When you look at the history of wargaming, there are all sorts of strange contraptions that various rulesets use to determine area effects of firing. Most of these come in the form of firing templates. Usually, the template is placed on the table, and all figures underneath the template have some probability of getting hit.
Templates are an old idea. Below you see a cannister template from The Wargame by Charles Grant, which was soldered together.
Here's another curious template, from Don Featherstone's Advanced Wargames.
Especially fantasy rules have used templates in all forms and sizes. Here you see a selection of templates from my "templates box", most from various incarnations of Warhammer. The "Fallen Drunk Giant" template is still one of my favourites :-)
In order to bring more variability (and randomness) in the fire effect, some rules also specify that the template can "deviate" from the original position. A random direction is determined, a random distance is rolled for, and the template "deviates" that particular distance in that particular direction, to determine the final area where the firing will have an effect.
There are various possible mechanisms to determine this random direction and random distance, and especially fantasy and science fiction rules have used a whole zoo of variations on this theme over the years.
It will be impractical to analyze them all in terms of probability and effect, but I have chosen one particular procedure using the "scatter die" and "artillery die", I believe first pioneered by Games Workshop back in the nineties (any corrections welcome!).
The idea is as follows:
- Place the firing template over the intended target.
- Roll the scatter die to determine direction of deviation (the direction of the face-up arrow). However, the scatter die also has 2 "Hit" faces, indicating no deviation at all.
- Roll the artillery die to determine the distance. The die can give 2, 4, 6, 8, and 10 as a result, along with a "misfire" that usually ends up in some hilarious effect for the crew firing the war engine.
Distribution of position of the template
Let's take a look at where the template might end up when there is a deviation. To make matters easier, we will consider a template with a 2" diameter (1" radius), and consider the artillery die distances in inches as well. This is a common application of the procedure, and can also be found in various Games Workshop games.
The diagram below shows the possible positions for the template, when the deviation would happen in the horizontal direction.
When taking into all possible random directions (a few are drawn below), a pattern starts to emerge.
It is obvious that the further you move away from the initial position of the template, there is less chance that any particular area will get covered by the deviated template. Indeed, the same number of possible template positions have to cover an ever-increasing larger circular area. Hence, the closer you are to the initial point of impact, the higher the probability your figure will be hit.
Let's look at this probability in some more detail.
Because the artillery die has equal probabilities for distances 2, 4, 6, 8 and 10 (remember, we ignore the misfire result), there is an equal probability for the template to end up in any of the 5 concentric circles.
Each of these 5 concentric range bands has its own area, which can easily be computed by subtracting the area of the smaller circle from the bigger circle. E.g., to compute the area of zone C, we compute
The areas of all zones, using the same method of calculation:
- Zone A: 8pi
- Zone B: 16pi
- Zone C: 24pi
- Zone D: 32pi
- Zone E: 40pi
- Zone A: 12.5%
- Zone B: 6.25%
- Zone C: 4.17%
- Zone D: 3.13%
- Zone E: 2.5%
Overall hit probability
Since each of the range bands has a 20% chance of occurring, the overall probability of a figure, located somewhere in the total possible impact area, is then as follows:
- Zone A: 12.5% *1/5 = 2.5%
- Zone B: 6.25% *1/5 = 1.25%
- Zone C: 4.17% *1/5 = 0.83%
- Zone D: 3.13% *1/5 = 0.63%
- Zone E: 2.5% *1/5 = 0.5%
In order to achieve this, we have to adjust probabilities in proportion to their relative areas. Since the areas of Zone A, Zone B, Zone C, etc. are equal to 8pi, 16pi, 24pi and so on, we simply will have to design a randomizer proportional to these areas, and making sure that all probabilities sum up to 1. The sum of 8+16+24+32+40 = 120, and thus:
- Zone A should be generated with a 8/120 = 6.67% probability
- Zone B: 16/120 = 13.33%
- Zone C: 24/120 = 20%
- Zone D: 32/120 = 26.67%
- Zone E: 40/120 = 33.33%
Some more mathematics
The analysis made above is also related to the problem of generating points in a circle with uniform probability, which is often a textbook exercise in many Probability 101 classes. Since the area of increasing concentric range bands goes up quadratically (a pattern that you can also see above), a random point can be generated by picking a random direction, and picking a random distance from the origin by taking the square root of a uniform random number generated between 0 and and the radius squared. Simply picking a random distance uniformly between 0 and the radius would produce a spread of points located closer to the centre of the circle.