# FiftyPageMaths

While not the most interesting part, some calculations can help you understand how much the specific unit you’re deploying is good at its job for the amount of points you’re spending. A word of advice is that while using statistics for army building is a useful device, too much reliance on it can easily bring the player in the min-maxing realm (and at that point, you might suck out the fun from the game).

The following explanations use Grimdark Future as a base, but they can be applied to any other mode as well.

## The “Exponential 6” method

The basis to understand how much your unit is likely to get hit or to hit someone rests on the game’s universal premise that all rolls are made with a d6. Depending on how many rolls you need, you raise the exponential of the 6 by that amount. For example, the basic interaction of Quality check => Defense check are 2 possible rolls, so 6^2 = 36. A simpler way to consider this is by just multiplying the fractions. Let’s say you want ot calculate how likely your Quality 5 dude can hit a Defense 5 model:

- since it is Quality 5, you need to roll 5 or more; the possible results that allow you to hit on a d6 are 2

- a Defense 5 means instead that you need the opponent to NOT roll 5 or higher, meaning the rolls which let the attack go through are 4

- we now have 2/6 chances to hit and 4/6 chances for the opponent to not resist the attack

- we now multiply the results and get (2/6) * (4/6) = 8 / 36 = 0.22

The end result is that if a Quality 5 model attacks a Defense 5 model, there is a possibility of 22% that the attack will deal damage. Let’s say now the unit on the receiving end has Regeneration, which means that on a 5+ they ignore the wound. In this case, we do the following:

- Regeneration is 5+, so we need a roll of 4 or below to deal damage; that’s 4 possible results

- we simply add another chance to the formula, which in this case is 4/6

- we get (2/6) * (4/6) * (4/6) = 36 / 216 = 0.16

By following such operations, we can say then that if a model with Quality 5 attacking a model with Defense 5 has a 22% chance to damage, if we buy Regeneration for the defending model that chance goes down to 16%. Let’s try now a more difficult calculus which requires considering the exponential 6 and suppose we retrofit our attacking model and give it Rending:

- Rending activates on a roll on 6, meaning there is only 1 possible result for activating its effects

- on a 6, Rending attacks have AP 4, meaning that the Defense of the attacked model becomes 6

- Defense 6 means that there are 5 possible results where the model fails to defend

- Rending ignores Regeneration, meaning we can consider all rolls for Regeneration as a success (so 6 successes)

We have 1 * 5 * 6 = 30 successes with a Rending result. We add these to the number of successes we get outside of Rending (which, with Quality 5 means the single result of 5), which are 1 * 4 * 4 = 16. The end result is that out of 216 possible result, 46 allow us to inflict damage. At this point we simply divide our number of successes by the number of possible rolls and we get 46 / 216 = 0.21. The end result is then that if a unit before had a 16% chance to damage a unit with Regeneration, then by adding Rending its chances to inflict damage now are 21%.

The process to consider penalties and buffs to hit is also easy, since by the rules rolls or 1 are always a failure, while rolls or 6 always a success. Let’s say the opponent’s unit has Stealth, which gives your rolls -1 to hit. To account for this in terms of probabilities, simply consider the Quality of your unit one step worse. In this case, instead of considering your unit Quality 5, with Stealth they now work as Quality 6 units.

## Binomial distribution

What was introduced so far can easily cover simplier calculations. Let’s say that you want to exstimate more precisely the capabilities of a unit, like 5 models shooting a target. For this kind of calculus, we need binomial distributions

To understand binomial dsitributions, we need two more mathematical concepts: factorials and binomial coefficient. To try and explain them in the easiest way possible, let’s make some examples.

A factorial of a number is equal to the multiplication of every single positive integers less than or equal to the number in question. Let’s say you need to calculate the factorial of 4. You write it down with a exclamation mark and get the following:

- 4! = 4 * 3 * 2 * 1 = 24

Remember that per convention, 0! is always equal to 1.

The binomial coefficient is usually written in another way, but can be rendered as nCk (for the correct representation, you can check the wikipedia page https://en.wikipedia.org/wiki/Binomial_coefficient)

The binomial coefficient represents the following operation:

- n! / k!( n – k )!

Another example is to have n as 2 and k as 3, giving us the following formula:

- 2! / 3! * ( 2 – 3 )! = ( 2 * 1 ) / (3 * 2 * 1 ) * 1 = 2 / 6 = 0.33

Now, here comes the question: what do we need all this stuff for? The reason is that what we did before to calculate the probabilities can work when we have no more than 2 different instances to consider, but with multiple ones things get tricky. To calculate the probabilities of achieving successes with multiple indipendent tries, a binomial distribution is needed. The formula is the following:

- B = ( nCk ) p^x q^(n-x)

Where n is the number of tries, x is the needed results on the dice, p the probability to deal damage and q the probability to fail to deal damage. Let’s take back our example with Quality 5 with Rending attacking a unit with Defense 5 with Regeneration. Let’s say our unit is made of 5 models, with each of them attacking a single time:

- n is equal to the number of attacks; since we have 5 models with 1 attack each, n is 5

- x is equal to the number of times we get a hit; let’s say we want to calculate the percentage of not hitting even once, so we put x equal to 0

- p is equal to the possibility of dealing damage; as we calculated before, such possibility is 46 / 216 or 0.21

- q is the inverse of p; simply subtract 46 from 216 and you get 170; q is then 170 / 216 or 0.78

The formula then becomes:

- B = (5! / (0! * (5-0)!) * (46/216)^0 * (170/216)^5 = 1 * 1 * 141.985.700.000 / 470.184.984.576 =0.30 = 30% to not hit even once out of 5 attacks

Let’s say we want to calculate hitting at least with 3 models. We simply exchange the 0 with 3 and get the following:

- B = (5! / (3! * (5-3)!) * (46/216)^3 * (170/216)^2 = 10 * ( 97.336 / 10.077.696 ) * ( 28.900 / 46.656 ) = 10 * 0.009 * 0.61 = 0.05 = 5% to hit with at least 3 models

The use of binomial distributions requires to run the calculus for each possible instance you want to test, but with a little patience it gives you all you need to know about how much likely you will hurt the opponent’s unit

## Points efficiency

The last calculations are to consider how much you’re paying for a unit and if the points are worth for what they bring. The first step is to consider how many attacks a unit needs to bring down another one. Let’s say you have a HDF Infantry Squad and you want to calculate how many attacks you need on average to kill another opposing HDF Infantry Squad. Taking from the calculations before, we know that the chance to cause damage with a Quality 5 model against a Defense 5 model is 22% or 0.22. Basic HDF Infantry Squads field 10 models and they need to lose at least 5 to start Morale checks (against ranged attacks, at least). To calculate the needed attacks, we simply divide the chance we got by the number of wounds we need to cause. Simply put, we need on average 5 / 0.22 = 22 attacks with our HDF Infantry to start forcing morale checks and 10 / 0.22 = 45 attacks to completely wipe out the unit

Another way to understand a unit’s potential is to consider how many attacks it needs to deal a single wound to a unit. Let’s use again our previous chance of the Quality 5 unit against a Defense 5 one. Now let’s also consider how does it fare against a Defense 3 unit. Given what we previously wrote, the formula is 2 * 2 / 6^2 = 4 / 36 = 0.11 = 11% to hit a Defense 3 unit with a Quality 5 one. To now calculate the amount of damage they cause on average on each turn against such units, simply multiply the chance to hit with their number of attacks. Against a Defense 5 unit, HDF Infantry Squads of 10 deal an average of 0.22 * 10 = 2 wounds, while against units with Defense 3 they deal 0.11 * 10 = 1 wound on average

Finally arriving at our point, we can pinpoint how many points we are spending to deal with the opponent’s points. A normal 10 models HDF Infantry Squad costs 120, meaning we are spending 12 points for each model. We know that against Defense 5 units they deal 2 wounds on average, meaning we’re spending 120 / 2 = 60 points for each wound inflicted against infantry per round. Since we’re targetting another HDF Infantry Squad, we know that each wound we cause will cost the opponent 12 points. By dividing the amount of points we spend for each wound we inflict by the amount of points the opponent loses with each wound his unit gets, we can pinpoint how many points we are expending with this unit to make the opponent lose points by playing that unit. In this case, the amount is 60 / 12 = 5. With our HDF Infantry Squad targetting another HDF Infantry Squad, we spend 5 points to inflict a loss of 1 point to the opponent.

What’s the reason of all this? Let’s introduce for example another unit now, an Assult Brothers unit which attack 10 times in melee at Quality 3. By using the already mentioned calculations, against a HDF Infantry Squad in melee each Assault Brothers has a 4 * 4 / 6^2 = 16 / 36 = 0.44 = 44% chance to deal damage. Assault Brothers have 10 attacks, which means they inflict 4 wounds on average against a HDF Infantry Squad. With 140 points, we are spending 28 points for each brother, but also 35 points for each wound inflicted against the 60 of the HDF Infantry Squad, meaning that we are spending 35 / 12 = 2.9 points inflict a loss of 1 point to the opponent against the 5 points we spend by using the Infantry Squad. On the other side, if both the Assault Brothers and the HDF Infantry Squad were to suffer 5 wounds from ranged attacks, with the brothers we lose 28 points for each wound taken and lose the squad, while with the humans we lose 12 points for each wound and must take a Morale check to see if they get Pinned or not.

## Conclusion

Now, the real question: is all this stuff necessary? To be honest, not really. OPR games are, after all, games and as such they are meant to be a fun experience instead of a number crunching simulator. However, if you find yourself in the position to be doubtful about which one between 2 units can fulfill better the role you want them for, the mentioned formulas can help you decide on that. Again, do not be overly reliant on such things, as not only you’ll waste your entire time behind infinite calculations, but min-maxing like hell can easily make you unbearable to play against.