“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

 

One thought on ““Average Dice” part 1 — Attack rolls

  1. Pingback: “Average Dice” part 2 — Damage Rolls | Ice Axe Miniatures

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