The Sliding Scale of 2-dice Probabilities

Warhammer 40,000 is a game built on rules and probabilities. Those who designed the game wanted it to be balanced, fair, but with enough complexities that the game would allow almost any situation to occur and be carried out. Not only that, but the game had to be designed with a certain flavour. This flavour is called “telescoping”.

Telescoping is how the game goes from small scale to large scale, or large scale to small scale, by orders of magnitude. A 500 point game lasts about 30min, start to finish. A 1000 point game lasts about an hour and a half though, 3 times as much time for only twice as many points. A 3000 point game can last the whole day if things start getting off course.

(note, the reason for the jump is that you are dealing with more models. A 1500pt game of two Grey Knights players will still go swiftly, a 1500pt game of two troop-heavy Ork players will take much, much longer)

However, telescoping also applies to the math and structure behind the game. The simplest reason is due to the properties of the dice.

A single die roll does not produce many variations. It’s an exact equal chance of the numbers 1 through 6 being rolled. However, when you roll two dice together (like in a Ld test), you can achieve an ARRAY of results:

Ld	2	3	4	5	6	7	8	9	10
Pass %	3%	8%	17%	28%	42%	58%	72%	83%	92%
% Change	-	+5%	+9%	+11%	+14%	+16%	+14%	+11%	+9%

Notice the % change column. In a single d6 roll, each roll is 16.7% greater or less than the ones next to it. If you need a 4+, it’s 50%, but 5+ is 66.7% and 3+ is 33.3%. The change is even.

Not so with 2 dice…

If you’re measuring something like Leadership, going up from Ld7 to Ld8 (+14% increase) gives you a bigger jump in your chance of success than going from Ld8 to Ld9 (+11%). Keep that in mind the next time you’re deciding how far away from that Junior Officer you’re going to be moving your Infantry Platoon.

Leadership is not the only situation where a single attribute change makes a larger than normal difference. The same applies to twin-linked weapons.

BS	5	4	3	2	1
Normal	83%	66%	50%	33%	17%
Twin-Linked	97%	89%	75%	54%	30%
% Change	+14%	+23%	+25%	+21%	+13%

As you can see here, it’s clearly a huge advantage to be twin-linked (a Twin-Linked BS2 weapon has an even better chance to hit than a normal BS3 weapon) – but that the amount that it’s worth changes from point to point. A BS5 weapon only gets a 14% increase in effectiveness from being Twin-Linked, but a BS3 weapon gets a staggering +25% increase in effectiveness.

The laws of probability affecting the roll of 2 (or more) dice is a great example of something called the Normal Distribution.

Normal Distributions are used by probability experts to determine just how likely, or unlikely something is. Essentially, it says that if you do something a whole bunch, 68% of the answers will be more or less within 1 Standard Deviation of each other. A Standard Deviation is a short way of saying “the square root of the average distance of any point in the study from the average of the study”.

Or in other words… with a d6, the average roll is 3.5, with a standard deviation of 1.5.

(To calculate standard deviations quickly, but not exactly accurate, find the difference between the average result, and then the higher or lower result, and divide by 3. Three standard deviations account for 98% of results, which for us effectively can be said to be all, or 100%, of the possible results. This makes the working deviation for a d6 1.2, not 1.5, as you can take 3 steps down from 3.5 to 0 at 1.2, and 3 steps up to 6 at 1.2.)

So, if you shoot 10 bolter shots, your average result will be that 6.6 will hit (upper SD = 1.1, lower SD = 2.2). That means that 68% of the time 4 or more or 8 or less will hit.

Also with Normal Distributions, we can see further and further into the probabilities. Go a SECOND standard deviation away from the norm, and you encounter 95% of all occurrences.

Knowing the standard deviation of a dice roll is very important. In the last probability analysis posted here, I said that if 24 shots came from Space Marine Bolters you can count on 12 hitting. Why 12? Why not 8? Surely you can count on hitting with 8 more than you can count on 12. Why not just 1?

I don’t count on these, because they’re too far away from the average (16). If I WERE to estimate a fast and dirty standard deviation, I’d say it were 2. I know it’s not, but 2’s an easy number to work with. I can say that I’m 68% confident (mostly confident) that I’ll hit anywhere between 14 and 18 times. If I hit more or less than that, then I really have encountered bad luck.

How is this useful to you? Once again, when judging a situation. Before we said, with the very basics, that you could count on killing 2 Space Marines with 24 bolter shots, but now we can say that we’re confident that we’ll kill between 2 and 4.

***PERSONAL NOTE:

While doing this example, 24 Space Marine Bolter Shots, I ran a quick probability study. I averaged the number of shots hit from the mean and 1 standard deviation away from the mean. In other words, I took all the possibilities that are likely to happen at each step in the way, and calculated the average of that.

Thus, 24 shots, average 16 hit, possibly 18.7 hit, possibly 10.7 hit. Then, 10.7 hit, average 5.3 wound, possibly 3.6 wound, possibly 7.1 wound. And so on.

Once finished, I arrived at an answer that was 68% of the time you’ll kill between 2 and 8 Space Marines. I knew this was wrong. Why? Because experience tells me that 24 shots will often kill 2 Space Marines. That’s just what I’ve noticed over the years, no math about it. I then looked through the equations and found where I was wrong.

Redone, the calculations show what likely happens, 68% of the time between 3.7 and 1.9 Space Marines will die from 24 Bolter shots.

Morale of the Story: Never let numbers dominate your thinking. If you KNOW something, and your calculated answers say differently, chances are your numbers are wrong. All too often people get caught up in the equations, but the fail to anticipate the outcome in order to really gauge how accurate it is. If I say 6 – 2 = 3, you know it’s wrong, because you already expected a different answer. If I ask what 529 – 315 equals, you might not know the exact answer, but you’ll know that an answer above 300 or lower than 100 has GOT to be wrong. Develop these instincts, they’re incredibly powerful tools when dealing with math.

***

Want to see this in practice? Here’s the 24-Shot experiment. Find a dice roller online (type “dice roller” into google, first one should be the D&D one, that’s fine). Now, pretend to roll the dice for 24 Bolter Shots at BS4 against a Space Marine squad (3+ save).

Keep track of how many Space Marines died each time.

Do these experiments 10 times.

Now, average out the number of space marines died (total space marines died divided by 10). I can guarantee you (98% certain) that this number lies between 3.7 and 1.8.

Want even more precise? 68% of you will have a value between 3.2 and 2.5.

Your most common scores for number of Space Marines died will be 2 and 3. You will have either one or two results of 1 or 4 Space Marines dying (most likely you’ll have more 4’s than 1’s, but I can’t be quite as confident). One or two rolls will be completely outside this (like 7 dying, or 0 dying).

When you understand probabilities, you can really give yourself an advantage by predicting the outcomes of near-future events.