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 Features
Statistically Speaking, It's Probably a Good Game, Part 1: Probability for Game Designers
 P R O B A B I L I T Y
Most games have one or more elements of probability incorporated into their base mechanics. Even chess requires the flipping of a coin to determine who takes white. Usually, we call probabilistic mechanics “random events”. Of course, the term random really might mean “completely random” or “sculpted random.” Regardless, whether you’re talking Texas Hold’em, World of WarcraftTM, or BombermanTM, random events are integrated into key game mechanics.
Probability: It’s not Just a Good Idea...It’s the Law!
You’ve probably heard the term “according to the laws of probability.” The key word in the phrase is “laws.” Probability is all about indisputable facts, not guesses. Ok, technically it’s all Probability Theory, but for the purposes of game design you can compute probabilities absolutely. When you roll a six-sided die, the chance of rolling a “6” is 1/6 = 16.7%—assuming a fair ‘throw’ and a perfectly manufactured die, of course. This 16.7% is not a guess, nor anything of the like. It is as good as fact*. Many of the most common thought errors that people make concerning probability have to do with the belief that probability is not based on laws, but rather on approximations or guidelines. Don’t fall into the traps! I’ll mention a few of the most common ones below, and try to draw some big DANGER! signs around them.
*I guess there might be quantum mechanical concerns that make the 16.7% not exactly fact. I mean, the die could suddenly warp out of existence or maybe your act of looking at it unfairly forces it to collapse its wave function (a severe inconvenience, to be sure).
Independent and Related Events
Let’s start our whirlwind tour of Probability’s Greatest Hits with a key distinction: whether events are independent or related. It’s vital to know before you can start calculating probabilities.
Independent Events: The chance of each event occurring does not depend in any way on what happened in the other event. For example, rolling a six-sided die (event #1) and then rolling it again (event #2) are independent events. The first and second rolls are not related in any way. The number you rolled in event #1 has absolutely zero influence on event #2 (see the “Fallacy of Equipartition”, later). Another example of independent events is drawing a card from a poker deck and then drawing a card from a second, totally different deck.
Related Events: the chance of each event happening is related in some way to the other event. For example, drawing a card from a poker deck (event #1) and then drawing a second card from the same deck (event #2). The chance of drawing a Jack on event #2 is affected by event #1—if you drew a Jack on event #1, then there’s a smaller chance of getting one in event #2 because there are less Jacks remaining in the deck.
Conditional Probability
One of the most useful bits of probability to know is how to calculate the chances of conditional events—that is, events that rely on other events occurring. For example, I used to play lots of old WarhammerTM tabletop games which are d6 based. According to the “to Hit” charts, if you had a somewhat unskilled warrior (with a low Weapon Skill) matched up with a superior enemy, you might have to roll a “6” followed by a “6” in order to hit. Just what is the chance of rolling a “6” followed by another “6”?
Well, first things first, you have to get the first “6” out of the way (a 1/6 chance). Then, you need to roll another “6” (a 1/6 chance again). Whenever one event depends on another’s success, you multiply the chances to get the cumulative chance of both occurring. In this case, it’s a 1/6 x 1/6 = 1/36 chance to roll a “6” followed by a “6”. (Note: If you have an irrational fear of rational numbers—har har—then you can always convert the fractions to decimals by using your calculator. In this case, 1/36 = .028 = 2.8%)
Armed with this newfound power of Conditional Probability, it’s very easy to calculate the chances of crazy dice throws. What are the chances that you can roll four “6”s in a row? The answer is 1/6 x 1/6 x 1/6 x 1/6. Or more simply, (1/6)4 = .0008 = .08%. How about ten “2”s in a row? (1/6)10 = AnIncrediblySmall%.
Incontrovertible Visual Proof that Four “6”s is Possible
Ratcheting up the difficulty, how about the chances of rolling a “3” or above followed by a “5” or above? It’s just 4/6 x 2/6 = 8/36 = 2/9 = 22.2%. Now we’re rockin’ the free world!
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