Probability is so pervasive in the RPG hobby that it’s important to have a basic understanding of how it works regardless of where in the hobby you are, designer, GM, or player. Yet in my experience, far too few people regardless of their role, have a grasp on the probabilities at work at the gaming table. I won’t name names, but I’ve seen systems where characters with what’s labeled as a “normal human level of competency” with a skill has a 0% chance to succeed at an “easy” task, GMs place opponents that their players had a 5% chance or worse to effect, and players taking actions with a 20% chance of success where failure meant certain death. You can argue that maybe these things are intentional, but I find it more likely that the individuals in question need a refresher course on how dice work.

For players, the most useful applications of probability is being able to gauge the effects of your actions in the game world given your current numbers and die rolls. GMs should have the same feel for the probabilities of certain actions, but may also occasionally want to better understand the chance their players have to overcome certain challenges or how their tweaks to standard numbers effect their game. Designers should more or less know the probabilities associated with their systems inside and out, which is beyond the scope of a single article for beginners.

**Probability Values and Notation:**

Probabilities are numbers from 0 (no chance) to 1 (absolutely certain). There can never be a probability less than 0 or more than 1. If you calculate one, you’ve done some math wrong somewhere. If you feel the need to convert a probability to a percent, simply multiply it by 100. We don’t use percent values in calculation because it would make extra steps in the math, but they’re easier for many people to understand, so converting your final probabilities makes sense if you’re sharing them. Probabilities are noted P(Event).

**The Sample Space and Events:**

In probability, we define the sample space as a set of all possible outcomes of an experiment. The sample space is denoted as S={outcomes}. The size of a sample space is denoted |S| and has a value equal to the number of outcomes in the sample space.

An event is an outcome or a group of outcomes we’re interested in. Events are commonly referred to by letters and denoted as A={outcomes}. The size of an event is denoted |A| and has a value equal to the number of outcomes in A.

If we were rolling a d6 and we wanted to find the probability of rolling higher than 3, our sample space and event would look like this:

S={1,2,3,4,5,6}

A={4,5,6}

|S|=6 |A|=3

**The Simple Sample Space:**

The simple sample space is a sample space in which all the possible outcomes are considered equally likely. Rolling most single dice is a simple sample space; drawing cards from some kinds of decks is a simple sample space. If an experiment uses a simple sample space, the probability of an event A is equal to |A|/|S|.

Thus, using our earlier example of rolling

higher than3 on a d6,S={1,2,3,4,5,6}

A={4,5,6}

|S|=6

|A|=3

P(A)=|A|/|S|=3/6=50%

Let’s look at a more useful example:

Grog the barbarian is fighting a highly armored foe, and needs to roll a 16 or better to hit him. If Grog rolls a 20, he crits. What’s the chance Grog hits without critting, crits, or does either?

S={all numbers 1 through 20}

|S|=20

H (just hits)={16,17,18,19}

|H|=4

C (crits)={20}

|C|=1

E (either)={16,17,18,19,20}

|E|=5

P(H)=|H|/|S|=4/20=20%

P(C)=1/20=5%

P(E)=5/20=25%So what should Grog do? Should he swing his axe? Run? Drink a healing potion? That depends on the situation Grog is in, but with this info you can make the right choice.

**Counting Methods:**

Let’s consider an example of rolling 3d6 for an ability score. We want to know the probability we’ll roll a 16 or higher.

S={all sequences of 3 numbers 1 through 6}

A={all rolls of 16 or higher}

P(A)=|A|/|S|

Figuring that out could be very daunting task. We could start counting the sample space manually: 111,112,113, etc… but luckily there are counting methods that help us find these numbers quickly and easily. The hard part of these counting methods is knowing when to use which one. Not only are they not always the same, but sometimes you need to calculate the sample space size with one of them, and the event size with a different one.

**Multiplication Rule:**

The multiplication rule states that if an experiment can be broken down into smaller, independent experiments, and that if, regardless of the outcome of the individual experiments, the **number of outcomes in the other experiments, though not necessarily the specific outcomes** remain the same then the total number of outcomes in the complex experiment is the product of the number of outcomes of all the simple experiments.

The sample space of our ability score example can be easily figured out with this counting method. Each d6 has 6 possible outcomes, regardless of what those outcomes are. Thus, the number of outcomes in 3d6 is 6*6*6= 216

**Permutations:**

Permutations are a way to count the number of outcomes in a decreasing pool of options. The number of ways you can draw cards from a deck, or the number of ways you can roll unique numbers in a die pool are permutations.

Permutations are noted Pn,k, read “number of permutations of n objects taken K at a time” and the formula for determining how many there are is n!/(n-k)!

Say hello to the factorial. The factorial, represented by an exclamation mark, is a mathematical symbol used to denote the product of an integer and all preceding positive integers. Thus 6! = 6*5*4*3*2*1 = 720

Permutations don’t come up in dice much because they usually model sampling without replacement (outcomes are removed from the pool of possible outcomes as they occur) which doesn’t apply to dice. They do, however apply to cards a lot.

You have to role 3d6 on the GM’s home brewed critical fumble table. Doubles result in permanent injuries, and triples result in various ignominious deaths and bards writing comedic songs about your ineptitude. What’s the probability you get off easy?

S={all sequences of 3 numbers 1-6}

A={all sequences of 3 unique numbers 1-6}We can use the multiplication rule to determine |S|=6*6*6=216. |A| however, is a permutation. You can roll any of the six numbers on the first die, any of the other 5 on the next, and any of the remaining 4 on the last. |A|=P6,3=6*5*4*3*2*1/3*2*1=120. Thus: P(A)=|A|/|S|=120/216=5/9=approx. 56%

**Arrangements:**

When you want to know how many ways you can arrange more than one type of otherwise indistinguishable elements (like how many ways you can get 6 successes on 9 dice, or how many ways three orcs and two ogres can be stationed at 5 guard posts, that’s an arrangement.

Arrangements are noted where n is the total number of object you’re selecting and n1,n2,n3… are the size of each different group you’re arranging. The sizes of all the sub groups must also sum to the greater number.

The formula to calculate the number of arrangements is n!/(n1!*n2!*n3!…)

Warning!if you’re using arrangements to determine the number of ways you can split things into like-sized indistinguishable groups, you have to then divide the above formula by x! where x is the number of indistinguishable like-sized groups. For example, the number of ways the party of 6 PCs could split into 3 groups of two, is (6!/2!*2!*2!)/3! unless those groups are somehow distinguishable from one another. This situation doesn’t come up much, but when it does, it can throw your probabilities off badly!

As an example, consider the probability of rolling exactly 3 6s with a die pool of 5d6.

S={all sequences of length 5 of the numbers 1-6}

A={all of the above that contain exactly 3 6s)Using the multiplication rule, we know that |S|=6*6*6*6*6=7776.

We can use arrangements and the multiplication rule together to determine the size of |A|. Break A into A1={rolling 3 6s} and A2={rolling 2 not sixes}. We can use the multiplication rule to figure each of these out individually, and then multiply their results together. The multiplication rulealsosays there are 5*5=25 ways to roll 1-5 on two dice and that there are 1*1*1=1 way to roll 6s on three dice. Now we just need to know how many ways we can order those 6s and not 6s. Since we have 5 total objects, and groups of 3 and 2, there are 5!/(3!*2!)=5*4*3*2*1/(3*2*1)*(2*1)=10 ways to arrange those two groups. Thus |A|=25*1*10=250Thus, P(A)=|A|/|S|=250/7776=approx. 3%

**Application:**

To use these tools, all you have to do is decide what your sample space and event are, and count them. It’s important to always count the same type of objects. If you’re counting die pools, for example, counting your sample space in terms of numerical rolls, and your event in terms of successes and failures can result in an error. This doesn’t however, mean that you need to use the same counting methods for both the sample space and the event. As you can see in the examples above. It’s necessary to mix counting methods on occasion. Think carefully about what you’re counting and the methods you use to count them. It’s very easy to over-count complex events, so it’s worth extra time carefully considering which method to use. Finally, it’s important to make sure you remain grounded in simple sample spaces. If every event in your sample space and event aren’t equally probable, than your result won’t be valid. As with many things, practice with these techniques makes perfect, so if you’re not getting reasonable numbers, keep trying.

Wow. When I saw the title of this article, I was prepared for something along the lines of “no matter how “lucky” your d20 is, you still only have a 5% chance to roll a 20″.

I am impressed! I’m going to have to go back over this when I’m more awake, because it’s a little dense, but it’s really nice to see probability explained as thoroughly (and as understandably – I only have to go over it again because it’s late!) as the other GMing stuff you usually discuss. Bravo!

I was doing fine right through the factorial, then…

++?????++ Out of Coffee Error. Redo From Start.Seriously, this is awesome material, and is extremely GM-centric, especially for when the GM tries to come up with a new dice mechanic.

There’s a lot more to probability too, but we’ll see how this goes over before I clog up the stew with another dense post like this one. So if you find it interesting or use it, shout out, because otherwise, parts 2 & 3 may never see light of day.

@Kurt “Telas” Schneider – Oh, and in case others have the same issue, a Factorial is just the product of a number and all integers smaller than it. Take the number, and multiply it by the next smaller number, then the next, then the next, etc… till you get to 1.

So 8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1

4! = 4 * 3 * 2 * 1, etc…

If you divide a factorial by another factorial, all the numbers they have in common cancel out, leaving you with only the unique ones.

So 8!/4! = 8 * 7 * 6 * 5

Factorials are the only thing in this chunk of probability that’s outside of basic algebra, but not because it’s harder than basic algebra, just because there’s few uses for them outside of applied calc.

@Matthew J. Neagley – I was good with factorials; the stuff afterwards was a bit fuzzy, at least until filtered through a second cup of coffee.

So what should Grog do? Should he swing his axe? Run? Drink a healing potion?

Well, if the player to whom Grog belongs is playing an RPG and has been reading this blog for any length of time, he or she should forget the character sheet, the stats and the sums and just decide what *feels* right, then do that and live with the consequences.

If the player is playing Wonkhammer 401K they should argue endlessly about the probabilities of every aspect of the game, especially if they have little or no feel for the maths involved, then – when they get kicked out of their LHS – go home, get onto the internet and do it some more on Yahoo!Groups.

If , on the other hand, the player is trying to figure out what needs to be done to make this situation “realistic” and “fair” in their own homebrewed RPG, they should indeed dig in and get the calculator out.

@Roxysteve – I agree except that in RPGs some players have no idea what kind of chance they have despite the fact that their characters probably would. It’s not so much an “anal retentive evaluate every action” thing for players, IMO it’s more of a “understanding the general feel” thing.

Unless you’re actually designing a gaming system, most of this information isn’t very useful. It’s a nice statistics lesson, but the most math involved is unnecessary. If players know the basic setup of the rules for whatever system they are using, they should have at least a fair guess at their success rate. Unfortunately, since a majority of the games we play use dice, and dice are fickle, you can throw out as many numbers as you want, every roll comes down to a 1:1 probability.

@evil – I disagree, because I do know the probabilities for the rolls that I am requesting. I use the math to adjust the tension. Easy tasks are likely, and difficult tasks are not. There is always a chance, but 1:1? Nope, that isn’t probable.

@Patrick Benson – No, I think the point evil is making is that a qualitative feel (like you get when looking at the spread of die rolls that get you where you want to be and comparing them by figuratively squinting at the whole range using your thumb and an outstretched arm as a sight) is enough to play the game. I agree.

Whether someone understands stats or not they can “get” that a 16 or better on a D20 is only going to happen 4/20ths of the time, and that if they can change the situation such that it becomes 15 or 14 or better that you have a better chance than you did before. In most situations, that’s all they’ll need.

As far as knowing he math, in my opinion it’s not going to achieve what you said in response to evil’s post unless everyone understands the stakes and the stats.

And while I admire the skill and facility that you and Mathew demonstrate with stats, and the obvious thought you’ve both used when applying them to your gaming hobby, the only time I’ve ever seen stats pulled out *in* a game has been the start of something ugly in the minmax department. Actually, that’s been the case out of game too.

One reason I no longer play Wonkhammer 401k is because of all the internet stats-fighting that goes on in that hobby.

I plan on reading Mathew’s article several times, and I thank Mathew for writing it. It’s a model of clear explanation of a sometimes very misunderstood science.

But the whole notion of using stats in a session to “game” your responses does rather fly in the face of the majority of articles decrying what is “wrong with RPGing today” that have appeared in the Stew this year, many suggesting strongly that the GM and players stop rolling dice and start using “cinematic” adjudication instead.

It’s nice to see that GS doesn’t have a “company line” or that if it does, people will go against the doctrine.

@Roxysteve – Again, I disagree.

A simple understanding of odds is fine for some games, but often I do know the exact probability for a success or failure for any dice roll in my regular games. When I run Fudge with my particular rules I refer to these numbers:

http://www.sinisterforces.com/SF_FudgeDice/SF_FudgeDice.html#table3

Refer to something like that often enough and you start to memorize it and can recall the exact odds quickly. So when I say “Roll 4dF and add a Power Die.” in my game I know that the player has a 20.3% chance of a +1 result and a 66.58% chance of a +1 or better. Understanding the science behind the dice has helped me to better understand the risks for the players. I don’t run into “Man, I thought you guys could survive that encounter!” type situations as often as I did before I took the time to learn these things.

BTW – I had no idea what Matt was writing until Saturday when he shared merely a sentence with the rest of us gnomes about his article. He was just trying to ensure that no one else had anything planned for today, so please praise Matt for his facility with the stats. I’m just commenting here. 😉

Yes, the stew does give a lot of advice about the art of being a GM. That is why I think Matt is doing a great thing by sharing this article with everyone, because while so much of GMing is a form of art or a performance of sorts that makes it even more important for GMs to understand these few areas where there are concrete facts to apply to the craft. This is the sort of knowledge that can really help you grow as a GM.

The comment that evil made of “Unless you’re actually designing a gaming system, most of this information isn’t very useful.” is something that I disagree with whole heartedly. It is this kind of information which can be used as a foundation for the more creative aspects of GMing to be developed from. Knowing a little hard math only makes me appreciate the mechanics of a system more, not less, and it also helps me to learn how to better apply those mechanics during my sessions.

Of course you don’t need to do this for your own games, but some GMs will benefit greatly from this lesson.

But how does *your* manipulation of the die stats translate into tension in the players unless they are in on what you are doing to the same degree you are?

I can see you can generate tension in the GM (or more accurately, reading your response, reduce tension induced by “will this kill them?” dithering). But if the players don’t understand what you did to the extent you do, how can they appreciate the stakes? Clearly we differ here, but I don’t understand what you are trying to convey in this example.

And I *WAS* praising Mathew. I did so by name. I assume he will run an eye over every post, and didn’t want to break my train of thought by posting three responses one after the other. Sorry for the confusion.

Perhaps if you expanded evil’s comment to include designing scenarios or encounters you’d see where (I think) he and (for sure) I am coming from.

I guess this is more important in your Fate games than in the games I play, but I thought Fate and Fudge were all about getting away from the “math” of gaming. Isn’t that why they have that daffy ladder of difficulty/skill ability and encourage people to use the adjective rather than the numeric modifier (an idea I find, at the core, counter-productive I should add)?

Don’t get me wrong. I think this article should be read by all gamers and at least the first four or five points absorbed until the reader understands what is going on. I just don’t agree that the resulting facility with stats should be the first recourse of a gamer making an in-game decision.

Either way, this was one of the more thought-provoking threads in recent Gnome Stewing. Thanks to you, Patrick, and everyone else for making the old neurons spark.

@Matthew J. Neagley – agree except that in RPGs some players have no idea what kind of chance they have despite the fact that their characters probably would.

I don’t think I agree with this statement, Mathew. I think that a qualitative assessment more properly represents what a character with experience of a given action will use in deciding whether or not to act that way again. The players get to make that assessment by looking at the raw numbers (“roll vs DC 15” – “I don’t think my chances are good”)

However, the article was an eye-opener in some ways for me and I’m thinking about it still, so well done.

But I respect your right to do it your way too.

Gah. That last sentence should have been before the line starting “However”. Drat my 1 rank in Cut-and-Paste editing.

I don’t think anyone worth their dice has a problem gauging a DC15 with a d20, but many games don’t have such a linear approach.

Dice-pool games often have an unintuitive probability curve, especially if there’s a ‘bad number’ as well as a ‘good number’. IIRC, some World of Darkness games actually had a worse chance of success if you add dice past a certain point. This is something I’d like to know, even if I was going cinematic with it.

My own beloved Savage Worlds has some odd quirks, such as how a d6 has a better chance of hitting an 8 than a d8 has. It’s a very small chance, but since the game is played with dice, knowing the probabilities is an important aspect of designing, modifying, and playing the game.

Frankly, if someone at WotC had read this before designing the first draft of Skill Challenges, they wouldn’t have had to reduce the target numbers by five points right out of the gate…

Long story short: Yes, probabilities are important.

@Roxysteve – Well I don’t manipulate the die stats. Because I took the time to research what the probabilities are I know what the stats are. No manipulation is necessary.

And that helps me to generate tension amongst the players because when I know what the actual probability for success is I can use my tone, body language, a look even to convey either the truth or I can deceive them into believing that things are worse than they actually are. It works great in horror games where deception is a common tactic used in the genre.

I know that you praised Matthew. But you wrote:

“And while I admire the skill and facility that you and Mathew demonstrate with stats…”

I was simply pointing out to anyone who might mistake that to mean that Matthew I and collaborated in some way that the work was all on Matthew’s part. Sorry if you took it to mean something else.

I believe that I do understand where you and evil are coming from, and I disagree with you. That was the reason for my response. I am not trying to reach consensus. I am saying that I believe your point of view on this matter is wrong. I do not expect you to agree with me. I am arguing with you, and I am trying to do so in a respectful manner. Again, if this offends you I assure you that is not my intention. This is a debate, and the audience will ultimately decide for themselves who is right.

I don’t play FATE even though that is a Fudge derivative. I have designed an play my own Fudge derivative. The reason for the Ranks Ladder is so that players can use adjectives to tell the story with. Believe it or not the Fudge rules from 1995 onward have always included the math behind the dice so that people would understand the mechanics of the ladder better. When you say that your character is a Great fighter well the dice rolls tend to favor the high point of the bell curve. That high point is a flat result so that your Great fighter achieves Great or better result the majority of the time, just like you intended the character to at the time of creation. Steffan O’Sullivan (the primary author of Fudge) understood that the math was important to using the mechanics of the game when running Fudge. I do not see that as counter productive at all. I see it as good design.

No one is suggesting that the math behind a dice roll is the first thing that a GM shoudl turn to in making a decision. I said that it helps you to be a better GM. Knowing how to read your player’s body language is important to making decisions as well. Knowing both gives you an advantage that you would not have if you only knew one or the other.

I read some of the comments as simply dismissing what I see as great advice, and when I disagree with something I say so and give my reasons for it. That is why I am defending what Matthew has written here, and while I respect that you do not see the same degree of benefit to this article that I am offering counterpoints to your comments because I want people who are on the fence to agree with me. Not because I desire that, but because I believe that things of worth must be defended when challenged. This article has worth, and I hope that you respect my right to defend it when others challenge it.

@Kurt “Telas” Schneider – Your point about 4e Skill Challenges is spot on. It is the first thing that I thought of when I finished reading this article.

When I created my rules system, I followed probability exactly. A new character has a 100% chance of success for easy tasks, 90% for average, 70% for difficult, and 40% for hard. I never explained to the play testers about the probabilities, but, as they became used to the system, they knew they were able to do more things and were willing to try complex actions. They also developed the feel that “the dice weren’t trying to get them” and accepted failures a lot better.

As a player, I’ve always tended to weigh my chances on a given action. This takes some of the drama away, but, I just can’t help it.

The article was enjoyable and generated a lot of thought. I, for one, would appreciate more along this line.

I’ll agree with the sentiment that was expressed that some systems are definitely easier to nail down a general feel for than others. Similarly, some systems are harder to crunch the actual numbers for as well. White Wolf’s exploding die dice pool system for example is particularly nasty and you wouldn’t be able to do the numbers with just the tools above. It requires infinite sums and knowledge of geometric power series which is probably beyond the scope of anything I’ll ever write for the stew.

This article would have helped me ohh, so much 15 years ago in my statistics class! Not that it’s not helpful now, mind you! But back then it might have helped me in terms of my final grade!

Personally, I find this information useful gaming-wise because I’m a bit of an amateur tinkerer with game mechanics. Any information I can get that will help me NOT do something that will blow up in my face is good.

Many thanks!

I’d like to see more of these, specifically talking about game design, and what the optimal probabilities should be for actions in dramatic, realistic, social, etc situations.

@Patrick Benson – I’m several responses down the line now, but I’d like to fix up something that might have been misunderstood. Earlier I said that the base probability for any outcome is 1:1. This is, in fact, true. At the basest level, any attempt has two outcomes: it works or it doesn’t. Hence, 1:1. While knowing probability is fine, I’ve seen too many rolls on the far end of the normal curve to base anything on a statistical basis. (ex. In a recent warhammer 40k game, an opponent threw 30 dice against me in an assault. 25 came up 1, 3 came up 2, and 2 came up 5. This was an unlikely result at best, but in the next turn a very similar roll happened.) My original point is that just because statistical information says something should happen, it often does not.

Then again, I live in a world where the weird happens all the time, so the above may just apply to me.

@evil – I understand what you meant now, but probability is not a statistic. Those are actually two very different things.

A statistic is a sample that we use to evaluate the whole. We poll a certain percentage of the population to determine how the whole will vote or what television show the whole will watch. Statistics are only as good as the sample from which they are pulled.

Probability applies to the moment. What the previous and the next dice roll were and will be are irrelevant to what the dice roll is at the present moment. Two rolls with a very low probability occurring back to back is odd, but one has no influence over the other.

So do you understand why I disagree with what you are saying about this information not being useful? If you are suggesting that statistics are not useful I am more likely to agree with you, but probability is very useful.

@Kurt “Telas” Schneider – Hmm, OK. Don’t see intuitively how chances get worse with a bigger dice pool but I’ll take your word for it. Must be a Bell/Poisson curve thing.

As for Savage Worlds, I’d say that’s a prime example of why concentrating on stats can blind you to the facts. Yes, there’s a better chance of hitting an 8 with a d6 (there’s actually *no* chance of hitting an eight on a D8 if you think about it – grinning big – ) but that fact blinds people to the more important one – there’s a bigger chance of a flub with a D6 than a D8. In SW you’ll get a success – 4+ – on a D6 1/2 the time. With a D8 that becomes 3/8ths (call it 50% and 36% respectively without running the math).

SW tends to suffer from what I call the Randi Effect. People remember the endless strings of aces some rolls get more than they do the repeated misses.

There was an argument raging in the web-presence of my LHS recently over why it’s better to leave skills at D4 based on the 25% chance of acing on a D4. No-one mentioned the fact that 75% of the rolls were going to be big, nasty flubs (not accounting for the wild die of course).

Thanks to Mathew I now can see that that same D4 will fumble 1/24th, about 4% of the time. Boost to D6 and that goes down to about 3%. Why that is more easy and intuitive than 1/24th and 1/36th isn’t obvious to me but again, I’ll take your word for it.

@Patrick Benson – I’m not offended, Patrick. I hope you aren’t either. I’m criticizing the premise, not the man standing behind it.

You manipulate the stats by adjusting the scores/dice pool size (to adjust in-game tension). Isn’t that what you were saying? That’s how it reads to me. Perhaps it would be better to say you slide the particular game facet around in the stat space until it is where you want it. A matter of viewpoint. Stand on the table of stats and the condition cursor is moving overhead. Stand on the encounter transparency and the stats are whizzing by underfoot.

I’m afraid your arguments all seem to me to be making my “designer-rather-than-gamer-tool” point. I must be missing something, somewhere. We can let it rest. I’m guessing it’s getting old for everyone.

@Roxysteve – I am fine with leaving this to rest as well.

To clarify there is no manipulation of the probabilities. There are no stats involved here, so I don’t understand what you are referring to there. I simply judge the target rank needed, rationalize if there are any bonuses or penalties to be applied according to the rules, and once those are calculated I know what the probability is for a success.

Now if you are saying that I am manipulating the game by adding bonuses and penalties and by determining the rank needed I can understand your point, but that does not manipulate the probability of the outcome of the dice roll that is decided upon. Manipulating the probability means that I would adjust the dice roll after it is made to guarantee a result.

Regardless, I am fine with leaving this where it is.

@Roxysteve – The “worse with a bigger die pool” thing comes from WWs “botch” system, where if you rolled a 1 on a die, you removed a die that would have otherwise been a success. Thus you often saw rolls of huge die pools where you got a few ones and completely ate up all your successes. Add to that the fact that some of the older systems required you to roll higher than a 10 on a d10 by rolling a 10 and then rolling again over a certain target number, and on a DC 11+ roll, you probability of rolling a botch and removing a success (.1) was greater than your chance of rolling a 10 then rolling a additional success (.01 to .09), meaning that the best chance you were ever going to have on a DC11+ roll was with a single die. At least that’s the intuitive answer. I’ve never crunched the numbers to see if it’s actually the case.

If that was for beginners, I’m a mewling babe. Brain not work no gud now.

That said, it’s interesting stuff. Do you really consider all of those probabilities (roughly) as you’re playing and as rolls come up?

@Rafe – I generally crunch some of the low hanging numbers in any system I play. I can’t really help it. Of course, there are shortcuts to some of these probabilities that makes that process easier. I use the rules or tables I come up with to guide me as I play or design encounters as a GM. I’ve never designed a system, but if I were I’d make an effort to understand it inside and out. Otherwise, I might deliver a sub-par product.

To number-grind or not to number-grind? I think a little bit is okay, and two much is a nuisance.

I feel it’s acceptable when:

You’re checking to make sure your monster is in fact killable.

You’re designing a character on a premise like “He’s able to cut through orcs with ease” (Hmm, looks like I need more strength or a bigger axe to bump myself to 90% chance of a one-hit kill…).

You’re reading the rules and trying to figure how things work before you play.

In other words, number-crunch as much as seems appropriate to you ON YOUR OWN TIME. And from a gaming perspective, I think instances like those above are legitimate (you can’t earn your spurs as a nerd until you’ve done at least a LITTLE mechanics-related obsessing). If a player is number-crunching at the table, they’re slowing everyone down for a meta advantage. That’s not acceptable, even if you carry the two. If I ever saw this happening, I’d want to ban calculators from the table.

I agree, however, that “intuitive” number-crunching is best at the table. As well as the practical reasons, it’s what we use in most games. Some will object that it’s not really accurate, and sometimes you want accuracy. Pish to that I say. Does a baseball player need to remember a set of physics equations to throw a ball where she wants it to go? Does an acrobat need to know their mass and velocity to do a backflip? Obviously, these are vastly more complicated systems than a handful of dice, in terms of what PRECISELY accurate equations would look like. Dice provide an artificial restriction to probabilities which a real swordfight wouldn’t have (again, you could do the physics equations for a real swordfight, but not without abstracting away a lot of the messy real world). But my point is that your brain is an immensely powerful calculator which, after enough practice, is capable of INCREDIBLY accurate calculations, assuming you know the factors at stake. I find, when I’m gaming, that my brain switches rapidly between in-game intuition and meta-game quantification.

An armoured ogre with a gemmed sword?

Well I think of his size. Intuitively, I know that makes him easier to hit, and a voice in my head says “Yep, in the rules. -1 AC.” (I think?)

I think of his armour. Kind of hard. I think of his sword. Maybe a more skilled attacker? Better be extra-careful.

So when I finally decide to run away, parts of it are due to my “theoretical” knowledge, but I haven’t done the maths. I’m just aware that there are quantities, “Estimations” in Sun Tzu’s terms, that affect how likely I am to succeed. Unless it’s a really new character, I know roughly what my capabilities are, I know how I’ve performed in past fights against comparable opponents.

People’s “common sense” sense of what is likely and probable is very often wrong. It’s what drives superstition and gambling.

I realize this reply is a month late .. just saying.