Mathammer: The Law of Averages

Today we are joined by Adam Lane from the Art of War, our resident math expert! He’s here to break down how math can be used in our games of warhammer. 

How a simple change in mindset (and some maths) can help you win more games

The famous philosopher and mathematician Nassim Taleb once said that you should never cross a river that is, on average, three feet deep. The reason is simple: the average tells you what to expect to happen 50% of the time and ignores what happens when things go wrong. As crazy as it may sound, this concept is a common human bias that was a significant contributing factor to the financial crisis of 2008.

I am sure you are asking, what does this have to do with the game of Warhammer 40,000? To this question, I would ask you: how many times have you heard a player exclaim, “That wasn’t the average” or “I am so unlucky,” or even something like “These dice are broken”? I believe we hear this so often because players often base their decisions on the expectation that they will achieve the average outcome and make no real account for the fact that the river (their dice rolls) may not actually be three feet deep.

Don’t get us wrong. We like calculating averages, and this is a great place to start. You can check out our Mathhammer article on how to calculate averages the easy way here: https://theartofwar40k.com/2021/01/mathammer-101

In this article, we will show you how to avoid the “flaw of averages.” It will also show you how top players apply a simple concept to tilt the odds in their favor, which will mean you win more games and keep a positive mindset, allowing you to be zen when all around are stressed.

A simple rule of thumb

The rule of thumb we want you to remember is “expect to roll badly and expect your opponent to roll like a god.” This sounds counterintuitive, but we promise you this is a fundamental idea that will lead to more consistent results at the tabletop.

An example is when you are considering threat ranges for your units and your opponent’s units. If you are charging your opponent, the question is what charge range you should assume. The average outcome of the sum of two dice is 7.5, so a simple answer would be around 7”. Using our rule of thumb, we would suggest you plan on making a charge of less than 7” and assume your opponent will roll something better than 7”. If you incorporate this into your plan, you will find that you make more charges, and your opponents will more far fewer charges. So far, so simple!

Another practical example here is when you think of committing a significant damage dealing unit to trade with your opponent. We would suggest that you calculate what might happen before you take this risk and then assume you will roll below the calculated average. If this conservative calculation does not achieve the desired result, we would suggest looking at ways the unit can be buffed or avoid engaging in the trade entirely. 

You can apply the principle above immediately, and we believe this will deliver better results. To take this further, we need some mathematics to answer the question by “how much” we should not knock off the averages we often calculate.

The concept of variance

Variance is an essential concept in statistics and is a measure of how likely a random event is to deviate from the average outcome. This is the metric we need to help us articulate “how much” we should knock-off from the average outcome to achieve a given level of certainty. Let me give you an example based on the average outcome from rolling dice. Below we have plotted the distribution of outcomes for rolling one, ten, and then fifty dice.

The charts show that each outcome is equally likely when you roll a single dice (i.e., pretty random). The charts also show that as you roll more and more dice, we can be more certain about outcomes both on the upside and downside (but we only can really “expect” to achieve the average if we roll lots of dice). As you roll more and more dice, the range of results starts to cluster around the average outcome.  

We have chosen a “1-in-6” measure of downside risk as it is easy to think about. When you roll one die, the “1-in-6” is a 2+ (no surprises there!). As you roll more dice, the “2+” starts to increase, and you can see if you roll 50 dice, then the “1-in-6” outcome is 3.3 (i.e., equivalent to a 2+ on a single die).  

If we can make decisions based on a 2+ outcome, this will lead to far more consistent results than basing it on the average (i.e., a 4+). It sounds simple, but this can be extremely powerful for ensuring you get good outcomes when you are at the tabletop.  

For those that are mathematically minded, we include a recipe for calculating this “1-in-6” outcome / 2+ chance of success for a given attack:

• Step 1: Calculate the probability of wounding = p
• Step 2: Multiply p by the number of attacks to calculate the average = np
• Step 3: Calculate the probability of not wounding = (1 – p)
• Step 4: Multiply it all together = np(1 – p)
• Step 5: Take the square root of the outcome np(1-p)

This result gives you the amount to knock off from the average to calculate the “1-in-6 outcome” for a given attack. To show you how this works, we provide a practical example below.

Efficient trading 

An important gut check is whether your army can eliminate 20 Necron Warriors in a single attack phase. This is important because if you don’t wipe the Necron unit, they will reanimate, leading to you trading down and allowing your opponent to play havoc with your game plan. 

The example below looks at the scenario where 10 Vanguard Veterans with lightning claw/ storm shield successfully charge into a unit of 20 Necron Warriors while in assault doctrine.

As you can see in this example, the Vanguard Veterans get 41 attacks on the charge, hit on a 3+, and then wound on a 4+ re-rolling with an AP of 3, leading to the Necrons not getting a save. This gives us an expected outcome of 20.5 wounds. In other words, on average, we expect the Vanguard Veterans to wipe the Necron unit. However, if you have been following along, we would suggest that this trade is not the best risk management decision as you have a 50% of wiping the unit but, more importantly, a 50% chance of not doing the needed damage (meaning potentially lots of Necron reanimations).

To calculate the 2+ / “1-in-6” outcome, we following the recipe above:

• Step 1: Calculate the probability of wounding = 1/2
• Step 2: Multiply p by the number of attacks to calculate the average = 41 ×1/2=20.5
• Step 3: Calculate the probability of not wounding = (1 – 1/2) = 1/2
• Step 4: Multiply it all together = 20.5 x 1/2 = 10.25
• Step 5: Take the square root of the outcome 10.25=c.3.2

So we know the average number of wounds is 20.5 (i.e., a 50 / 50 outcome), and we can calculate the 1-in-6 / 2+ outcome, which is the average less the variance (i.e., 20.5 minus 3.2 = 17). In other words, if we are conservative, we would not commit the Vanguard Veterans as there is a reasonable chance that they wouldn’t wipe the unit.

If you are the Space Marine play in this example, you should look to tilt the odds in the favor by adding buffs to the Vanguard Veterans to make the outcome more certain. For example, if we gave them Chapter Master re-rolls, the “1-in-6” / 2+ outcome would exceed 20, meaning that such a trade has a very high chance of working. By doing these types of calculations, we can get more confidence about in-game decisions, allocate resources more appropriately and ultimately win more games.

How to incorporate this thinking into your game plan

The calculations above are quite a bit more complicated than calculating averages, and maybe hard to do these numbers in game without a lot of practice.   

We would say that having this information will provide you with a significant advantage over opponents who have not done the numbers. To make the most of this advantage, we would suggest doing some pre-game homework for your significant damage dealing units. Armed with this sort of information, you can build this into your game plan and then look to build on the technique over time.

There are also ways you can build this thinking into your gameplay without having to do complicated mathematics, for example:

1. Try to be conservative in your game plan; 
2. Expect your opponent to roll well when thinking about charges, spells, or a given attack sequence.  
3. When committing resources, don’t just rely on averages. Start with the average but assume you will do worse than that calculation!

Would you like to know more?

Come check us out in the War Room or reach out to me to go deep into meta army lists using techniques such as those outlined above. We also take subscriber requests to dive deep into the analysis and provide them with cheat sheets using the analysis above.