What Is Random?

What is random chance? What does it really mean that a dice has 6 perfectly equal sides, or that you have a 66.66% chance, or that a dice encountered interference?

Randomness pervades our lives. Things happen on a daily basis that we cannot explain, and thus we attribute the event to being “random”. Like what time does the bus arrive at the bus-stop when you go to school? You know that it’ll come around a certain time, but it’s rarely exactly on time – often being a little early, or a little late.

We would call the exact time that it arrives at a completely random event – because we have no influence over it.

However, that’s not exactly true. The time the bus arrives at ISN’T random at all, in fact. The time it arrives at is completely predicated by the events leading up to the exact time it picks you up. The bus driver woke up and left for work early, but then encountered heavy traffic. The first two bus stops were emptier than usual, but the next two were a bit more packed. He stopped for a quick coffee later, but then went a bit faster than normal to make up the time.

You could argue that the different circumstances leading up to each event were random, but they aren’t either. All have a basis in the actions of people. Someone decided not to go to work and take a holiday, another person took a later bus, road-construction slowed down traffic, etc.

The roll of a die too is a perceived, but false, random event. The way you cup your hand, the force put behind the throw, the smoothness of the table, the Rhino the die bounces from. If you had a strong enough computer, and accurate enough cameras to record the exact conditions immediately after the throw, before the die has touched anything, you could predict and determine how (and where) the die will land. It’s simple physics really.

For this reason, many physicists and mathematicians have been trying to discover something truly random – and its hard work! If anything, it’s so hard because Physics as we know it implies cause and effect. One event leads to another. If I drop a ball, I KNOW it will fall downwards. In the same way, even at especially small scales, where more and more apparently truly random events are occurring, we still can say with certainty that if one thing happens, so will something else.

A great example of this is the decay of a single atom. Normally decay rates are very easy to predict. An object will lose a certain % of its mass to decay in a given amount of time. For example, Radium has a half-life of 1602 years, meaning that in one thousand, six hundred and two years it will lose half of its mass (decaying into Radon gas). At masses of trillions of trillions of atoms of radium, this exercise is very predictable – so predictable that you can ascertain how old the radium is if you know how much there was at the beginning and how much there is now.

However, individual decay rates are a % chance that a single atom will decay at any given moment. Just like the school bus, you know roughly when it will, but not exactly when – sometimes it’ll be early, and sometimes it’ll be late. However, when you repeat the process a hundred thousand million times, you arrive at an average that will likely apply VERY CLOSE to the next hundred thousand million times. But it is still POSSIBLE that the entire bar of radium could go poof all at the same instant, or that it practically never decays.

The decay of a single radon atom is like that kind of situation. It could decay at any moment, and in truth you really can’t predict when. It could be one of the atoms that decays today, or it could be one that won’t decay for another 50,000 years. You can only give a percent chance that the particle will decay within a given time.

And even this, like the school bus, is not really random. As electrons and neutrons and preons and muons all fly around subatomic space, they’ll be hitting each other. At a large enough level it seems to be random, but it isn’t.

At the end of the day, true randomness is impossible, and all we can function on is perceived randomness. The less you can control the outcome of a die roll, the more random it is. So the next time your die rolls off the table, let it finish rolling – it’s actually more random now since you didn’t anticipate it going off the edge.

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.

Probabilities made Easy

How I Stopped Guessing and Learned to Love the Fraction:

Throughout this Strategy and Tactic’s blog, you’ll see me referencing numbers, percentages, and probabilities. Sitting down typing, I’ll be a lot more tempted to work out the exact probabilities of what you want (or what you don’t want) occurring, and I might even get into some more complex stuff too (like saying “you can be 95% confident that you’ll score between 3 and 5 hits).

However, for most of us involved in the fast-paced games where stuff actually happens, or those of us playing a tournament, you're not going to have the luxury of time and paper or a graphing calculator.

So much of the time I see people discussing probabilities online. They say that a certain event is so exactly probable. Like, a bolter shot fired from a Space Marine Tactical Squad has a 66.6% chance of hitting. If the Marine fires 2 shots, he has a 50% chance that both hit and a 75% chance that at least 1 hit. Etc., etc.

This ISN’T helpful for the majority of people, mostly because it’s too in-depth. A lot of people just want to know if what they want to happen has a favourable chance of happening, or not. Heck, most people don’t even know what a favourable chance is.

So, I’m going to give you a quick and dirty run-down of how to judge your chances on the fly. This method should help you size up your chances within 10 seconds. If it takes longer than this, then you probably don’t have a favourable chance, or you’re dealing with a very complex situation.

Effectively, to do this you are going to break the rules of standard probability calculations. You’re going to wind up with numbers over 100%, which normally is not allowed, you’re going to add when you should multiply, and you’re going to estimate when you should calculate.

Step 1: Know the Odds

Memorize this.

6+ = 16% or 1/6
5+ = 33% or 1/3
4+ = 50% or 1/2
3+ = 66% or 2/3
2+ = 83% or 5/6
1+ = you never have a 1+, but it’s 100%

It’s imperative that you know these numbers or fractions. I put down both, because I like working with fractions (seriously, when you understand fractions and can do them in your head, all of math becomes easier to handle).

Step 2: Add

Any time that you have a chance of doing something, add these together. If I fire two bolter shots from a Space Marine Tactical Squad, I have a 130% chance of hitting with at least one. Or, in other words, often 1 hits, sometimes 2 hit, but rarely will none hit. Why? Well, 100% means that you have a good chance of it happening. It’s not perfect (perfect does not exist in this system) 100% merely means a good chance that at least 1 of what you are trying to determine will happen. 30% means that sometimes, though not often, 1 MORE will hit. Now, you know that there’s a chance that neither shot will hit, but that’s a different chance calculation (specifically, 33% + 33% = 66%, is less than 100%, so not a good chance, but it’s there).

When dealing with larger shots, knowing fractions will help. Say it’s 24 shots. Well, 24 * (2/3) will give you 48/3. Don’t worry if you can’t do that division, you can guess that 3 goes into 48 at least 12 times. That’s quite a bit lower than in reality, but you don’t need to know that it’s exactly 16 times (besides, exact figures are misleading). 12 is a figure you can count on – meaning that you should be able to hit 12 times. You may get more, but it’s unlikely that you’ll get less since you under-estimated.

If you do shoot for exact estimations (like 16 in this case), you will find that 50% of the time you score this or more, and 50% of the time you’ll score this or less. This isn’t a probability study, this is a tactical survey.

Step 3: Determine Best Decision

So you know that you should hit more than 12 times. Will that be enough? At this point you should be able to know (shooting at Gaunts, you’ll kill a good number of them, 7 or 8 probably, shooting at Space Marines, you’ll get maybe 1 or 2… maybe none at all). If not, continue you’re fast calculations.

Guants: 12 * (2/3) = 24/3 = 8, no saves, 8 dead

Space Marines: 12 * (1/2) * (1/3) = 12/2 * (1/3) = 6 * (1/3) = 6/3 = 2 dead

What will this accomplish? Will the Gaunts be neutered? Will the Space Marines be forced to make a Ld test? Will this even the odds in Assault? That tactical decision will still be up to you, but you are now armed with info that can be compared to other information and your strategy in order to determine the best choice.

Remember, however, that this system is for BASICS only. Things get MUCH more complicated than this, but your understanding of this basic concept is rather critical. You need to know this math in order to make statistically good decisions. Just remember, for best comparisons, guess on the low side. This game involves a strong element of chance. Sometimes, even a statistically favourable event will go catastrophically wrong.

For example, one time I had a squad of genestealers assault a Land Raider back when Rending gave you an extra D6 (not an extra D3). By all probability, the Land Raider should have been toast – but it didn’t, and its hurricane bolters then proceeded to munch apart my Genestealers – a devastating blow to my army (I was already getting low on units that could deal death-blows to my enemy).

A Note on Unfavourable Situations:

Sometimes an event is unfavourable, but you should take it anyways. A great example of this is when shooting at tanks. Technically speaking, a single Lascannon has a poor chance of killing any vehicle, simply because over half of the Vehicle Damage table gives a result that is NOT destroyed. Even the dreaded Tau Railgun has a less than 50% chance of destroying even an Armour 10 vehicle (1/2 to hit, 1/2 to destroy… remember that being AP1 gives it +1 to its damage roll = 1/4, or 25%... and we’re not even counting the vehicle as hull-down for an additional 1/2 chance that it shrugs off the hit).

However, it still makes sense to include these in your army because, when it DOES destroy a tank, it will often deal a very large blow to the enemy. Tactically speaking, it has the best chance-to-impact ratio out of your whole army by targeting the enemy’s vehicles. Impact is the tactical blow that losing that unit will cause to your opponent. It cannot be directly measured (destroying that Demolisher in range of your Terminators is a much greater blow than destroying that Demolisher that’s 48” away from anything).

Determining that ratio will largely be determined by you, as a tactical decision governed by your strategy.