None of this is my work but I think its well worth making as widely available as possible so this reposting is by way of helping it along
Dice Statistics
Why Statistics matter
Most roleplaying games are played with dice, which job is to generate
a random number; these random numbers represent chance, which – without
initiating a philosophical discussion – is that part of life we can’t
do anything about. These numbers are further modified by our skills in
the different tasks are most honoured and revered Dungeon Master /
Hackmaster / Game Master – the names of this esteemed individual are
many, and all rightfully said with the utmost respect when uttering this
magnificent individual’s title and name – sees fit to give us, the
players. Now, any Dungeon Master worth his salt would cast the evil eye
upon any player trying to make decisions on, say, which weapon to
choose, based on the arithmetical data provided in the books, rather
than making this decision based on the colourful, vivid description
given by his or her highly respectable person. This is where I, a
Dungeon Master and Hackmaster, provide you with the exact knowledge you –
the player – need to figure out everything you need to know to make
just such choices (all the more to your Dungeon Master’s distress).
Why would you do such a thing?, you’d might ask. Well, the
more you know, the better you can play; the more you understand the
mechanics of the game, the better you can have informed discussions with
your game’s esteemed leader, when you want to contribute and make the
game even better for everyone. And, of course, all such discussions are
done during breaks or between sessions. Metagaming is
disrespectful to players and Dungeon Master alike, who are trying to
create a mood and feel for the story everyone are participating in, and
should be avoided at all costs.
For your convenience, at the end of this article, I will provide you
with a link to a pdf document showing the formulae and numbers for some
normal die pools. The document may be shared freely, as long as it is
done so in it’s entirety. Should you quote me, feel free to let me know.
Nomenclature and the mechanics of different die rolls
Let’s get things straight right away. The correct way to name these polyhedra we love to hate, is one die, many dice.
If you are interested in learning everything you need and do not need
to know about dice, take a look at this excellent document by Kenzer and
Company, called On Dice.
It discusses nomenclature, dice etiquette, the nature of dice (they are
inherently evil, you know), how to choose which dice to purchase, what
to do when good dice go bad, how to roll dice, and so on. Take a look at
it; it’s a good read.
Shorthand for writing what kind of dice a player should roll, is quite simple: m dice, d, of n sides each, for example 2d6, means that a player should roll two six-sided dice and add the numbers. In many games, however, exploding dice
are used. This means you get to keep rerolling the die as long as you
get the maximum number, adding all together. In this article, i will
denote this dice as mdnx, for example 3d6x. Using the
above example, rolling 3, 6 and 5 on these three dice, and getting the
series 6, 6, 6, 6, 4 on the mid die (= 28 – you get the full sum) would
yield a total of 36. Now that is painful.
In Hackmaster, most die rolls are so-called penetrating dice. These are a special case of exploding dice, and are denoted by adding a p
at the end, which means that a player should keep rolling as long as
the player gets the die’s maximum, but deduct one for every roll past
the first. An example notation would be 3d6p; if the player rolls
these three six-sided dice, and got the same numbers as above (3, 6 and
5) on these three dice, and got the same series of numbers with the mid
die as above (6, 6, 6, 6, 4), he would get the following result:
Die 1: 3. Total: 3
Die
2: 6 → penetrating roll, the whole series of rolls being 6, 6, 6, 6, 4)
→ resulting sum of this die is 6 + (6 − 1) + (6 − 1) + (6 − 1) + (4 −
1) = 6 + 5 + 5 + 5 + 3 = 24.
Die 3: 5
Result: 32.
It
is perhaps not as painful as with the regular exploding die, but three
six-sided dice yielding 32 damage still hurts like hell.
The different die rolls
Regular die rolls, dn,
rolls with a regular die, where n is the number of sides of the die
Everyone probably already know the basic formula for calculating the expected result of a standard die roll. If your die has n, numbered 1 through n, your expected result will be (n + 1) ÷ 2. For a six-sided die (from now on: d6), this yields the following: (n + 1) ÷ 2 =
(6 + 1) ÷ 2 =
7 ÷ 2 = 3½
But, wait a minute? Half of 6 is 3, isn’t it? Yes, of course, but half
of a d6 is not the same. Just take a look at the numbers a d6 can yield:
1, 2, 3 [halfpoint] 4, 5, 6
As you can see, there are three numbers to the left of half-point, and
three numbers to the right. What’s midway between 3 and 4? The number 3½
of course. To find the average for any regular n-sided die, add 1 to n and divide the sum by two.
Exploding die rolls, dnx,
where you get to keep rolling and add the numbers, for as long as you roll the maximum
Eric T. Dobbs wrote a great article about these dice a couple of years ago, in which he shows how he came to a formula correctly giving the expected result for an n-sided exploding die. Here’s his conclusion:
For
any N-sided die numbered 1 to N with all sides equally likely, the
exploding modifier will increase the die’s expected value by a factor of
N ÷ (N − 1).
In other words, you get the following formula:
((n + 1) ÷ 2) × (n ÷ (n − 1)) (Eric Dobb’s)
which may be shortened to
(n² + n) ÷ (2n − 2) (by me (any mistakes are mine))
Penetrating die rolls, dnp,
where you get to keep rolling and add the numbers, for as long as
you roll the maximum, but deduct 1 on every roll after the first
Surprisingly, the formula for figuring out the expected value for a
penetrating die roll, is incredibly simple. Now, Mr Dobbs didn’t have
the time to do it mathematically, so he did some number crunching
instead, and ended up with this:
½n + 1
It’s quite beautiful in it’s simplicity, isn’t it? With this simple formula, anyone can do the math in their heads.
The special case of the thief’s backstab ability in Hackmaster
In the game Hackmaster, a thief may make a backstab if using a knife or dagger. This allows him to make damage amounting to 2d4p, but his or her dice penetrate on the maximum and second highest number.
This surprisingly makes the backstab attack preferable to wielding
larger weapons. A thief’s backstab is in fact better than weapons that
do d12p damage, and matches weapons that do 2d6p or 2d12p+1 damage. The
formula for calculating a thief’s special backstab damage, is as
follows, again with courtesy of Mr Dobbs:
(n² + n − 4) ÷ (2n − 4)
Expected damage with the puny dagger is in fact 8 full points of damage; that’ just average damage…
How to abu… use this power
Finally you can figure out how to do the most damage, as
effectively as possible and with mathematical precision! Are you better
off going for the 1d12 og 2d6 damage weapon? Should the Hackmaster thief
opt for the simple dagger, or put his wagers on something more
stalwart, such as a sword? I hope the formulae provided will be
benefitial in your search for the best weapons and tactics. I did the
calculations, so any errors are mine. I also did the shortening of ((n +
1) ÷ 2) × ( n ÷ (n − 1)) to (n² + n) ÷ (2n − 2); if anything was wrong
in how that was done, the error is mine. And here is the document i
promised: Terningstatistikk – Dice Statistics
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