system of potential for failure, is a wash. But getting lucky maximizes the potential of any number of attacks. If we think of the totality as divided into thirds (not really, but close enough), we can consider bad luck as happening 1/3 of the time, the expected value 1/3 of the time, and good luck as happening 1/3 of the time.

So because the weighted expected value accounts for 2/3 of the time rather than 1/3 like the expected value, we have a more accurate gauge of the value of various configurations, and a reason to account for the number of attacks twice (once when they increase the likelihood of something happening, and once again when they are what has happened).

To me the use of mathhammer is in deciding which units belong in my list. What am I playing against? Is it an all-comers list? How do I intend to use the unit? What is it delivering versus what does it cost to field?

When you get to a game, I follow the role I have for that unit in my army - If my primary AT unit is a devastator squad with 3 ML and 1 LC I am firing it at the enemy's biggest threat armor - albiet I may ignore the landraider and go at the predator but I am using mathhammer to develop an effective force mix not to figure out turn by turn what to shoot at or should I charge. That is by instict and experience. As an example, based on situations I may do something that mathhammer says I will lose but it may serve a vital purpose for the rest of my army... 10 tacticals with no PF sgt may just charge that wraithlord or ork boyz unit because I need to tie them up for a turn/turns to hold that objective even if mathhammer says I will lose.

WEV = EV x PV

The idea here is sound, but this specific implementation strikes me as being a bit skewed.

At various points in 40K, there are situations where you'd have a huge number of trials that have a very poor chance of success. Say, lots of S4 models trying to hit a Skimmer on a 6+. This result would end up looking far better than it should. It also places a lot of emphasis on hitting, rather than "getting a result."

What I'd suggest instead is going with the standard deviation, and then giving the values that describe either edge of the standard deviation.

If you don't like the standard deviation, you could use some arbitrary percentage, and then use the edge values such that you have that percentage for results between your two values.

To put it more clearly, you might say "when X shoots at MEqs, 90% of all results will be greater than 2.5 dead MEqs, and less than 3.75 dead MEqs." This will give you a range of results that you can be pretty certain will cover whatever result you get.

Actually, i have a question regarding mathammer.


Has anyone figure out how to apply rending rules as far as fractions/percentages go?



Automatically Appended Next Post:

Nurglitch wrote:

The easiest way to explain it is to consider two Bolters and a Twin-Linked Bolter. While they may be equally reliable, having the two Bolters is better because they have a greater potential for damage.

Sorry nurglitch, I mean no flaming, but that actually is wrong, or maybe I am misreading your post... this seems more likely, because you said a whole bunch of intelligement sounding stuff and I confused me.


When accounting for re-rolls, you take the original percent chance of something missing, and square it, and subtract this total from 100% or 1 if you prefer to use fractions. I prefer fractions myself, but I'll use your example of the two bolters and one twinlinked bolter against say... another space marine.

the original miss chance of an SM firing a bolter is 2/3, so the miss chance is 1/3. 1/3*1/3 = 1/9 so the miss chance of a TL bolter is 1/9.

therefore the equation for a TL bolter firing and wounding a space marine is 8/9*1/2*1/3= 4/27

The equation for one bolter is of course 2/3*1/2*1/3= 1/9 because there are two bolters, the value doubles to 6/27.

So two bolters aren't less reliable than a twin linked bolter.

Again, apologies if this is what you were trying to say


Btw, what you're saying about the luck factor is spot on

There is absolutely no reason to dislike off-table mathammer. Sorry OP, but that's the truth, and if you don't like it, ignore it. It's perfectly fine to calculate your chances of getting a certain number of kills or termegant spawns or whatever in a given situation.

However, when you whip out a calculator mid-game, that's just lame. Let's face it: you're not a general, you're playing with little plastic men on a table. When someone takes out a calculator mid-game because "real" generals do it, that just ruins the whole atmosphere and completely takes any fun out of it. It's a fun game. So keep the fun in it, and don't be that guy that has to calculate his chances of killing the whole squad of terminators with a single Firewarrior assault when any n00b can see that the skinny anime guys are gonna get destroyed.

Shrike78 wrote:Has anyone figure out how to apply rending rules as far as fractions/percentages go?

I can do it, but I don't have the formulas for rending in my program, so I have to do it all by hand.

Basically, I calculate the normal wounds like normal, but wounding on one higher (to remove the 6 for wounding), then add in the rending results after-the-fact. It takes a few moments, but it is doable.

I have two programs. One actually takes seed numbers and calculates the binomial distribution. The other just does 100,000 iterations of the dice rolls and plots a histogram of the results. The second I am modifying to allow for rending and other effects. I just haven't had the time, but next week looks clear.


Automatically Appended Next Post:
Overall I have two rules of thumb when it comes to Mathammer.

The first is this: happens. I account for this by rating probabilities. Average is "average." One standard deviation above average (84%) is "likely." Two standard deviations (98%) is a "sure thing." One standard deviation below is "unlikely."

For instance, when Tau was my main army there were certain rolls I NEEDED. I need to pop transports. Well, one squad of Deathrains has a "likely" chance to pop the Rhino, but I needed a "sure thing." Well, two Broadsides also has a "likely" chance to immobilize a Rhino, and when you put them together you have almost a "sure thing." So, at 1500 points that is what I had vs. a Mech army. It had the added benefit of being able to go after a second target if the first "likely" result succeeded.

Similarly, if I knew a certain load-out had an "unlikely" chance vs. MEQ, I just wouldn't bring it unless it had a secondary purpose (like a Pathfinder squad.)

The second is this: Dice Gods favor the odds. It sounds like a "duh" thing, but if you look at a probability distribution for a moderate number of shots with a low likely-hood of success for each individual shot, you will see the opportunity to roll poorly is more devastating than the equivalent rolling higher. That is, there is a larger number of wounds separating the standard deviation above the distribution curve than below. The exact opposite is true of low count shots with higher probability. Yes, the deviation is technically higher, but the chance for rolling exceptionally well is better than rolling like . That is why you should never let you terminators out of cover in range of my Helios suits with Markerlight support, lol.

That is how I plan my lists pre-game.

Of course, there is some stuff that just *doesn't* make sense on paper, but is phenomenal on the table. That is one thing a good player can recognize and why experience is still necessary in army list construction.

Fizyx wrote:

Shrike78 wrote:Has anyone figure out how to apply rending rules as far as fractions/percentages go?

I can do it, but I don't have the formulas for rending in my program, so I have to do it all by hand.

Basically, I calculate the normal wounds like normal, but wounding on one higher (to remove the 6 for wounding), then add in the rending results after-the-fact. It takes a few moments, but it is doable.

so the formula for say sniper scouts vs MEQs would be

Normally 1/2*1/2*1/3

with rending X*(1/2*1/3(?)*1/3) + X/6(?)

My friends and I actually figured out the equation a few months ago, but none of us for the life of us can remember what it was... I blame super glue fumes

IF you wanna play the game the way its supposed to be played, you dont use a calculator. Doing it in ur head is different in most cases...