Custom Dice Worthy of Odin Himself...or Not

Those two squigglies towards the left are actually sea serpents, by the way.

Why have Axes and Double Axes instead of say, Axes and Swords? User interface issues, honestly. Rather than introduce another symbol, I believe it's more thematic and easier for the player to count number of axes, instead of axes plus swords. It's also easier for me to write tight rules. For example, "You succeed on a 2 or more axes" is easily understood and processed, whereas "You succeed on any of the following combos: 2 axes, any number of swords, or any combination of axes and swords" is not as simple.

I arrived at the above face distribution by process of design iteration. Iteration -- you know, trying something and seeing if it works.

Yeah, I hadn't heard of iteration either until I worked on some paper games.

SFX: RIM SHOT

The writer smiles and briefly envisions a successful stand-up career.

MONTAGE: STAND UP COMIC LIFE, ULTIMATELY ENDING IN HUMILIATION.

But I digress.

In this case, the design iteration for arriving at a good face distribution was mainly just math. That is, I did all the calculations below for some different face distributions and then adjusted the faces and the game rules until I liked the results.

Let's take a look at the numbers and mechanics associated with these new fancy custom dice. Probability, per die:

• pBlank = 1/6 = 16.7%

• pMonster = 2/6 = 33.3%

• pAxe = 2/6 = 33.3%

• pDAxe = 1/6 = 16.7%

Armed with this info, let's finally dive into some of the dice-related mechanics in the game. Obviously you don't have a full copy of the rulebook, but I'll try to cover enough detail to give you some context for the rules.

SEA MONSTER CHECK:
Roll both dice. If any MONSTER symbols are shown, then the ship has encountered one.

_{A Sea Monster in a Sea Zone}

Any time a Longship sails through a sea zone with a MONSTER in it, any players without Shamans onboard their ship must make a check to see if a vile sea beast is encountered. Hitting a monster results in loss of a crewman and the player must draw a damage card and apply the damage markers to the ship. I wanted this to happen approximately half of the time. (It's always good to have a design goal before you get mired in the numerical analysis.)

To figure the chance of this happening, it's easiest to use the concept of "converse probability." Essentially, we just have to calculate the chance of a monster not occurring on either die, and then subtract this from unity (100%) to see how likely it is for at least one monster to show.

• pNOMONSTERS = 4/6 * 4/6 = 16/36 = 4/9.

In plain language, there is a 4/6 chance on each die that a monster will not appear. In order to roll both dice and have no monsters, each die must satisfy this requirement. In this case, we multiply the probabilities as shown above. There is a 4/9 chance of rolling no monsters at all.

• pAtLeastOneMonster = 1 - 4/9 = 5/9 = 55.6%

So there is a 55.6% chance of the player hitting a monster. This number is close enough to my design intent for the monsters and shouldn't really require any major balance tweaks.

Since there is a 2/6 chance of rolling a monster on each individual die, it's very easy to make the mistake of trying to calculate the chance of rolling at least one monster on two dice this way:

• pAtLeastOneMonster(WRONG) = 2/6 + 2/6 = 4/6 = 2/3 = 66.7%

To see why this is erroneous, consider rolling 2d6. What are the chances of rolling a 4 or above on at least one die?

• p4orAbove(WRONG) = 3/6 + 3/6 = 6/6 = 100%.

Hmmm, suspicious. Rolling 2 dice, you certainly are not guaranteed to roll 4 or above on at least one die, yet this calculation says you should be. As a general rule of thumb, if you are doing probability calcs that say an event is 100% likely to happen, most of the time something has gone wrong.

It gets even worse if you look at the chance of rolling 2+ on at least one die:

• p2orAbove(WRONG) = 5/6 + 5/6 = 10/6 = 167%.

Danger, Will Robinson! You've invented the equivalent of the perpetual motion machine! (http://en.wikipedia.org/wiki/Perpetual_motion_machine). It's just not valid to add probabilities when you have two independent events going on. Here, each die is an independent event.

You can only add probabilities when you have a single event with multiple success criteria -- for example, checking for a King or an Ace by drawing one card from a deck. In this case,:

• pACEorKING = 4/52 + 4/52 = 8/52

This is a totally valid way of adding probabilities, because you are just adding success criteria for a single event.