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Ayo

So I was wondering where to get the equations to do this MathHammering business? I'm not the flashest at the ol' mafs, but wanna give it a crack to get a better understanding of my units and their capabilities.

Shanshu

It's math based on the limitation of the chance of a d6 coming out out a certain way.

Simple example. You need 4+ to hit and a 4+ to wound. You have one roll. Assuming perfect averages, you have a 50/50 shot to hit, and then a 50/50 shot to wound. So you have 25 percent of causing an unsaved wound going by pure average assuming they have not save. This is the best way to objectively look at how something will perform in a given situation.

As a long term player this is just a rough guide to decision making, an important one no doubt, but the dice like chaos are fickle.

Yes dice will ruin your averages lol. 3 games and tons of potential 4+ saves today. I didn't make more than 40 total saves lol.

My understanding, low as it is, is that one uses fractions, specifically, multiples of 1/6th, to estimate average whatever from a unit.

For example, you have 20 Necron Warriors in rapid fire range, trying to glance down a tank. Two shots each, so 40 shots total. Necron warriors hit on 3's, so remove 2/6ths (1 and 2 rolls) of 40. We now have 26 and 2/3rds shots. It glances on 6's, so take 1/6th of 26 and 2/3rds. We have about 4 and 1/36th glances. Thus, assuming no saves, a 20-man squad of Necron Warriors will deal an average of 4 hull points worth of damage.

Yeah, it's, assuming even probabilities, the likelihood of something occurring. Once you get good enough to memorize calculations, such as damage you'll likely do against SM, Terminators, Necron Warriors, etc you can start to work out which units should be attacking which units.

Xafilah wrote:
My understanding, low as it is, is that one uses fractions, specifically, multiples of 1/6th, to estimate average whatever from a unit.

For example, you have 20 Necron Warriors in rapid fire range, trying to glance down a tank. Two shots each, so 40 shots total. Necron warriors hit on 3's, so remove 2/6ths (1 and 2 rolls) of 40. We now have 26 and 2/3rds shots. It glances on 6's, so take 1/6th of 26 and 2/3rds. We have about 4 and 1/36th glances. Thus, assuming no saves, a 20-man squad of Necron Warriors will deal an average of 4 hull points worth of damage.

Cool,this I understand and can work with, but what throws me is saves etc. like an armour save and then a FNP or RP. Do we start cracking out BIMDAS

Klowny wrote:

Cool,this I understand and can work with, but what throws me is saves etc. like an armour save and then a FNP or RP. Do we start cracking out BIMDAS

Damned if I know what BIMDAS is, but let try another example.
Let's say those 20 Necron Warriors from last time have come under fire from a group of Tactical Marines. The Marines have scored 10 wounds, and the Necrons roll to save. They have a 4+ armor, so that means they tank exactly half (the 6, 5, and 4 rolls to save) of the wounds. Now for the glorious REANIMATION PROTOCOLS. Assuming no boosts, it is a 5+ roll to ignore whatever hit these Necrons, so they reduce the wounds to 3 and 1/3 wounds. So ten AP 5 or worse wounds on a squad of Necron Warriors results in about 3 Necrons slain.

Killing us zombie Egyptian terminators in space is quite tough.

What about reroll's etc like preferred enemy and target designated, one's rerolling ones, the others everything. Also master crafted, tesla etc. so many variables

Klowny wrote:

Xafilah wrote:
My understanding, low as it is, is that one uses fractions, specifically, multiples of 1/6th, to estimate average whatever from a unit.

For example, you have 20 Necron Warriors in rapid fire range, trying to glance down a tank. Two shots each, so 40 shots total. Necron warriors hit on 3's, so remove 2/6ths (1 and 2 rolls) of 40. We now have 26 and 2/3rds shots. It glances on 6's, so take 1/6th of 26 and 2/3rds. We have about 4 and 1/36th glances. Thus, assuming no saves, a 20-man squad of Necron Warriors will deal an average of 4 hull points worth of damage.

Cool,this I understand and can work with, but what throws me is saves etc. like an armour save and then a FNP or RP. Do we start cracking out BIMDAS

Just keep going with the math. Nothing changes.

In his example, he did 4 hull points. However the vehicle is in ruins with a 4+ save. So you remove 3/6s of the possible 4 hull points. Leaving you with 2 HP damage.

15 Necron warriors Rapid fire at Tac Marines. 30 shots. only 2/3s will hit, giving you 20 hits. S4 vs T4 means you wound on a 4+ or 50% of the time. Those 20 hits generate 10 wounds. The marine saves on a 3+, so only fails 1/3 of the time. So 10 wounds x 1/3 failure = 3.33 unsaved wounds.

Add in FNP. 3.33 unsaved wounds x 2/3s failure on FNP = 2.22 unsaved wounds.

rerolls are where it gets pretty tricky. Here is the formula.

%chance of success=%chance of passing +(%chance of failing x %chance of passing.)

This obviously looks more intense then it really is. Lets take an examples.

Say you are bs3 twin linked. You hit on fours rerolling. To calculate the %chance of each hit we plug the numbers into the formula.

%chance of passing is 3/6 because 3 sides of the 6 sided die are passing. This can be simplified to 1/2 or 50%

%chance of failure in his case is 3/6 because on 3 sides of a 6 sided dice result int failure. It's worth mentioning that this is always 1-%chance of passing as a decimal.

Now when plugging percents into formulas its important that you always do it in decimal form. so 50%=0.50

so %chance of success=0.50+(0.50X0.50)

First we do what's in the brackets so
%chance of success=0.50+(0.25)
Then the addition
%chance of success=0.75
Then you can convert our answer back into percents, and get an answer of 75%

Now to add that into other math hammer stuff you want to use the %chance of success as your too hit percentage.

So bs3 twinlinked str4 vs T4 no armor ends up being
0.75because that's your chance to hit multiplied by 0.50 because that's your chance to wound.
So you get 0.75*0.5=0.375 or 37.5%

The Basics
Your basic math to determine a wound delivered is as follows
(To Hit) * (To Wound) * (Failed Save) * (Failed FNP) * (Number of attacks)
It is easiest to express this as a fraction. Fractions are grade school math, so even if you are not great at maths, you don't need to worry about the calculus. Basic fractions will take you far.

So lets say you want to determine how well a bolter fired by a marine can kill a CSM.
2/3 * 1/2 * 1/3 * 1 = 1/9 chance to kill a CSM.

Now, to compare, lets look at a BS 3 inquisitor henchman shooting the same bolter.
1/2 * 1/2 * 1/3 * 1 = 1/12 chance to kill a CSM.

This lets you see that the henchman is much worse at killing that CSM, just because of the BS. About 33% worse, actually.

Killing more than one?
The chances to kill a marine double when up close and getting 2 shots per gun. This might wind up with odd fractions like 88/12 models killed, which means that ~7.3 models will be killed.
Another example is the he punisher cannon....
20 * 2/3 * 2/3 * 1/3 = ~3 dead MEQ per turn.

Rending
It gets a bit more tricky when talking about rending weapons. When shooting a rending weapon, you need to calculate the odds of wounding on everything except the 6, then add another calculation with the six. This is an example of what a warp spider exarch shot looks like vs a marine.
(31/36 to hit) * (4/6 to wound) * (1/3 failed armor save) = 31/162 chance to kill per shot. ~19.1% +
(31/36 to hit) * (1/6 to wound) = 31/216 chance to kill per shot. ~14.3%

Melee Weapons
In assault, you follow the same formula. This is comparison TH/SS terminators vs striking scorpions in and our of an aspect host.
(2/3 to hit) * (2/3 to wound) = 4/9 chance to kill per swing. ~44.44%
Under the aspect host they look like this.
(1/2 to hit) * (2/3 to wound) = 1/3 chance to kill per swing. ~33.33%
Again, we see that same damage reduction, so the scorpions take 75% of the damage they normally would.

Why Mathhammer
The law of averages are just that .. averages. There will be times when you will fail 10 3+ saves in a row. There will be times you get four 6's in a row. Just because you can expect your grav guns to kill 8 models does not mean they will.

There are enough dice thrown in a game to where the law of large numbers will come into play, but variances in that law will cause your game problems. IE -- while there may be 500 dice thrown during a game, those six ones in a row came at just the wrong time to cause your terminators to wipe. The odds of you rolling 500 ones though get to astronomically small levels, so your game will wind up with average dice rolls. You probably have noticed this in your game. You might roll an 11 for leadership then a 4" for the distance ran.

That being said, you should plan your game around averages. If you know that attacking unit A will result in 6 casualties, and attacking unit B will result in 3 causalities, then it's better to attack unit A with that weapon. Some weapons are better at specific targets. Swooping hawks murder GEQ, but do moderately against MEQ. As you play with the numbers in your army you will get a better understanding of what tools to use at which targets.

The mathhammer goes deeper than just shooting and assaulting. You can also calculate averages for charge distances, psychic powers, etc. As you play more with the mathhammer you will be amazing at all the things you pick up about your game. Enjoy!


Automatically Appended Next Post:

 coblen wrote:
Say you are bs3 twin linked. You hit on fours rerolling. To calculate the %chance of each hit we plug the numbers into the formula

You pick these up pretty quickly, and there is a lot of insight.

A 1/3 roll, rerolled will result in a success 55% of the time. This means ork twin-linked weapons are slightly better than BS3.
It means that if two MEQ are making saves, there is a 55% one of them will fail...

Now with something like preffered enemy my above formula is too simple because it assumes that all the failures get to be rerolled. A more fully functioning formula would be.

%chance of success=%chance of passing +(%chance of a reroll x %chance of passing.)

The difference in this case is that the chance of a reroll is not the same as our chance of failure. Instead only side of the dice lets us reroll making it a 1/6 chance. This is roughly a 17% chance, or 0.17. You can find this by dividing 1/6 on a calculator. If you want to not have any rounding errors you can do the whole thing in fractions, but int his case I don't think being that exact is necessary. Using our above example of bs3 but now with preferred enemy we can solve as follows

Plug in the numbers.
%chance of success=0.50+(0.17X0.50)
solve the brackets
%chance of success=0.50+(0.085)
Now addition
%chance of success=0.585 or 58.5%

You can see from this that in this case preferred enemy is one third as effective as twin linked. This is to be expected as your rerolling one third as many dice.

Thanks guys! I never looked into it, it just seemed so daunting when thrown around dakka dakka. It's not so bad after all! now to get some practice in

Yeah, doing calculations for Preferred Enemy is obnoxious.

One good thing about Mathhammer is that it often reveals stuff that you'd have assumed otherwise. For example the Vindicare's Exitus rifle has the Turbo-Penetrator for dealing with multi-wound models and tanks, but in most cases you'd be better off just using the hellfire round for multi-wound models, as reliably getting in 1 wound a turn is much more statistically sound than trying to get 3 wounds in on 2 dice rolls. In the long run, the Hellfire round actually does more wounds over the same number of turns than the Turbo-Penetrator.

A long time ago, I compared a wolf priest vs a rune priest casting Prescience on a unit. Here are the results.

Shooting
A bolter shot by a wolf priest has on MEQ has...
77.72% to hit * 58.30% to wound * 33.33% failed save = 15.1% chance to kill
A bolter shot from a rune priest on MEQ has....
89% to hit * 50.00% to wound * 33.33% failed save = 14.8% chance to kill

A PG shot by a wolf priest has on MEQ has...
77.72% to hit * 97.2% to wound = 75.5% chance to kill
A PG shot from a rune priest on MEQ has....
89% to hit * 83.3% to wound = 73.9% chance to kill

Assault
In assault, the rune priest edges out. This is because the wolves go from hitting 2/3 of the time to 1/2 of the time against MEQ targets.
Against GEQ targets, the results would wind up similar to the shooting, where the rune priest would be more effective.
A normal attack from a wolf priest squad has on MEQ has...
58.30% to hit * 58.30% to wound * 33.33% failed save = 11.2% chance to kill

A normal attack from a rune priest squad has on MEQ has...
75.00% to hit * 50.00% to wound * 33.33% failed save = 12.3% chance to kill

Summary
If you are looking to just boost a squad's damage output, the wolf priest is slightly better. For better or worse, it's not psychic -- so it cannot be denied or perils. On the other hand, he does not bring extra dice...
You can take those same results and apply them for the effects of preferred enemy.

I just use excel spreadsheets when I'm list building.
During a gam I'll use fractions to get a reasonable idea.

The only thing that continues to trip me up is how to factor mastercrafted into an excel equation.

 Farseer Anath'lan wrote:
The only thing that continues to trip me up is how to factor mastercrafted into an excel equation.

Break out the attacks in a similar fashion that we break out the rending attacks -- where the first one gets a reroll.
Lets say you have a chapter master with 5 attacks and a master crafted weapon, you are looking at something like this.

8/9 (to hit) * 1/2 (to wound) * 1 (# of attacks) = 4/9 dead MEQ (~.44)
=sum(2/3)*(1/2)*1
plus
2/3 (to hit) * 1/2 (to wound) * 4 (# of attacks) = 8/6 dead MEQ (~1.3)
=sum(2/3)*(1/2)*4

Excel will automatically convert the fractions to decimals for you, meaning you don't need to muck about with LCDs.
I usually use multiple cells. One for (to hit), one for (to wound), one for (failed save) and another for (fail FNP). This lets you easily alter the values to see changes.

Mathhammer works over many, many die rolls, but many games have critical die rolls with sometimes only a single die or a pair of dice. That's why some games still come down to sheer luck. Also, if you get all your high rolls on your boltgun shots, and roll low for your grav guns, it has a disproportionately high effect on the outcome of the game. So the devil is really in distribution of average rolls.

The thing to remember with mathhammer is that its great for figuring out the average potential of a unit but anomalies happen. When an anomaly happen and its during one of your first 3 shooting phases all those numbers mean nothing. Use mathhammer for figuring out potential but do not get upset when your rolling below average during a game. Basically relying on mathhammer can lead to the darkside so to speak and you can get upset and lose because of it.

 CKO wrote:
Basically relying on mathhammer can lead to the darkside so to speak and you can get upset and lose because of it.

I agree that getting upset when random chances happen is not healthy. It does happen to us all though. Just last week I failed 15 out of 17 4+ saves and got really frustrated.

When playing, you should expect that one scatter laser should kill about one marine, so to clear a 5 man squad, expect about 5 scatter lasers to clear them. You might need more, you might need less -- but the averages are that is how many will be needed.
Remember that between those five weapons, about 44 dice will be rolled. (20 to hit dice, 13.3 to wound dice, 11.1 saves made,) This will help the law of large numbers come into effect giving us a bell curve. Sure, you might get all 1's and 2's for saves, but the averages are the best way to plan your game.

When playing, however, it's best not to think about what a unit can do. Instead think about, what is this unit likely to do. Does that make sense?

For example, one scat bike might kill 4 marines in a round of firing, but the chance of that happening are ...
The chances for one scatter laser shot to kill a marine is 5/27, or 18.5% (2/3*5/6*1/3).
Pulling off four such shots lowers the odds to 625/531441, or 0.001%. (5/27*5/27*5/27*5/27).
I'm not saying it won't happen in your lifetime. I am saying that odds like that are the reason casinos are in business.

 labmouse42 wrote:

 Farseer Anath'lan wrote:
The only thing that continues to trip me up is how to factor mastercrafted into an excel equation.

Break out the attacks in a similar fashion that we break out the rending attacks -- where the first one gets a reroll.
Lets say you have a chapter master with 5 attacks and a master crafted weapon, you are looking at something like this.

8/9 (to hit) * 1/2 (to wound) * 1 (# of attacks) = 4/9 dead MEQ (~.44)
=sum(2/3)*(1/2)*1
plus
2/3 (to hit) * 1/2 (to wound) * 4 (# of attacks) = 8/6 dead MEQ (~1.3)
=sum(2/3)*(1/2)*4

Excel will automatically convert the fractions to decimals for you, meaning you don't need to muck about with LCDs.
I usually use multiple cells. One for (to hit), one for (to wound), one for (failed save) and another for (fail FNP). This lets you easily alter the values to see changes.

Mastercrafted is actually a bit more complicated than that. You need to account for the possibility that the first attack hits without needing the re-roll, so the second attack then can use the re-roll if IT misses, etc, etc. It works out to a geometric series that with enough attacks is equivalent to a full extra attack.

Math works out that expected number of hits with N mastercrafted attacks at hit ratio R
= (N+1)*R - POWER(R, N+1)

eg
with 5 MC attacks hitting on 3's, you should expect 3.9 hits, rather than the 3.5 suggested above.

of course this kind of calculation kind of defeats the purpose of mathhammer - which should be easy enough to estimate on the fly

Mathhammer really isn't all that complicated, and can be hugely helpful when your trying to determine the best course of action.

Generally speaking, it relies on a simple principle of probability:

The odds that two connected events will happen in sequence are the odds of each event happening multiplied by each other:

IE wounding a marine with a boltgun from another marine:

Event A: hitting the marine successfully
Event B: wounding the enemy
Event C: the enemy failing an armor save

Event A: 4/6 chance of success (BS 4 needs a 3 + to hit)
Event B: 3/6 chance of success (Str 4 vs T 4 wounds on a 4+)
Event C: 2/6 chance of success (marine needs a 1 or a 2 to fail armor save)

(4/6)*(3/6)*(2/6) = 4/36 or 1/9 chance (11%)

This process can be applied to any sequence of events. You just need to keep going through the process. Subsequent saves (like FNP) are really no large task, as you just add an extra "event D - 4/6 chance of success (fails FNP on 1,2,3 or 4)" and multiply 4/36 by 4/6 and get 8/108 or 2/27 (7.4% chance of that marine taking a wound).

Sometimes the probability might get a little wonky, but if you think logically through it, it works all the same and helps you figure out what the best course of action really is.

Trasvi wrote:
Mastercrafted is actually a bit more complicated than that. You need to account for the possibility that the first attack hits without needing the re-roll, so the second attack then can use the re-roll if IT misses, etc, etc. It works out to a geometric series that with enough attacks is equivalent to a full extra attack.

Math works out that expected number of hits with N mastercrafted attacks at hit ratio R
= (N+1)*R - POWER(R, N+1)

eg
with 5 MC attacks hitting on 3's, you should expect 3.9 hits, rather than the 3.5 suggested above.

Thanks for the catch. You are right, you can use the reroll on any attack.