Post by Adam on Mar 24, 2009 6:03:22 GMT -5
When you're designing games, units or even army lists, it's always useful (often imperative) to use a bit of maths, often known as 'mathhammer' (obviously derived from the injection of maths into discussions of Warhammer or 40K). Mathhammer's slightly forbidding, but actually pretty easy.
Mathhammer, otherwise known as probability, is all about chances. Chances are represented as numbers, usually fractions but often also percentages. A chance of 1 or 100% means that something is definitely going to happen, and 0 says that it cannot happen.
The chance of rolling any particular number on a D6 is 1/6 (I'm sure you can guess why). Correspondingly, the chance of passing a roll is the total number of results that'll give you what you want, divided by the number of sides on the die. So the chance of getting a 3 or less on a D6 is 3/6, and the chance of passing a 5+ to wound roll is 2/6 = 1/3 (you can divide the top and bottom by the same number to cancel a fraction down).
To calculate the chance of passing multiple rolls in a row, multiply their chances together. Let's say I'm playing 40K, and I charge a brood of my shooty Warriors into some Space Marines. To score a kill, each attack has to pass a 4+ to hit roll and a 3+ to wound roll, and the victim has to fail an armour save. That's 1/2 x 2/3 x 1/3 (or 3/6 x 4/6 x 2/6).
To multiply fractions, the number on the top of the result is the multiple of all the top numbers, and likewise for the number on the bottom: 1/2 x 2/3 x 1/3 = (1x2x1) / (2x3x3) = 2/18 or 1/9. So, if my Warriors have 12 attacks between them, on average, they'll kill 12 x 1/9 = 12/9 = one and a third Space Marines. Not that amazing.
When you need to average the result of a dice roll, take the number of sides, add one, and halve it. For example, the average result on a D6 is (6+1)/2 = 3.5. You can't actually roll a 3.5, but if you roll a load of D6s, add all the results up, and divide by the number of dice, you should statistically get a figure close to 3.5. Similarly, 2D6 average (and most common result) is 7, 3D6 average is 10.5, and so on. Very useful to know in Warmachine, Duel of Steel and other games that use a lot of such rolls.
To deal with stuff like re-rolls and rending, you do multiple 'cases' and add them up. Say I'm mathhammering the results of a brood of Stealers attacking some more Space Marines. To get a kill, either I can hit and rend, or I can hit, wound, and hope the Marine fails its save. To hit and rend requires a 3+ to hit roll then a 6 to wound, and to hit hit and wound normally takes a 3+, then a roll of 4 or 5, then a roll of 1 or 2 on the armour save.
Work them out separately and add them up. Like so:
Rend: 2/3 x 1/6 = 2/18
Wound: 2/3 x 1/3 x 1/3 = 2/27
When adding up fractions, if they don't have the same number on the bottom, multiply the each one by a certain number so that they have the same bottom, then add up the numbers on top. I accept that that was a bad explanation, but an example might help. So here I multiply the top and bottom of 2/18 by 3 and the top and bottom of 2/27 by 2. I get 6/54 + 4/54 = 10/54 = about a fifth.
This means that if I sink 30 attacks into those Marines I can expect, on average, to kill 300/54 or 5.5555555 of them, which rounds up to 6.
Re-rolling is similar: the chance of passing the roll is the chance of passing + the chance of failing times the chance of passing.
So a 3+ with a re-roll has a (2/3) + (1/3x2/3) = 6/9 + 2/9 = 8/9 chance of hitting.
That's pretty much all there is to it.
Mathhammer, otherwise known as probability, is all about chances. Chances are represented as numbers, usually fractions but often also percentages. A chance of 1 or 100% means that something is definitely going to happen, and 0 says that it cannot happen.
The chance of rolling any particular number on a D6 is 1/6 (I'm sure you can guess why). Correspondingly, the chance of passing a roll is the total number of results that'll give you what you want, divided by the number of sides on the die. So the chance of getting a 3 or less on a D6 is 3/6, and the chance of passing a 5+ to wound roll is 2/6 = 1/3 (you can divide the top and bottom by the same number to cancel a fraction down).
To calculate the chance of passing multiple rolls in a row, multiply their chances together. Let's say I'm playing 40K, and I charge a brood of my shooty Warriors into some Space Marines. To score a kill, each attack has to pass a 4+ to hit roll and a 3+ to wound roll, and the victim has to fail an armour save. That's 1/2 x 2/3 x 1/3 (or 3/6 x 4/6 x 2/6).
To multiply fractions, the number on the top of the result is the multiple of all the top numbers, and likewise for the number on the bottom: 1/2 x 2/3 x 1/3 = (1x2x1) / (2x3x3) = 2/18 or 1/9. So, if my Warriors have 12 attacks between them, on average, they'll kill 12 x 1/9 = 12/9 = one and a third Space Marines. Not that amazing.
When you need to average the result of a dice roll, take the number of sides, add one, and halve it. For example, the average result on a D6 is (6+1)/2 = 3.5. You can't actually roll a 3.5, but if you roll a load of D6s, add all the results up, and divide by the number of dice, you should statistically get a figure close to 3.5. Similarly, 2D6 average (and most common result) is 7, 3D6 average is 10.5, and so on. Very useful to know in Warmachine, Duel of Steel and other games that use a lot of such rolls.
To deal with stuff like re-rolls and rending, you do multiple 'cases' and add them up. Say I'm mathhammering the results of a brood of Stealers attacking some more Space Marines. To get a kill, either I can hit and rend, or I can hit, wound, and hope the Marine fails its save. To hit and rend requires a 3+ to hit roll then a 6 to wound, and to hit hit and wound normally takes a 3+, then a roll of 4 or 5, then a roll of 1 or 2 on the armour save.
Work them out separately and add them up. Like so:
Rend: 2/3 x 1/6 = 2/18
Wound: 2/3 x 1/3 x 1/3 = 2/27
When adding up fractions, if they don't have the same number on the bottom, multiply the each one by a certain number so that they have the same bottom, then add up the numbers on top. I accept that that was a bad explanation, but an example might help. So here I multiply the top and bottom of 2/18 by 3 and the top and bottom of 2/27 by 2. I get 6/54 + 4/54 = 10/54 = about a fifth.
This means that if I sink 30 attacks into those Marines I can expect, on average, to kill 300/54 or 5.5555555 of them, which rounds up to 6.
Re-rolling is similar: the chance of passing the roll is the chance of passing + the chance of failing times the chance of passing.
So a 3+ with a re-roll has a (2/3) + (1/3x2/3) = 6/9 + 2/9 = 8/9 chance of hitting.
That's pretty much all there is to it.
