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mhsellwood

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  1. mhsellwood
    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.
  2. mhsellwood
    Fundamentally to calculate a probability you need to work out how many possible outcomes there are, how many outcomes represent a desired result, and then doing a fraction based on this. Quick note on jargon: Numerator means the top number on a fraction - therefore with 2/6 the numerator is 2. Denominator is the bottom number on a fraction - therefore with 2/6 the denominator is 6.
    For a simple example of calculating probabilities:
    You roll a 6 sided dice, and you want to roll a 4, 5, or 6. There are 6 possible outcomes and 3 outcomes represent what we want. The likelihood is therefore 3 (the number of outcomes that we are interested in) over 6 (total possible outcomes) which translates to a 50% chance. N.b 1/6 is roughly 16.6% but is not exactly this amount.
    The next step from this is how to calculate multiple probabilities (i.e. if you roll a dice needing a 4+, then a dice needing a 5+, what is the chance of succeeding?).
    Following on from the example above we can work out that a 4+ is a 3/6 simplifying to 1/2, and a 5+ is a 2/6 simplifying to a 1/3. To calculate the chance of both succeeding we multiply both the likelihoods of success together. Maths below:
    Numerator 1 (1) x numerator 2 (1) = 1. Denominator 1 (2) x denominator 2 (3) = 6. Final result is therefore 1/6. From a practical point of view, the fraction tells us that on average if you hit on 4+ and wound on 5+ you will need around 6 attacks to inflict one hit.
    Again, remember the difference between theoretical and experimental probability. That is, theoretical says for every 6 attacks, you get one wound.  Experimental, i.e. actually rolling the dice, says if you start with 6 attacks, you may end up with no wounds, or you may end up with 6. Over the course of many thousands of rolls you will end up with 1/6 of your attacks doing a wound, but any given set of rolls is not likely to be exactly 1 wound per 6 attacks (in fact the most likely result on any single roll of 6 dice is no wounds).
    An extended example, and to provide some additional discussion:
    A Liberator with a Warhammer attacks a Blood Warrior.
    Calculation:
    2 (attacks) x 1/2 (to hit on a 4+ therefore 3 (number of desired outcomes) / 6 (number of potential outcomes) simplified to 1/2) x 2/3 (to wound on 3+ therefore 4 (number of desired outcomes) / 6 (number of potential outcomes) simplified to 2/3) x 1/2 ((note this is the chance of the Blood Warrior FAILING their save - a high save will lead to a lower chance of doing a wound, low save the reverse) to save of 4+ therefore 3 (number of desired outcomes) / 6 (number of potential outcomes) simplified to 1/2).
    Calculation for numerator 2 x 1 x 2 x 1 = 4
    Calculation for denominator 1 x 2 x 3 x 2 = 12
    Final result: 4/12 simplified to 1/3.
    Final point related to these calculations: note that absent external modifiers, order makes no probability difference as all rolls are independent. That is, if the liberator hit on 3+ and wounded on 4+ the end result is the same. Similarly, if the Liberator hit on 4+, wounded on 4+, but the Blood Warrior saved on 5+, the final result is the same. 
    Next blog post will be about most likely result, and how we can think about bounded or capped results.
  3. mhsellwood
    The intent of this blog is to do some really basic discussion on probability, and how that can inform thinking about your games, and a bit of thinking about how we can model outcomes.
    First then, what this blog is. I will aim to keep this relatively straightforwards in terms of what I am describing, and I will aim to not use excessive amounts of jargon or technical terms. Although I intend to focus on Age of Sigmar most of the logic is applicable to any dice rolling game and I would like to do some looking at Underworlds as well.
    What this blog is not. Full disclosure I am not a mathematician. I have been playing Games Workshop games for about 25 years, I am post graduate qualified in accounting, and I do a lot of work with numbers, but I do not hold a tertiary mathematics degree. Therefore it is very much focused on real world probability and application rather than academic level discussion.
    For this first blog then, let's look at a really basic concept that helps to put some context around mathhammer (by this I mean the discussion around the 'hard' aspect of the game being the numbers versus the 'soft' aspects such as the social contract, positioning, decision making etc.) and how it is useful or is not.
    Discussion of probability within games must be considered using 2 different lenses: theoretical probability versus experimental probability. Theoretical probability is the probability that is produced through pure calculation. As an example, the probability of rolling a 6 on 1 dice is 1/6 (~16.6%) so if we roll 6 dice theoretical probability indicates 1 of these dice will be a 6. Experimental probability is the result of actually doing the thing we are discussing. So to contrast the example above, for the experimental probability of rolling a 6 on 6 dice, we would roll 6 dice, count the number of 6's and that would give us the percentage of 6's rolled. 
    The importance of this is probably obvious; intuitively we know that 6 dice 'should' include a 6, but from personal experience I know this is all too often not the case. Thus, when we talk about probability and what 'should' happen always keep in mind that we are discussing theoretical probability, and that experimental probability is quite significantly different.
    The other thing to consider is the concept of independent versus dependent probabilities. Independent probabilities are probabilities that are in no way influenced by another probability. I.e. if you roll a dice the chance of getting a six is 1/6. If you roll a six, then the next roll has a chance of rolling a six of... 1/6. Dependent probabilities are where probabilities change. As an example the chance of pulling any given card (let's say Great Strength) from your deck in Underworlds is 1/20. After you draw your first card, the chance of the next card  being Great Strength is 1/19.
    Why is this important? If a thing is an independent probability then a player has no control over the outcome (you rolls your dice and you takes your chances). If a thing is a dependent probability you have a degree of control - if you draw and discard cards then in order to increase your chances of drawing a specific card you need to draw more cards as each card drawn increases the odds of drawing the card you want.
    Next blog post will be basic mathematics of working out the possibility of something happening.
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