*This week’s Hammer of Math we’re heading into a smaller scale and looking at the mechanics of wounding in the Warhammer 40,000 game of skirmish combat called Kill Team. This article will review a lot of the information on probability way back when this series started, and apply the unique twists of Kill Team’s Injury mechanic.*

Kill Team is one of the most accessible games available for players who want to do something at a smaller scale than 40K. If you have a squad, you have a Kill Team. It’s also a fun way to expand your painting repertoire and get a taste of factions that you might not have the time to paint a full army for. 40K players will recognize many of the mechanics used by Kill Team, but the smaller scale allows for a slightly more nuanced level of damage that exists somewhere between 40K and Necromunda. Like any other other game system the result is that certain aspects and modifiers will have a larger impact than others.

### Kill Team Mechanics

The basic kill team attack sequence is a series of independent probability events based on D6 rolls. Like in 40K there are a series of gatekeepers and drivers. As a review, gatekeepers are simple pass/fail criteria like the probability of making a hit. Drivers are multipliers that significantly impact the final outcome. We’ve covered this before in our **Kill Team Injury Guide**, but the sequence is as follows:

- The attacker determines the number of attacks, which is a driver. This can either be a fixed value or determined randomly.
- The attacker passes a gatekeeper hit roll by rolling equal to or greater than their Ballistic Skill characteristic on a D6.
- The attacker passes a gatekeeper wound roll; the ratio between the Strength and the Toughness characteristics determines the probability on a D6.
- The defender attempts a gatekeeper of their own with the saving throw on a D6, which is generally a fixed probability modified by the Armour Penetration of the attack. If the save is failed the sequence continues.
- The attack inflicts damage on the target equal to the Damage characteristic, which is another driver.
- Certain abilities (such as Lucky Escape for Demolition specialists) allow for a gatekeeper roll to avoid losing a wound; a D6 is rolled for each point of damage inflicted and if the roll is successful a wound is not lost.
- If the resulting damage exceeds the Wounds characteristic of the target then an Injury roll is made. The attacker rolls a number of D6 equal to the Damage characteristic of the attack (which could be modified by something like the High Explosive Tactic). A 4+ takes the model out of action, a 3 or less inflicts a flesh wound that improves the probability of future Injury rolls resulting in an out of action result. Regardless of the number of dice rolled during an attack only the highest value is used. Because this is effectively a pass/fail criteria the Injury roll is a gatekeeper. Note that the characteristic, not the damage inflicted, determines how many dice to roll. This reduces the efficacy of abilities that allow wounds to be disregarded. Thanks to Patreon supporter BCPSlave for bringing up this important point to our attention.

Note that while attacks are performed sequentially, if a model’s wounds are reduced to 0 any further attacks directed to the model are not resolved. This results in a very interesting and nuanced difference between Kill Team and 40K. In 40K the expected value for an attack is the same regardless of whether your double the number of attacks or the damage inflicted. In Kill Team the expected value changes depending on the ratio between the Wounds characteristic of the target, the number of attacks, and the Damage characteristic of the attack.

### Influence of Modifiers and Re-Rolls

Because Kill Team is D6 based the impact of modifiers and re-rolls on gatekeeper rolls is the same as with 40K. For re-rolls the influence is either fixed if the re-roll is based on a number (re-rolling 1s is a flat 16.7% increase in expected value) or a function of the probability of failing the roll (re-rolling failures on a 4+ is a 50% increase in expected value). For driving rolls we can reference the chart below.

Roll | Initial Probability | -1 Modifier Probability | Delta | +1 Modifier Probability | Delta |

2+ | 83% | 67% | -20% | 83% | +0% |

3+ | 67% | 50% | -25% | 83% | +25% |

4+ | 50% | 33% | -33% | 67% | +33% |

5+ | 33% | 17% | -50% | 50% | +50% |

6+ | 17% | 0% | -100% | 33% | +100% |

The impact of a modifier is dependent on the probability being modified; the lower the probability the greater the effect. While every faction has their own abilities and tactics, there are a few universal abilities with modifiers definitely worth considering:

- If a model is obscured (a section of the model is blocked from the point of view of the attacking model) then the hit roll and Injury roll are subject to a -1 modifier. This is a cumulative effect. The Headshot Sniper tactic enables an attack to ignore the obscured condition.
- Several conditions (long range, flesh wounds, broken kill team) and tactics on the attacker will inflict a -1 penalty on the hit roll of a shooting attack.
- Intervening terrain, flesh wounds, a broken kill team, and certain tactics will also impact the attacker with -1 penalties on the hit roll of a close combat attack.
- The
**Custom Ammo**demolition specialist tactic provides a +1 modifier to wound rolls, while the Painkiller Medic tactic adds 2 to a model’s Toughness characteristic. - If the target is already injured then each flesh wound will provide a +1 modifier to the Injury roll.
- The
**High Explosive**tactic for Demoltions specialists increases the Damage characteristic of an attack by 2, but the weapon only fires 1 shot. - The
**More Bullets**Heavy tactic provides 1 extra shot to an attack and Quick Shot doubles the number of shots fired by a weapon (at the expense of a -1 hit roll modifier), while the Stimm-Shot Medic tactic increases a model’s Attack characteristic by 1. - The
**Killing Frenzy**Zealot tactic enables the attacking model to make an extra attack against the target every time a hit roll of 6+ is made, meaning this tactic is influenced by modifiers to hit rolls. - The
**Overwhelming Firepower**Heavy tactic allows the attacker to shoot twice. Because they are separate shots Overwhelming Firepower the second shot occurs even if the target is reduced to 0 wounds, making it particularly deadly. **The Roll With The Hits**Veteran tactic forces an attacker to only roll a single die for an Injury roll regardless of the Damage characteristic of the attack.

### Damage and the Injury Roll

The rules surrounding the Injury roll and the fact that any additional attacks are nullified once a target loses its last wound makes for an interesting combination of effects. The attack sequence can be divided into two components that are multiplied together to get the final result; the probability that an attack will reduce the target down to zero wounds, and the probability that an Injury Roll will result in an out of action. The latter is easier to deal with than the former, so let’s start with the Injury roll. The probability that a target will be put out of action is the inverse of the probability that none of the Injury rolls will be successful. In other words it’s (1 – P(OOA)^D) where D is the Damage characteristic. The chart below shows the probability of success given the various characteristic values and the possibility of several modifiers to the Injury roll. In terms of affecting the outcome increasing the characteristic is always more effective than increasing the Injury roll. The values of 100% aren’t actually 100%, but the probability of failure is so small that it might as well be.

Damage Characteristic | Probability of Out of Action | |||

Modifier | -1 | 0 | +1 | +2 |

1 | 33.3% | 50.0% | 66.7% | 83.3% |

2 | 55.6% | 75.0% | 88.9% | 97.2% |

3 | 70.4% | 87.5% | 96.3% | 99.5% |

4 | 80.2% | 93.8% | 98.8% | 99.9% |

5 | 86.8% | 96.9% | 99.6% | 100% |

6 | 91.2% | 98.4% | 99.9% | 100% |

7 | 94.1% | 99.2% | 100% | 100% |

8 | 96.1% | 99.6% | 100% | 100% |

Dealing with the probability of reducing the target down to 0 wounds is more complicated as it is a binomial distribution problem. If the Damage characteristic of the attack is equal to or greater than the Wounds characteristic of the target then the solution is the inverse of the probability that none of the attacks will pass through all of the gatekeeper rolls.

P(Injury Roll) = 1 – [1 – (P(Hit)*P(Wound)*(1 – P(Save))]^n

Where P(Hit) is probability of passing the hit roll, P(Wound) is the probability of passing the wound roll, P(Save) is the probability that the target will make its armor save, and n is the number of attacks directed at the target.

Things get even worse if the Damage characteristic is less than the number of wounds of the target. In that case you have to look at how many attacks must get through in order to reduce the target to 0 wounds, and then calculate the probability that at least that many attacks will get through. This can be a major pain to figure out, but the approach is to sum the probabilities of all of the combinations that result in failure, and then subtract that value from 1. For this we’ll need to use the binomial distribution formula provided in the link above.

P(X = k) = [n! / (n – k)! k!] * (p^k) * (1 – p)^(n – k)

Where n is the number of options, k is the number of successes, and p is the probability of success. The first value in square brackets is the number of combinations that result in k successes out of n options, the second value is the probability of exactly n successes, and the third failure is the probability of the remaining outcomes failing. The final step is to add the probabilities of each failing outcome together and subtract it from 1.

#### Example: Volley Gun vs Primaris Space Marine

Let’s look at the example of a Militarum Tempestus Scion Gunner (4 shots, BS 3+, S4, AP -2, 1D) shooting at a Primaris Space Marine (T4, 3+, 2W). The Scion is a Demolitions Specialist and has to decide if he should use the High Explosive Tactic, which only lets him fire 1 shot but adds 2 to the Damage characteristic. The probability that the Scion will hit, wound, and that the Marine will fail his save is 22.2%. Using High Explosive the resulting Damage characteristic of 3 would give the Scion three Injury rolls which has an 87.5% chance of producing an out of action result. Multiplied together the probability of killing a Primaris with High Explosive is **19.4%**.

Assuming the Scion chooses not to use the Tactic, we would need to determine the probability that at least two shots get through. In order to do this we’ll look at the probability that none or only one shot gets through and take the inverse. Using the binomial distribution function we have 4 shots (n = 4) and an individual probability of success of 22.2% (p = 0.222).

P(X = 0) = [4! / (4 – 0)! 0!] * (0.222^0) * (1 – 0.222)^(4 – 0) = 0.366

P(X = 1) = [4! / (4 – 1)! 1!] * (0.222^1) * (1 – 0.222)^(4 – 1) = 0.418

P(X > 1) = 1 – P(X = 0) – P(X = 1) = 0.216

Amusingly enough the 21.6% chance that the Scion has of wounding the Space Marine at least two times over four shots is nearly identical to the probability that a single shot would wound the Marine. The final part of the problem is to multiply the result by the probability that a single Injury roll will result in an OOA (50%) for a final probability of **10.8%**.

In this case the Scion is better off using the High Explosive Tactic.

## Wrapping Up

That was a pretty interesting problem to figure out!. The math behind Kill Team is surprisingly complex thanks to the presence of the Injury roll and the elimination of further attacks once a target is reduced to 0 wounds. I plan on revisiting this topic in the future, and if you have any particular combinations of attacks and targets you would like me to take a look at please **submit them here**.

As always, thanks for reading. We know that things are really challenging lately, and hopefully this article and the rest of our content provides some levity, useful information, and entertainment.