## Dice Probability

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Warning: The following is heavy on statistics (Albeit easy to understand ones.) If you have an aversion to math, believe in astrology, or simply like to play by your gut feeling then you may want to stop reading now. I am completely open to criticism if I've made a math error but arguing something like the "But I don't roll that way" or "What about luck?" approaches are pointless: The numbers do not lie. This article is to discuss how probable a dice roll is, and factors like luck aren't taken into consideration- only math!

## The d6

Lets begin by looking at the chance of obtaining a particular result when rolling 1d6. As is obvious, a six-sided die has 6 possible outcomes (1-6) which each outcome having an equal chance of appearing on any given roll. That means the chance of rolling a particular result is 1/6 or about 16.67% and that we have an equal chance of getting any particular result on a single die roll. See Figure 1. Notice that the distribution of results is flat - each outcome has exactly the same chance of occurring. That's the nature of a fair die.
Figure 1: Probability of a particular outcome on 1d6
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Since probabilities add arithmetically, the chances of getting a particular result greater than a desired number is found simply by adding up the separate probabilities of getting any of the desired outcomes. E.g. the chance of getting a 4+ (4, 5 or 6) on 1d6=1/6+1/6+1/6 or 3/6 (Exactly 50%.) See Table 1 below.
 Result Out of 6 Probability Cumulative Probability 1 1 16.67% 16.67% 2 1 16.67% 33.33% 3 1 16.67% 50.00% 4 1 16.67% 66.67% 5 1 16.67% 83.33% 6 1 16.67% 100.00%
Cumulative Probability is the probability of getting this particular result or less. Reverse the order of the #s if you want the chance of getting a particular result or greater (i.e. a 3+).
When we talk about results from any singular event (i.e., a single die roll) we always have exactly the same chance of obtaining a particular result on that roll - 16.67%. However, in trying to make decisions people will refer to what the "average" result would be if we rolled a die many times - this gives us an idea of what to expect but does not predict the future with any great accuracy.
Myth 1: The average roll on 1d6 is a "3."
This is patently false and comes from using simple logic rather than thinking about the problem and using actual math. The average (Or more correctly the arithmetic mean) is found by adding up all of our observations and dividing by the number of observations. So, in this case we add up all the possibilities and divide by 6.
1+2+3+4+5+6=21. 21/6=3.5!
Notice, the average is 3.5, not 3. Wait a minute, you can't roll 3.5 on a die. That's right, the average isn't an actual outcome - instead if you look at Figure 1 again, you'll see what's happening - half the time you will roll a 3 or less. The other half of the time you'll roll a 4 or more (Notice the 50% point is at "3" meaning you've got a 50% chance of rolling a 3 or less.)
Obviously when gauging the outcome of a single roll "averages" aren't very useful. Why? Because, averages are meant to summarize data you already have collected - in other words, you know the results already and want to describe and summarize them. We're instead trying to predict future outcomes which means we need to focus more on probabilities and the probabilities are equal for each result meaning on any given roll all possible outcomes have an equal chance of occurring.
A side note: This doesn't mean that averages aren't useful. Calculating the average number of models wounded or armor saves failed is a very useful exercise. It is in fact based on the collection of hypothetical data ("If I get 12 hits, how many wounds will I generate on average?"). However, you need to be careful that you don't think that the average equals the expected outcome - as illustrated above, probability dictates that in most cases you're going to fall above or below that average on any given roll.
So, how do we use this information? Armed with the knowledge of what our chances are of obtaining a result we can make decisions. E.g. - You know that in using Fleet of Foot, you have a 50% chance of getting a 4 or better - this could effect whether you decide to move your unit up to try and get a charge in or whether you hold them back. If you know you're 16" away to start, you now know you have a 50/50 chance of actually charging an opponent. However, if you're 18" away you now that you have only a 16.67% (1 in 6) chance of succeeding with reach your opponent. The same principle can be applied to deciding on what the impact of an improved BS Definition: Battle Sister or Ballistic Skillor WS skill will be against a particular foe or what the chances are of someone failing a 3+ armor save.

## 2d6

At first glance, you wouldn't think adding a 2nd die to a roll would change things greatly, but, in reality, it does. This is because we are adding the results together which causes certain results to be much more common than other results. For example, there are 6 possible ways to generate a "7" on 2d6 (e.g., 1+6, 2+5, 3+4, 4+3, etc.), but only a single way to obtain a "2" or a "12" (i.e., double ones and double sixes, respectively). Thus, we need to generate an outcomes table that shows all the possible results and then use these to calculate individual probabilities.
If we then arrange these into a histogram, we get an unusual distribution. See Figure 2. Notice, this generates what is considered a perfect bell-shaped distribution - in other words, you are much more likely to get scores closest to the middle of the range (6- and scores approaching the extremes become more and more unlikely. It's also perfectly symmetrical, meaning that the chances of getting a result on either side of the most common value are equal (e.g., the chances of getting a "3" or an "11" are equal).
Figure 2: Probability of a particular outcome on 2d6
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Once again we can work out the probabilities of getting particular results to construct a table as well. See Table 2.
 Result Out of 36 Probability Cumulative prob1ability 2 1 2.78% 2.78% 3 2 5.56% 8.33% 4 3 8.33% 16.67% 5 4 11.11% 27.78% 6 5 13.89 41.67% 7 6 16.67% 58.33% 8 5 13.89% 72.22% 9 4 11.11% 83.33% 10 3 8.33% 91.67% 11 2 5.56% 97.22% 12 1 2.78% 100.00%
Cumulative Probability is the probability of getting this particular result or less. E.g., the chance of getting a 4 or less on 2d6 is 16.67%. Reverse the order of the #s if you want the chance of getting a particular result or greater (i.e., a10+).
Myth 2: The average result on 2d6 is a "6."
Once again, this probably arises from use of faulty reasoning or the mislead belief that you simply take the maximum result and divide by 2. If we calculate the true average it comes out to be "7."
2+3+4+5+6+7+8+9+10+11+12=77/11=7
However, here the average is once again the wrong number to be using (although as you will see it actually works), because we do not have an equal probability of getting each result - a "7" is much more likely to come up than a "2" and thus you actually need to calculate a weighted average if you want to know the true average result. All that said, the average isn't what we're interested in anyways because averages are dealing with known data and don't let us "gamble" very effectively.
Instead, we want to know the most likely (probable) result from rolling 2d6 and that's available just by looking at Figure 2 or Table 2 and choosing the result with the highest percentage of outcomes. In other words, the most likely result of rolling 2d6 is a "7." So, you are most likely to roll a "7" when rolling 2d6 and the chance of getting a result goes down as you move farther away from "7." The chances of getting either a 2 or 12 is only 1 in 36 (2.78%). In other words, betting on getting "snake-eyes" is largely a sucker's bet because the house will win that bet 97.22% of the time.
Cumulative probabilities are of more interest with 2d6 because typically in a WH40k game we need to obtain a particular result or less rather than a specific value on 2d6. However, unlike the results of a d6 you can't easily calculate these in your head without having memorized the individual probabilities, because the don't go up linearly (i.e., they don't go up an equal amount with each desired result), but instead form a S-shaped curve where the probabilities rise slowly at first and then quickly accelerate before slowing down again. You can see this in Figure 3 below.
Figure 3: Cumulative probabilities on 2d6 (chance of getting a particular result or less)
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Once again, how is this knowledge applicable to WH40K? The first area that should come to mind are LD checks (morale, targeting decisions, pinning, etc.). Leadership is taken on a 2d6 and looking at the results discussed above (and shown in Table 2), you can see exactly what percent of the time you can expect to pass a particular LD check. E.g., - if your unit has LD 7, you have a 58.3% chance of passing a LD check. However, adding a unit upgrade which boosts your LD to 8 raises the chance to 72.2% which means you'll make your LD checks a little less than three quarters of the time.
Specific example: Grots have a LD of 5 meaning they will only pass a LD check about 28.8% of the time. Add in a slaver and they more than double their chance.
You can also use the tables to gauge the overall outcome of a particular event.
Specific example: A SM Definition: Space Marineslibrarian hitting four LD 8 units with Fear of the Darkness will result in at least two units failing morale on average (they take their LD at -2 so that's a 41% chance of passing a single morale check).
Another place this information can be used is as it applies to anticipating fall back and deep strike scatter distances. Looking at you table, you can see that you are most likely to get scores closest to the middle of the range (6- . In fact you have about a 42% chance of getting a result between 6 & 8 anytime you roll 2d6, and a ~66% chance of getting a result between 5 & 9. This can be of some help figuring out how far you want to space your units from the back table edge if you're going to stand & shoot (weighing the benefits with the risk) or how close you want to risk placing a unit you're deep striking.

## The End

Hopefully you now have a basic grasp of the probabilities involved in roll a single 6-sided die or a pair of them. I am not trying to suggest that knowing these numbers will make you into an unbeatable opponent or that they can make up for bad tactics. Instead you should look at the information as one more piece of knowledge that you can use in making decisions, weighing the cost/benefit of particular units or upgrades, and in forming strategies.

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