# How to calculate a probability

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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