“Average Dice” part 2 — Damage Rolls

In the last installment of this series, I looked at the math behind attack rolls in Warmachine.  This time, I’m going to move onto damage rolls, as once you scratch the surface and get past the notion that “average dice equals seven,” there is some interesting stuff going on in the probabilities.

seven1.jpg

Fascinating.  Tell me more…

For the uninitiated, Warmachine uses 2d6 to resolve most attack and damage rolls.  Generally, when making an attack against an enemy model, you must roll 2d6, add your attack modifier, and equal or beat the opposing model’s DEF to score a hit.  Once you have hit, you determine how much damage you do by again rolling 2d6, adding the POW/P+S of the weapon you are attacking with, and comparing it to the ARM value of the opposing model.  For every point that this roll exceeds ARM, you do one point of damage to the opposing model.  Models in Warmachine have anywhere from one to sixty or more hit points or “boxes,” and to make it easier, players usually mentally subtract the opposing player’s ARM from their POW before rolling damage — calculating that, for example, with a POW of 16 up against an ARM of 20, they will do 2d6-4 or “dice minus four” damage.

Single wound or multi-wound?

Before I get started, I would like to point out that there is a difference between single wound and multi-wound models.  For single wound models, you only have to do a single point of damage to kill them, which means that you get the same result (usually, a dead enemy model) whether your roll beats their ARM by one or by a lot.  Since we don’t care how much we overkill a single-wound model by**, our math actually resembles the to-hit rolls we discussed last week in that we’re essentially rolling to hit a target number, which in this case is one higher than their ARM value, in order to do at least one point of damage and kill the opposing model.

Additionally, for a multi-wound model, you can also use a similar principle to calculate your odds of one-shot killing the model, by simply adding up the number of boxes and the ARM to figure out what you need to roll to one-shot kill the enemy model.

Since the math on trying to roll a target number or better and not caring how much you beat the target number by was discussed extensively last week, right now I’m going to focus on a case where you are up against a multi-wound model, and assume you are just trying to do as much damage as you can.

Widowmakers vs. Forge Seer

A couple weeks ago, I was playing a game and having some Khador-on-Khador action. My opponent had a Greylord Forge Seer marshalling a warjack, and had left it in the open and within striking distance of an entire four-man squad of Widowmaker Scouts and a single Widowmaker Marksman.

F_Khador_WidowmakerDue to the accuracy of the sniper rifles on the Widowmakers (who says the Khadorans don’t have a knack for precision engineering?), hitting the target wasn’t a big problem.  At anything but snakes to hit, each individual shot had a 97.2% chance of connecting and I have an over 85% chance of hitting with all five.

The problem, however, comes when it is time to do damage.  The opposing Greylord Forge Seer has an ARM of 16 and eight boxes.  On the Widowmakers, I have a choice.  Due to their Sniper rule, with the Scouts, on a hit, I can choose to either do one point of damage automatically or roll a POW 10, and with the Marksman, I can choose to either do three points of damage or roll a POW 12.

Adding up the automatic damage, we see that if I take advantage of the Sniper rule, doing one point of damage four times and three points of damage once, I will end up doing seven points of damage — just shy of what I need to kill this Forge Seer.  Since I would rather him not survive until the next turn, that won’t do.

So, if I want any chance of killing him at all, I use my POW 10s and POW 12s and roll it out.  Here is where the math gets interesting…

Dice minus X…

“Average dice” theory states that since “average dice” on 2d6 equals seven, my Scouts, doing 2d6-6 damage, will on average do one point of damage each, while my Marksman, doing 2d6-4, will on average do three points.  Again, this adds up to seven damage, which means that I will probably not manage to kill the Greylord Forge Seer.

Fortunately for me, the math of the “average dice” theory is wrong.  To calculate the expected value of a random discrete variable such as a series of die rolls, it’s a simple matter of, for each possible outcome, multiplying the probability by the outcome.  The formula for the expected value*** on a damage roll, thus, is:

E(x) = x1p1 + x2p2 + x3p3 + … xnpn

Where, for each possible outcome on the dice, p is the probability of said outcome and x is the damage.

When we start calculating this for 2d6, we start to notice something interesting…

expected_value_damage.png

Here is the calculations, with the main part of the table showing the value of each term in the equation for all cases from dice-2 to dice-11, with the EV in the bottom row

From dice minus two upwards, for every +1 you add to the roll, the expected value of your damage increases by one.  In this area, the “average dice” theory seems to more or less work out, in that the expected value of 2d6-1 is equal to 6, the expected value of 2d6 is equal to 7, and so on.  However, as we go into dice minus three and lower, the curve starts to flatten out and trail off, only hitting zero when we get to dice minus 12 and can’t possibly roll a thirteen or better on two dice.

expected_value_damage_graph.png

The blue line is correct math; the orange line is wrong

This is because you can’t roll negative damage.  Think of it this way — if you are rolling damage at dice minus seven, and you roll snakes (2) once and boxcars (12) once, you have rolled perfectly average over those two rolls, but you still did five points of damage.  That time you rolled snakes, you didn’t heal the opposing warjack for five points of damage, you simply failed to crack ARM and did nothing.  As such, you did five more points of damage than the “average dice equals seven” theory would imply because your low roll didn’t “cancel out” your high roll in any way.

Looking at the blue curve, even at dice minus seven and below, you still have a chance of rolling high and doing damage.  In fact, the expected value on 2d6-7 damage is almost a full point higher than what the “average dice” theory states!

So, what does this mean?

Remember my situation with the Widowmakers and the Forge Seers?  “Average Dice” implies that I would do seven damage, but if we add up the expected values of four shots at 2d6-6 and one shot at 2d6-4, we get an expected value of 9.333.  With the Forge Seer having eight boxes, that means that assuming I hit all my shots, I have a pretty good chance of killing it.

Which is what I did — failing to crack armour on the first two shots, rolling boxcars on the third for six damage, using the fourth to kill some random single-wound mook, and then using the Marksman to do an automatic three points to finish him off.

To-hit and damage

Of course, this is Warmachine and you can always miss.  If you want to factor in your odds of hitting, all you have to do is multiply the expected value on your damage roll by the probability that your attack roll will hit.  In this case, since I need not snakes on all my shots, I simply multiply the EV by my the probaility of hitting.  In the Widowmakers case, it’s an easy calculation, multiplying the EV by 0.972, as I have a 97.2% chance of not rolling snakes, but this can be done for any to-hit value and dice plus/minus value on damage

EV of damage.png

Memorize this chart to be good at Warmachine…

Final thoughts

  • Against single-wound models, or when you want to calculate the probaility that you will one-shot something, you can simply adapt last week’s to-hit results
  • Below 2d6-2, the “average dice” theory of calculating damage breaks down
  • The expected value on your damage roll at this point is higher than you might otherwise expect, because low rolls do zero damage instead of negative damage that might cancel out the high rolls
  • To factor in your chance of missing, simply multiply the EV of your damage roll by your hit probability, and you can figure out the EV of the amount of damage you will do before you even make the attack roll.
  • For multiple attacks, simply add up the EV of each individual attack, factoring in your hit probability.  Doing so, you can figure out the “DPS” of a multi-attacking model such as a warjack or warbeast against models of a certain DEF and ARM value.
  • As always, the dice gods are fickle and literally any possible outcome, even some extreme outcome with very low odds, can happen.

Stay tuned next time for when I discuss what happens when you start throwing additional dice into the mix, and maybe if you’re lucky, why Vlad1 is really really good!

 

**Except maybe for bragging rights and to laugh at how badly some guy got splattered.  Like last weekend, where I overkilled a Cygnaran warcaster by 34 points…

***Again, I’m using the term “expected value” in the mathematical sense to represent the mean outcome, weighted by probability.  Over hundreds and thousands of rolls, you would expect your average outcome to converge on this value.  But on one individual trial, you can still roll anything from snakes to boxcars.  Just because something is the expected value, don’t count on it, because about 50% of the time, you will be disappointed.

“Average Dice” part 1 — Attack rolls

When it comes to calculating probability, humans are notoriously terrible.  Our brains just aren’t wired to be good at figuring out odds, and are prone to numerous fallacies. It’s what keeps casinos and lotteries in business, and while gamblers may be particularly prone to these sort of mathematical errors, Warmachine players are no different. Especially when we’re on the clock and have to rely on gut instincts and inferences rather than detailed calculations.

One of the little catch phrases you will hear emanating from our tables and one of the ones that irritates me the most is the concept of “average dice.”  Warmachine players regularly say things like “Okay, so on average dice, this guy will kill this guy,” or “you got lucky; average dice says your assassination run would have failed.”  So, as a nerd with two degrees in math-related subjects and an employee of a national statistics agency, I figured it would be a good idea to do a deep dive into probability as it pertains to the game of Warmachine.  As this is a heavy subject with a lot of graphs and fun mathematical equations, I’ll be splitting this up with the goal of making it into a series.

To-hit probabilities

Let’s start with attack rolls.  In Warmachine, to hit an opponent with an attack, you need to roll the dice (usually 2d6), add that number your attack stat (MAT, RAT, Magic Attack, or Focus/Fury), and get equal to or higher than your target’s DEF stat. Since it doesn’t matter how much you beat the target’s DEF stat by, Warmachine players tend to do the math up front and only concern themselves with the target number.  So, if Kommander Sorscha, who has a MAT of 6, is swinging her Frostfang at an enemy model with a DEF of 13, I will say “sevens to hit,” and roll the dice, knowing that if I roll better than a seven, I hit.

(Note that one minor quirk of Warmachine is that all ones is always a miss and all sixes is always a hit, which I will be taking into account in all my calcualations in this post)

And speaking of sevens, that is a critical number for Warmachine players because when you roll 2d6, seven is not just the most likely outcome, but also the probability-weighted average of all possible outcomes.  As you can see in the following probability mass function, seven is the central value of 2d6, and your odds of rolling a specific value are highest for seven and taper off as we go lower or higher.  This is because there are six different ways to roll a seven on 2d6 (1 and 6, 2 and 5, and so on), and only one way to roll a 2 (snakes) or 12 (boxcars).

2d6_pmf.png

This graph tells us that the odds of rolling a seven are about 16.7%, but in the context of Warmachine, that’s not a very useful factoid.  Almost never do we need to roll a certain number exactly; generally we need to roll that number or higher.  So, I’ve taken the liberty of summing up these probabilities to give us a table, or for the more visual among us, a graph, showing our probabilities of rolling a given number or higher.

To-hit Value Hit Probability
3 97.2%
4 91.7%
5 83.3%
6 72.2%
7 58.3%
8 41.7%
9 27.8%
10 16.7%
11 8.3%
12+ 2.8%

2d6_hit_probability.png

As we can see, one of the reasons why Warmachine players tend to be somewhat infatuated with the number seven is that it is a break point on 2d6 between having better odds of missing than hitting and vice versa.  You have a 58.3% chance of hitting on a hard seven, and a 41.7% chance of hitting on a hard eight.  This can be very useful in a game where when you don’t have time to do detailed calcuations, you can use your knowledge that seven is a break point to make a quick guess about my probability of hitting and make tactical decisions accordingly.  If I hit on a seven more than half the time, I know that if I need less than that, I’ve got pretty good odds of hitting, and if I need more than a seven, my odds aren’t so great.

7of9_s7.jpg

Did someone say “infatuated with Seven”… okay, I’ll admit it, I’m desperate for a way to spice up an article about math.  You’re welcome.

You will note that above, I opted for the wordier term “probability-weighted average” over “expected value” because this is where a lot of the time the “average dice” fallacy starts to creep in.  People can start thinking that if the “expected value” or (ugh) “average dice” on 2d6 is a seven, then they should expect to consistently be able to hit a target that they need sevens for.  The catch with that is that dice are random, and that if you expect sevens or better, then 41.7% of the time, you will be sorely disappointed.  Especially if those sevens are what a key attack roll such as an assassination run was relying on.

Now, if you also need a seven to crack armour, then your odds become even worse. “Average dice” says that you should be able to roll a seven to hit and a seven to kill because seven is your expected value.  But your odds of rolling two sevens in a row are actually much less than that.  With a probability of rolling one seven being 0.5833, the probability of rolling two sevens in a row are 0.5833^2, which comes out to 0.3403 or a mere 34% — a far cry from what the “average dice” theory would imply!  No wonder dice make people tilt…

Juggy vs. Typhon

One example of this logic was an argument that I saw on the Lormahordes forum a little while ago.  Someone was arguing that Khador warjacks need to be nerfed, because I guess he lives in a parallel universe where Harkevich is winning tournaments left, right and center.  To support his case, he applied the “average dice” theory to a matchup between a fully-loaded Juggernaut and a Typhon and vice versa where he assumed that both players always roll a seven, and came to the conclusion that a Juggy can one-round a Typhon while the Typhon can’t one-round a Juggy, therefore Khador warjacks need to be nerfed.

(of course, as a Khador player, I can’t pretend to be unbiased on this question.  And when comparing models, we need to take into account things like SPD, special rules, the existence of ranged weapons, and a whole host of things that go beyond just damage output and defensive stats. And it’s hard to compare cross-faction anyways because of the different support available, doubly so when you’re comparing Warmachine with the Focus mechanic to Hordes with Fury… but let’s ignore all that for now)

Anyways, if you’ve been paying attention, you can see where this logic fails.  A Typhon, at MAT 7, hits a DEF 10 Juggernaut 97.2% of the time, missing only on snakes.  So, the Typhon will almost always hit the Juggernaut, but thanks to Typhon’s superior DEF, the Juggy needs sixes to hit a Typhon. While a 72.2% chance of rolling a six or better isn’t bad, over five attack rolls, the Juggernaut (ignoring the crit stationary) actually has a less than 20% chance of landing all five.  Which means the assumption that you always roll sevens and therefore the Juggernaut will always hit massively skews the results in favour of the Juggernaut.

Upping your chances with multiple attacks

Of course, as a complex tactical combat game, Warmachine is filled with ways to adjust those odds.  Terrain features, spells, and many other effects offer bonuses and penalties to both attack rolls and DEF.  With numerous ways to get a +2 on your attack roll, it can be fairly easy to go from needing eights to hit to needing sixes, which increases your hit probability from 41.7% to 72.2% — a whopping 30% increase.

(Note: The percentage increase in your hit probability generated by a +2 on your attack roll isn’t uniform due to the shape of the probability mass function, but it can range from about 15% on the edges to about 30% in the middle, where most of our attack rolls tend to be)

But if you don’t have access to those fancy-schmancy spells and you’re duking it out on a parking lot, there is one more way of increasing your odds of killing stuff:  shooting it more.  It’s why the army has burst-fire features on their rifles, and the same logic applies in Warmachine.

If I need a nine to hit, my odds, at 27.8%, aren’t that great.  But if I have a ten-man unit, each of whom fires one shot and needs a nine to hit, then I can start doing some damage. Intuitively, we can see that if we have a 27.8% of hitting, and fire ten shots, we will on average score two or three hits.

But say we want to go deeper — we want to see how those probabilities are distributed. It is easy enough to say that we will on average score three hits out of ten, but that can lead us back down the trap of thinking that we “should” score an average result and getting bitter when our dice don’t cooperate.  What if we want to know the odds of scoring two hits, or four?  Given that we know the probability of scoring a hit for any given value from the tables and graphs above, calculating the chances of scoring a certain number of hits is a simple matter of calculating the binomial distribution.

binomial

Got all that?

Thanks to the magic of Microsoft Excel, these calculations aren’t that hard, so I’ve taken the liberty of creating some tables and graphs, based on the assumption that you’re firing away with a ten-man unit, so we can better understand the odds.

probability_of_hits.png

This table tells you the odds of scoring a specific number of hits, depending on what you need to roll to hit.

pmf_numhits.png

Here, I’ve graphed the above table for a few common to-hit values.  As you can see, increasing the to-hit value pushes the probability distribution to the left, meaning that fewer attacks hit.  For each to-hit value, the distribution for the number of hits has a bell curve like shape to it, centered on the point np, or the probability that a single shot will hit multiplied by the number of shots.

cdf_nohits.png

Summing up the probabilities is maybe a more useful graph for Warmachine probabilities.  Here the odds of hitting at least a certain number of times is graphed, so if you really need to kill those four dudes over there…

So, what can we see here?  Even when you need something like a nine or a ten to hit, if you take enough shots, you’ve got a pretty good chance of landing one or two.  This has some implications for high-DEF models like Kayazy Eliminators.  When an opponent attacks your eliminator with a whole unit, and you smugly respond with “10s to hit,” they have about a 50-50 chance of killing two of them, so don’t be surprised when they manage to successfully take them out (albeit at the cost of an activation for the entire unit).  You might want to say “damn, I would have survived if you didn’t spike your dice, you lucky bastard” but as this graph shows, individual dice spikes happen and should be expected when you start chucking ten or more dice at something.

Also, the shape of the curve really shows the power of that +2 to hit at certain points.  Say you want to shoot five targets with ten shots.  Going from a 9-to-hit to a 7-to hit boosts your odds from around 10% to around 80%.  Alternatively, it changes your 50% break point from 3/10 to 6/10.

Finally, while the mean value for the number of hits is equal to np, or the probability of a hit on a single shot multiplied by the number of shots, there is a bell curve going on here. In some cases, this bell curve can be fairly wide.  On a seven to hit, your middle 80% on the probability curve goes from about four to eight.  Even though with a 58% chance of hitting on each attack roll, your gut tells you that you “should” get six out of ten hits, you’ve got a 14% chance of hitting eight or more targets — something that, over the course of a three-round tournament when you’re activating that ten-man unit multiple times per game, has a pretty decent chance of happening at least once.

Conclusions

As mentioned earlier, before we even start accounting for things like confirmation bias, and how rolling snake-eyes on your key assassination roll tends to burn itself into your brain more than when you roll average to well, we suck at probability.  As Warmachine players, we try to guess using rules of thumb, but while these simple rules of thumb are helpful, there are a lot of nuances underneath.  In the worst case scenario, not understanding the underlying probabilities can cause a player to get frustrated that they didn’t get a result that they “should” have gotten because of “average dice,” and cause the player to go on tilt, which can cause them to make further mistakes and lose a winnable game, plus be a miserable opponent to play against.

A few key takeaways here are:

  • “Average dice” does not equal always hitting a seven — if you need a seven to hit, you will miss almost half the time.
  • Dice spikes happen, sometimes more often than you think.  Good players plan for the possibility, and don’t tilt when they do.
  • +2 to your attack roll is huge when you’re in that nice meaty part in the middle of the bell curve.  Going from needing a nine to needing a seven, or from an eight to a six boosts your odds of hitting by about 30%
  • By making enough attacks, you can overcome a harsh to-hit roll.  If you need a nine to hit, even though “average dice” states that you will miss, make enough attack rolls and eventually you will hit one.
  • Sacrificing a small animal to the dice gods before your game can greatly increase your chances of making that key assassination run.

That’s all for now!  Tune in next time while I explain why dice minus seven damage does not equal zero!

jeri ryan.jpg

Also maybe some more gratuitous pictures of Jeri Ryan