For this entry we will consider the concept of most likely result, and standard distribution.

Most likely result  is important when thinking about the difference between a theoretical result and an experiential result. As an example: a single attack that is 4+ / 4+ versus a model with no save has a mean (often simply described as average) outcome of .25 wounds. Needless to say you will never roll .25 wounds. Instead what is the most likely result? Clearly 0 wounds as 3 out of 4 times the result is 0 wounds. 

How do we though work this out on a rough basis sufficient to give us useful game information without necessarily having to work through an entire map of outcomes to calculate probabilities and the most likely single result? As a rule of thumb, do the maths hammer for an attack round against a 4+ save model* and look at the mean damage. Use the calculated damage as a numerator, and the damage characteristic as the denominator, and this will tell you how many attacks you can expect to deal damage. As an example using the single attack above, the outcome is .125/1 - so very close to 0. Expected outcome from 1 attack is... nothing. For a Stormcast Liberator Prime with a Greathammer though the number is 1.33/2 = .66 or more likely than not to actually inflict damage, with 0 damage being the next most likely result, then 4 damage, and 6 damage at a very small chance.

Why is this useful information in a game of Age of Sigmar? Basically it applies to adequate application of force without over committing or wasting force. As an example of this, lets consider two units: Rockgut Troggoths and Fellwater Troggoths. For the sake of our example lets consider that there is a Stormcast liberator blocking a path, and only one model can charge. Would you send in a Rockgut or a Fellwater? By quickly doing the calculations we can work out that a Fellwater does an average of 2.4 damage, and a Rockgut 2.2. On average both kill the Stormcast so does it matter? Using the rule of thumb from above we can see that there is a difference - the Fellwater will most of the time inflict at least one hit, while for the Rockgut a fairly substantial amount of the time they will do 0 wounds. If we need to inflict 2 damage, choose the Fellwater as it will more consistently achieve this. Other examples of when this kind of maths is useful: you have two models in a position to inflict the last wound or 2 on a character, who do you choose first? Does a unit need a buff to achieve what you want or should you place it somewhere else).

Finally a brief discussion of a normal distribution (also known as a bell curve). Basically this is where there is a single high point on the Y axis (horizontal in our discussion likelihood of an outcome) which reduces in a linear fashion the further from this you travel. A simple example is the distribution of a 2d6 roll;

The most common result is 7, and then the further you move from that the less likely the result is. What is relevant here is that when you consider how much you benefit from a plus or minus within the context of a normal distribution the closer you are to the mean the smaller the impact. This means that if you are rolling a 2d6 charge, a +1 to your roll triples your chance of rolling a 12, but is a improvement of about 25% when you are trying to roll a 7 or more.

Next post will be about how you can use excel to perform a Monte Carlo simulation.