Monday 31 October 2016

Of Math and Medikits

The MediKit is an unusual piece of equipment that is found on all Paramedics and many Doctors across all the factions in Infinity. As the Paramedic skill can be commonly found as a profile option for basic infantry units you may well have come across it before.

I've recently been finding myself using MediKits more often, likely due to the lack of competent Doctors and Servant Remotes in Ariadna, but I've encountered only limited success with them. I'm not sure that I should even bother trying to make use of them.

It's always difficult to work out odds on the fly, so I thought it'd be interesting to spend some time calculating some different probabilities so I can be better informed for future games.

Warning: There be charts ahead!

Let's start at the beginning. What does the MediKit do? I recommend reading the full rules on the Wiki, but here is a quick summary.

When in base-to-base contact with an unconscious target trooper you can give them the chance to recover a wound by passing a PH-3 roll. You can also activate this ability at range by firing it like any other weapon - although you aren't allowed to target opposing troops with it!

Note that if the target fails their PH check then they immediately die!

MediKit Profile - I'm fairly sure that the non-lethal trait is just one of CB's jokes.

Most (all?) troops with a Wound characteristic have a Physique of between 8 and 16 so that's what I'm going to show in all my tables going forwards.

With that out of the way, let's start with some simple stuff. The chance of making a successful MediKit check on a model whilst in base-to-base contact looks something like this.

Chance of a target passing their PH check.

Easy, right? As the target's PH increases, so does their chance of recovering a wound.

However, using a MediKit is almost always strictly worse than using an actual Doctor in base-to-base contact instead. The -3 penalty to PH is pretty harsh, WIP is higher than PH on average and you can't use Command Token re-rolls on MediKit checks.

Learning Points:
  • PH 13 is the break-even point. This is an outright coin-flip.
  • Even at PH10 you have a 35% chance of passing, this can certainly be worth it if your Paramedic isn't going to spend lots of orders getting up close.
  • There is only a pass or a fail. Regardless of the outcome, the target will always either be healed or dead after one order.
  • Use a Doctor instead whenever possible.

So far so good.

Let's move onto something a little more complicated, like what happens if you shoot an unconscious model using your MediKit instead?

Keep in mind that when firing your MediKit you still get penalties for cover but camouflage, ODD and other similar skills all switch off if the unconscious model is in a null state (i.e. it doesn't have Valor: No Wound Incapacitation).

From here on I'll also highlight, using a box, what I think is the most common range of values when using a MediKit. This is when firing with a modified Ballistic Skill of between 10 and 16 (i.e. at a +0 or +3 range bracket) and at a target with a PH of between 10 and 14.

A success in this case is calculated as the chance of successfully hitting the target multiplied by the chance of the target passing the ensuing PH roll. A table of success rates looks something like this:

Chance of an unconscious target trooper regaining a wound when shot at by a MediKit.

It's worth noting that shooting a model with a Medikit in this way is never better than using the skill in base-to-base contact. At best you break even with a BS of 20 (or more) as you can't even get any benefit from a critical hit.

On the other hand, as long as you are willing to keep spending orders if you miss it's no worse than being in base-to-base contact either. The chance of missing is always 5% for each point of Ballistic Skill under 20.

The results are pretty intuitive at this point. You want the highest possible PH target and the highest possible Ballistic Skill.

Learning Points:
  • Using a MediKit at range is never more efficient than using one in base-to-base contact.
  • In fact, it can potentially be rather order intensive if you keep on missing so work out in advance how many orders you can afford to commit.
  • Given infinite orders, the odds of healing/killing a target are the same as if you were in base-to-base contact.
  • The chance of succeeding in a single order is substantially lower than when using the MediKit in base-to-base contact. In the former all you need to get to a 50% chance is a target with PH13, but at range it's very unlikely that you'll ever have odds that good.

Of course, with my USAriadna I have another complication to worry about. My Paramedics are often part of a Fireteam so I have the option of using the +1 Burst bonus as well. But is this the right thing to do and how does it compare to being base-to-base?

Keep in mind that if a trooper is hit twice by a MediKit then a success and a failure on the PH rolls will cancel each other out and there will be no change in the target's status - just as if you had missed completely.

The chart below shows the chance of a target trooper successfully recovering a wound when being shot at by a MediKit with Burst 2.

Chance of an unconscious trooper regaining a wound when being shot at by a MediKit with Bust 2.

This is where things start to look a bit counter-intuitive.

At lower Ballistic Skills it seems better than firing with Burst 1 but after a certain point your chance of healing the target actually starts going down as your Ballistic Skill goes up! So what's going on?

Let's see if we can look at the data another way. The next table shows the chance of a target dying by being hit at least once and failing all the resulting PH checks. Note that in this the red colour denotes a high chance of death whereas green is generally good for survival.

Chance of an unconscious trooper being killed when shot at by a MediKit with Burst 2.

In this chart, the column for PH 13 remains unchanged whereas the values on either side flip over. you can see that, in much the same way as for the previous chart, after a certain point the chance of failure also decreases as Ballistic Skill goes up.

This must mean that it's the chance of nothing at all happening that is increasing. With modified PH values typically being around 9-11 there is a very significant chance of getting one pass and one fail if the target is hit twice. This is what seems to be causing the falloff.

Learning Points:
  • If you want to resolve the situation (either by killing or by healing the target) as quickly as possible then you want to hit your target exactly once.
  • Hitting your target twice is better than not hitting at all if you want to resolve the situation as quickly as possible.
  • A higher BS reduces both failure rate and success rate.
  • At some point, it actually becomes less order efficient to first at Burst 2 than to fire at Burst 1.
  • It's still nowhere near as efficient as just being in base-to-base contact.

It's also interesting to note what the chart looks like when you assume that you can spend an infinite amount of orders on firing your MediKit.

Chance of an unconscious trooper regaining a wound after being shot at by a MediKit with Burst 2 indefinitely.

With Burst 2, if you are willing to spend an infinite amount of orders on the task you'll eventually end up at exactly a 50% chance of healing a PH13 model. For any trooper with a PH higher than 13 it's actually better to keep firing at Burst 2 than it is to use the MediKit in base-to-base contact. The inverse is true for any trooper with a PH of less than 13.

In comparison, if you did a similar thing with Burst 1 then you'd just end up with the same table as for being in base-to-base contact.

Learning Points:
  • If you have a lot of orders to spare and the target has PH14 or better then you are better of firing with Burst 2 than you are using the MediKit in base-to-base contact.
  • At PH 12 and under, you are always better off being up being base-to-base instead.
  • If you have a lot of orders to spare and are firing at Burst 2 you are better off using a low BS soldier than a high BS one against targets with PH12 or less. The lower the target's PH the more pronounced this effect is.
  • With infinite orders, you are always better off shooting at Burst 1 for targets with PH12 or less and Burst 2 with targets with PH14 or more.
But nobody ever actually has enough orders to spare to make this worthwhile, unless of course they are desperate to complete an 'Experimental Drug' classified objective. So what if you need to maximise your odds with a single order, but can't make it into base-to-base contact?

Below is a chart showing whether shooting at Burst 1 or Burst 2 produces a better chance of success from a single order.

Table comparing success rate between Burst 1 (blue) and Burst (red).

In this chart, white and red shows a leaning towards using Burst 2 and blue shows a leaning towards Burst 1. Broadly speaking, as PH goes down and BS goes up you should feel more inclined to use a lower Burst value.

The values are an additional flat percentage change of success based on using Burst 1 or Burst 2 for each combination of attributes.

Here's a simplified version showing what Burst value you should use without all the percentages.

Chart showing whether is better to use Burst 1 (blue) or Burst 2 (red)

Again, this chart looks a bit counter-intuitive at first glance. It shows that if all you care is success rate - so you want the highest chance of healing a trooper in a single order - that at around effective BS13 or 14 you are better off firing only once.

This is because you want to maximise your chance of hitting at least once but preferably only once. Passing one PH check is often hard enough and having to pass two can reduce the chance of success by half or more!

So what can you take away from all this?

Learning Points:
  • As a rule of thumb, if your effective BS is 13 or better than you should fire at Burst 1 instead of Burst 2. You are hitting efficiently enough that your odds of hitting at least once and your target passing a single PH check gives better odds than trying to hit twice and trying to pass all resulting PH checks.
  • For difficult shots, especially against high PH targets, you should always fire with Burst 2.
  • For easy shots you should only ever fire at Burst 1 unless you have a lot of orders to spend and value having the chance to try again later.

Final Figures

Lastly, here are some average (rounded) figures for you for the following range of stats.
  • BS between 10 and 16 (inclusive)
  • PH between 10 and 14 (inclusive)
These can be some useful numbers to keep in mind if you want to ballpark your odds of successfully healing someone. 

Burst 2:
34% (range of 33% to 37%) chance of not accomplishing anything at all.
37% (range of 27% to 48%) chance of killing the target outright.
29% (range of 19% to 37%) chance of healing the target

Burst 1:
35% (range of 50% to 20%) chance of not accomplishing anything at all.
36% (range of 32.5% to 52%) chance of killing the target outright.
29% (range of 17.5% to 44%) chance of healing the target.

55% (range of 65% to 45%) chance of killing the target outright.
45% (range of 35% to 55%) chance of healing the target.

Trauma Doc:
100% (range of 100% to 100%) chance of killing the target outright... (Silly PanO). 

Well, there you have it. I hope some of this has been interesting and/or useful! If you think anything looks amiss or could be explained better then please let me know!