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

This article simultanously posted on Canned Blog.

Mr. K.