Up until this point, I haven’t spent any time on the common tools of table top gaming: dice and cards. For the most part I’ve been avoiding probability on purpose. Work has gone into examining how important game balances along with systems that are impartial. But the tools for how to achieve this have been left to the wayside. One of the reasons for this is the sheer complexity of the material. As I hope to show in this piece, game design is an act that is far from rational and masochistic all the while being somehow rewarding.
The first, and least complicated feature, is linear probability. One rolls a die and the chance of anyone number resulting is one out of however many sides there are on the die. The standard six-sided die gives a 1 in 6 chance. Pretty simple, right? If you are designing board games or other fairly simple random systems, that is all you need to know. Fortune is a harsh mistress and the universe hates you. Okay, that is a bit snarky. The universe does not hate you; it just does not care because you give up all power (and choice) to random, stupid chance.
In a standard deck of playing cards, the chance of any one card coming up is 1 in 52. What about a particular suit, color, or digit? 1 in 2, 1 in 4, and 1 in 13. Great, now what, you ask? What is the chance of drawing a specific hand in a game of five card poker? 1 in 52 x 51 x 50 x 49 x 48 if nobody else gets a card first. And, yes, that is the easy stuff.
Fractions suck for a lot of aspects of ratio in probability, though. Unless you are expressing a very large value, like the chance of being struck by lightning or winning the lottery, you are better off sticking with percentages. Less conversion that way for those of you who have not descended into this madness. Not to mention the ease when multiple systems of probability collide and you are forced to do a lot of math.
Knowing things like how likely a sequence will occur or the chance something will appear by a roll or random draw makes design easier. Remember, chance is not your friend. Just because you have a good idea of how likely a situation will occur in the rules as invoked by probability does not mean you can count on it happening at all. That said, it does tell you how much you have to pay attention to it during play. As a result, knowing percentages of any particular value arising in your randomizer is crucial. Below is a list of percentages of dice and a standard deck of playing cards for any one result to occur, values are approximate.
d2: 50% d3: 33.3% d14: 7.14% d%: 1%
d4: 25% d5: 20% d16: 6.25% 1 card: 1.92%
d6: 16.6% d7: 14.29% d30: 3.33% 1 suit: 25%
d8: 12.5% d12: 8.33% d34: 2.94% 1 number/face: 7.69%
d10: 10% d20: 5% d50: 2% 1 color: 50%
These numbers can be added for games that use ranges rather than discrete values. For example, a four or less on a d6 is 66.7%. Depending on the situation, that represents the chance of success or failure. That said, these are linear mechanics and only reflect single-die or discrete probability systems where multiple rolls are distinct and independent of one another. The use of dice in Risk, for example, use this model.
For other uses of multiple dice, one must employ the bell curve for determination of the most likely, or frequency of, results. To do this, begin with the average of the dice used: (x + y) / 2, where x = the lowest value and y = the highest where each face of the die has a unique, sequential value. The formula does not work for dice with repeating values on a single die, because multiple results with the same weight form their own frequency distribution (bell curve) on the die. This will be covered another time.
Let us stick with some familiar dice rolls to see the frequency distribution in action. The most familiar use of the formula for beginners to familiarize themselves with the problems inherent in probability mechanics is the roll of two six-sided dice. The two most common results are throwing doubles and getting the number 7. This is because both happen one sixth of the time. There are 36 possible outcomes from using a pair of d6s: 1,1; 1,2; 1,3; and so on. One can find the answer by adding up the values for each combination, which may be necessary in the case of doubles (or just look at the numbers of sizing question), or take the average found using the formula from the previous paragraph: 3.5. So for 2d6, the average is 3.5 x 2, or 7. The 3d6 roll for attributes common in Dungeons & Dragons or OGL games has the average of 10.5. This is the most common results for Ability Score generation. Okay, there are a few things point out. I am well aware that the OGL rules use a 4d6, throw out the lowest value rule. That said, the range of likely values has historically been 9-12, or ± 1.5 from the mean. This, for the more mathematically inclined is half of the standard deviation for the 3d6 roll.
A strange thing about that 10.5 value is that it is also the average roll for a d20. Granted, this is a linear expression for each throw the die, but over the course of the campaign (a series of ongoing scenarios for RPG neophytes) a lot of rolls are made using the d20 (this is the core mechanic). Hence, it averages out to 10.5. The bonuses for experience in regards to level advancement moves the average result further afield so that a 10th level fighter, for example, has a mean of 20.5 when making an attack roll. Using the 4d6, drop the lowest skews the mean towards 12, just below the threshold for positive Ability Score adjustments.
Some games use probability in conjunction with target numbers. These represent the breakpoint in a system where the difficulty of an action determines what a player must roll or exceed in order for the action to succeed as plan. One such system that uses this method is BESM Third Edition. The d6 System does this by using a dice pool, which means the action is determined by a collection of dice adjusted up or down in quantity before rolling against the target number. The average adventurer has a 3d6 pool for an attribute and easy tasks have a target value of no more than 10. Difficulty levels scale in increments of five, marking their upper and lower bounds. Depending on the situation and the like, it is possible to roll 50d6. Or, to reduce the number of dice needed, one can just roll a few and take the average for each die not rolled, rounded up. What the system does is essentially give players a 1 in 3 chance per die per difficulty level to succeed in completing a task (e.g. a die is required for each level of difficulty to maintain the 1 in 3 chance).
Let us look at something more complex. White Wolf uses d10 dice pools. They are not as easy to figure out. I spent two days trying to use my own limited knowledge of probability to deduce the formula without any success before giving up and searching for someone else’s answer. I felt a bit dumb when I realized that what I should have done is figure out the chances of failure based on the number of dice thrown. All you need is one die to read 8 or higher to succeed in any given task. That is a 70% chance of failure on each die. When you throw 2d10, though, failure drops to 49% (.7 x .7 = .49). If you try to measure success this way, it looks dismal to roll multiple dice (.3 x .3 = .09). But it does not make sense that way unless you are trying to determine the likelihood of rolling only successes. This proves you can figure out results by their negatives.
What did I do wrong? In this instance, I forgot that each die was mutually exclusive. The system is interested in the number of successes, not the totals on each die. Furthermore, unless the task occurs over time and needs multiple checks for success, it does not matter how many successes are achieved unless they result in what is called “exceptional success.” With five dice, that has the probability of .243%, meaning it probably will not occur. This is the chance if one ignores the 10-again rule where 10s are rerolled for extended chances of success. That basically gives you the chance to roll even more dice. The same is true of rote actions. These are events people are trained to do without thought. As such, any failure is rerolled once.
Unless one is at a casino, the theoretical aspects of all of this do not matter. Most dice are not made to have the tolerance precision casinos require, let alone a surface conducive to protecting that precision. The theoretical is a benchmark, but the dice and a kitchen or coffee table skew the reality of the rolls. So what the designer shoots for is an ideal situation that approximates the outcomes experienced by players. Playtesting is crucial for this reason. The more people who contest the rules, the better the feedback to ensure the math supports the experience and tolerance of the formulae used to create uncertainty.
Consider deck building games. Depending on the number of cards one uses, the frequency of any one event to occur decreases as the number of cards in the deck increases. This is why players include multiple copies of a card in their decks. An important lesson here for any fledgling game designer is the probability mechanic involved. For game like Magic: The Gathering, the limitation on the size of a deck means that players must balance the number of resources required to bring any one card into play with the need for cards that allow the player to act on their turn every turn of the game. The general formula used is one third resource cards and four of each card that allows a player to perform some action. The action cards also have their own frequency distribution with fewer resource intensive cards to further ensure a player has usable cards in his hand.
More complex aspects of probability will be discussed later. For now, it is sufficient to show how basic aspects of cards and dice work for game design and how to use them to create the balance one seeks.
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