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How to calculate a probability


mhsellwood

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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.

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Very good. A basic understanding of probability, I think, is fundamental in list building and general gaming. Knowing that, on average (theoretical) my 10 ard boyz with 20 attacks will only (on average) hit 10 times, wound 5 times and, against a 4+ save do 2-3 wounds on average, well, that makes their offensive output limited, so helps make the decision about the type of battlefield role they will get. This is obviously with no buffs or bonuses, but as a starting point, knowing what is probable makes a big difference. 

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Thanks for the comment, and agreed with your point about it being a great start point.

Something I intend to get to is how to basically work out a units offensive output and defensive output per point, to allow some basic comparisons. Important thing to keep in mind for AoS is as you touched on how much important the availability of buffs and debuffs is to how well an army works in total.

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