# How do I approach the probability of a d100, using 2 d100's?

We're running an almost 100% homebrew system, which uses skills in order to craft objects during play. Allow me to first formulate an example to explain my question better:

Jack is quite the mason and has his Stonecarving at skill level 50. He wants to carve a big stack of stones for his house. He rolls a d100, and must roll equal to or lower than his Stonecarving skill level. This quite obviously gives us a 50% chance of Jack succeeding. No problem here.

Now Jack has also been brushing up his artistic skills and has his Artistry skill also at level 50. Jack wants to carve an artistic statue out of stone. He rolls 2d100, since making the statue would arguably use both the Artistry and the Stonecarving skill. This is where the problem arises.

How do I make this 2d100 roll equal to the d100 Jack rolled earlier for the stone? If I say "you need to roll equal to or below your skill level for both rolls", it gives Jack (theoretically) a 25% chance of succeeding. If I say "you need to roll equal to or below your skill level for one of the rolls", we get a 75% chance of Jack succeeding.

Ultimately, what I am looking for: How do I arrange a 2d100 roll, in order to make it equally fair as a d100 roll?

To further clarify, how do I arrange the dice or what mathematical process should I follow in order to attain the probability of a d100, when rolling 2 d100s? In case of the example, I am looking to have the 50% probability of success, in the paragraph with the two skills being used.

In the case where Jack were to have differing skill levels, I am looking for a success rate based on the average of the skills. 30 Stonecarving and 60 Artistry should then result in a 45% chance of success.

• @D-zap Why are you having people roll a second dice? You're then asking us how to negate its existence, so the situation is quite bizarre. What then is the point of the second die? This is a real serious question, given the simplest answer to this question - and the easiest thing on the players - is simply "don't do that thing you are doing, and just roll 1d100." I hope you understand where I'm coming from here - I understand you want a second dice rolled, but given you want it to have no effect, I do not understand why. Oct 20 '14 at 8:51
• He answered his own question. He wants to average the skills. "In the case where Jack were to have differing skill levels, I am looking for a success rate based on the average of the skills. 30 Stonecarving and 60 Artistry should then result in a 45% chance of success." (Skill A + Skill B) / 2 = your % chance (round up or down as you wish, I recommend down). Alternatively, your 2d100 roll must be <= Skill A + Skill B. Seems overly trivial to me? He has other options, but this isn't really a question any more. Oct 20 '14 at 10:07
• @doppelgreener In retrospect, it was an error on my side for unnecessarily insisting on two dice, while some of the answers below have pointed out that the issue above can be resolved by simply omitting one of the two dice. In short, I never truly needed two dice. Do I rewrite the question or do I flag for removal? I feel the answers, including the one I accepted, have ironically invalidated my initial question. Oct 20 '14 at 13:22
• @D-zap Since this question is all about how to get 2d100 to equal 1d100, if you're intending to drop that like it sounds like, you'd probably then be asking an entirely different question - so that ought to be asked separately, and will benefit from not already having these answers on it. Oct 20 '14 at 13:32
• @D-zap Ah. Well if you already know the answer, there's probably no need to ask another question. :) Oct 21 '14 at 0:19

## Average The Skills

If he has to use two skills, average the two skills together and then make one roll. In this case, that'd be a single roll to get 50 or below, since he has 50 in both skills (so the average is 50).

If he was better at one skill than another, it'd look slightly different. Say he has a 50 in Stonecarving and 25 in Artistry. That makes the average of them 37.5, so he'd have to get a 37 or below (or a 38, depending on how you want to round).

That basically treats it like he's using both skills and has to succeed on using them in combination, rather than having to succeed on separate rolls for both. It also keeps it to a single roll with similar odds, and is relatively simple to implement for players.

## Alternative - Geometric Mean

The downside to averages is that if you're really good at one skill (say 100 in Stonecarving) and really bad at the other (0 in Artistry), you still have a 50 in the combined skill. That might not be what you had in mind, as someone with no artistic talent doesn't suddenly gain it just because they are working with stone.

In this case, an alternative method is to take the Geometric Mean. For two skills, that is this formula:

$$\sqrt{skill_1 \cdot skill_2}$$

So, if you have 100 in Stonecarving and 0 in Artistry, you do $$\100 \cdot 0\$$, which is 0. The square root of that is 0. As a result, you now need to at least have 1 skill point in Artistry in order to attempt the combined result. If you did have Artistry 1, you'd get $$\100 \cdot 1 = 100\$$, the square root of which is 10. As you add points in Artistry, your chances will quickly increase.

For my previous example of 50 and 25, you'd get $$\50 \cdot 25 = 1250\$$, the square root of which is 35.3.

The main downside to this method is that in a tabletop game, it's extremely hard to calculate without a calculator. Even with one, it requires a more complicated understanding of math and is more time consuming. If you put this in a rule book, there will be people who won't understand what you want them to do. For something like a video game where it's calculated by the software, that isn't a problem.

(Thanks to Peteris and Vatine for the suggestion!)

## Alternative - Minimum/Maximum

A very simple method for combining skills is to use either the minimum skill in the two of them, or the maximum skill in the two of them. The maximum means you're just using the skill you're better at, while the minimum means you're using the skill you're worse at.

In the case of the minimum, it simulates the idea that you have to succeed on what you're weaker at in order to accomplish the goal. This lets you do it in a single roll, and is very easy to understand. It also has some issues, in that if you're extremely good at Stonecarving and so so at Artistry, your Stonecarving gets ignored in this system as you only roll on your lower one (Artistry).

Because of that, I don't think it really accomplishes what you intend very well, but it's ease of use is a significant upside over the other suggestions.

(Thanks to Neil Slater and Ellesedil for suggesting.)

• A twist on "Average the skills" would be to roll the two dice, and require the sum of the rolls to be less than the sum of the skill values. This gives you the same average chance of success, but makes it a bell-shaped success distribution, which means that as you combine more skills, it becomes harder to fail at tasks that are (on average) easy, but also harder to succeed at tasks that are (on average) difficult. Oct 21 '14 at 20:33

You don't. You just roll one d100.

As you understand, rolling multiple dice is a useful tool for achieving different result spreads. But rolling multiple dice is a tool with a time and place for when you want various advantages: you can take highest or lowest, you can create a bell curve effect, or do other interesting things.

However, you're not interested in any of the effects of rolling multiple dice. All you want is the result spread of 1d100.

If you want the result spread of one d100, just roll one d100.

Otherwise, you're creating extra effort for no benefit: you're making people roll a second dice, then engage in some method to negate the point of its involvement.

The results you're describing, after all, don't need a second dice:

• In case of the example (Stonecarving 50, Artistry 50), I am looking to have the 50% probability of success, in the paragraph with the two skills being used.

• In the case where Jack were to have differing skill levels, I am looking for a success rate based on the average of the skills. 30 Stonecarving and 60 Artistry should then result in a 45% chance of success.

You have your mechanism for modifying difficulty, and it sounds fine. But there's no reason for a second dice getting involved in the success roll — this is a tool that has no place here. If you had a perfectly servicable hammer available you probably wouldn't ask us how to bash in a nail with a screwdriver; this is kinda the same situation where you should just leave the second dice out of it.

I imagine you might have some reason for getting both those dice involved. Maybe on a result of >90 on one of them, the corresponding skill gets trained, or something special happens, or something like that. But you're clearly not interested in this affecting the probability spread of the results, so it's not appropriate to involve a second dice. I suggest if you're interested in something like this, keep it separate from the success roll. Have them roll one d100 to determine success, then another d100 for each skill involved to determine if special things happen.

• Rather than targeting the average, you might consider targeting the lower skill, with maybe a slight bonus for the delta value of the higher skill. For example, with Artistry 30% and Stonecarving 50%, you might give +4% for having 20% better Stonecarving, and then the difficulty for the roll is a 34%. In this way, having one skill high doesn't negate the difficulty of having the other skill low entirely, while still making having a high skill matter. (Or use teamwork - let someone else make the roll the player is bad at with their good skill...) Oct 21 '14 at 0:09
• Alternate strategy: roll 2d100 and ignore the die that lands most to your left. Oct 21 '14 at 17:16

If you have a task that requires more skill, you would expect less success rate.

Roll twice, first roll decides if the statue looks like what it's meant to, second roll determines if it falls apart.

If rolling two dice is interfering with fun by bogging things down, just roll once against the lowest skill.

Using max or average is likely to lead to some odd situations, where someone can be so good at basketweaving that they are able to build a castle. This will require a lot of 'once off' thought from the game master to make it work.

As an example, my driving skill would be 99.9%. My shooting skill might be 5%. My chance of shooting a target whilst I am driving a car? It's going to be less than 5%, not 52.5%.

In the same way that 2 d10s can replicate a d100 if you use decimal positioning, you could use a d2 (i.e. "coin") and a d100. d2-1 gives you the 100s place, d100 gives you the next two digits.

Voilà: a deeply dissatisfying but perfectly uniform generator of numbers from 1 to 200. You can generalize this approach to generate uniform distributions with any range you want by selecting the proper combination of dice and rules to allow you to select each digit individually.

That said, you are probably better off with Tridus's answer of approaching this from the skills end rather than the random number generation.

• I double-taked initially at rolling 1d2−1. Oct 20 '14 at 5:20
• Oddly, I didn't. I have a druid that does 1d4-3 damage with most of his phsyical weapons :/ Oct 21 '14 at 19:40

## What does success mean to you?

I think the most important question here is how you are defining success. Jack wants to carve an artistic statue, so you need to figure out what successfully doing that means. I think that you're original idea of rolling a d100 for each skill check would work just fine. The only change I would make is in how you are looking at the problem. Instead of saying Jack has a 25% chance at success and a 75% chance of failing, I would add in some degrees of failure. There are three scenarios you should consider. Jack succeeds on both skill rolls, fails on both, or fails on one and succeeds on the other. If you want to get very specific you could treat both combinations of passed and failed rolls separately to get four scenarios. But lets ignore that for now and say there are three.

Critical Success: Jack passes both skill checks, and has successfully carved his statute. Good job! Chance of success = 25%

Moderate Success: Jack has failed one of his skill checks. The end result is not quite what he wanted, but it is still passable. Nice try! Chance of success = 50%

Critical Failure: Jack has failed both of his skill checks. His statute breaks, looks terrible, or is otherwise too screwed up to use. Too bad! Chance of failure = 25%

So going off of these scenarios, half the time Jack tries to carve a statue it will come out alright but not great, a quarter of the time it will come out exactly how he wanted, and a quarter of the time it ends up a hot mess. If Jack tries to make a complex skill check like this and one of his skills is much lower than the other, he still has a chance of getting at least a Moderate Success.

This method should let you keep the skill check mechanics relatively easy to understand, since it doesn't require any truly complicated math, and also give you a good idea of how to describe the different degrees of success or failure. Failing the Artistry check might mean the statue isn't as pretty as it could have been, while failing the Stonecarving check might mean the statue was carved a little rougher than it could have been.

The simplest way to do this would be to add the two skills together (50+50 = 100) and then compared that to a roll of a larger die (1d200). This maintains the 50% chance, despite making it a rounded curve instead of a flat curve. It also allows for character with different skill levels to perform something difficult and maintains simplicity. A character with 60 skill in one and 40 skill in the other would also have a 50% chance, but no division is necessary as it would be in getting a mean or average..

A character with a 50 in one skill but only a 10 in another would have a tough time 30% chance of getting it right. While a 50 and a 70 would succeed 60% of the time.

Skill 1 vs 1d100
Skill 1 + Skill 2 vs 1d200
Skill 1 + Skill 2 + Skill 3 vs 1d300


All of these are possible without breaking out the graphing calculators or even using anything but the standard dice used for a most RPGs. Standard d100 uses 2d10 with one designated as the tens place and one as the ones place. Just use a d20 for the tens place instead to create a d200 (Note that a 20 = 0). A d300 is a little trickier. Roll 2d10 and 1d3. Add the (result of the d3 - 1) * 100 to the result of the 2d10 to get a 1d300.

EDIT:Major changes made to fix the massive statistical errors in my previous answer.

• Now imagine the two skills are 10 and 10. With your system the user now has a 1% chance of success, rather than 10%, due to the rounded curve. Oct 21 '14 at 19:44
• This has a compounding effect on trying to pass high skill checks due to the bell curve. See this anydice example considering the example of 3 skills at 80 each, with a combined score of 240. With 3d100, there's only a 3.78% chance you'll score above that score. With 1d300, there's a 20% chance, the same chance as 1d100 with one of those skills at 80. The odds are equally distorted at the other end (where three skills are at 20, you're extraordinarily unlikely to roll beneath them). The bell curve makes this not seem like a good method. Oct 22 '14 at 7:42
• @MooingDuck Fixed to have flat curve. Oct 22 '14 at 13:15
• @Red_Shadow much better! Oct 22 '14 at 13:26

## Roll a single d100

and compare the single result to both skills.

• If it fails both, then it is a total failure
• If it fails masonry but passes artistry, it is a beautiful but fragile sculpture
• If it fails artistry but passes masonry, it is a robust but ugly block
• If it passes both, then it is a total success
• Welcome to rpg.se! Please take a look at the tour and the help center; they're a useful introduction to the site. Can you please expand your answer to explain the general rule this would follow? In particular I'm not sure how you intend it to work for tasks where both skills are necessary for even basic success.
– BESW
Oct 21 '14 at 13:24
• @BESW: If the single roll beats the higher skill but doesn't beat the lower skill, then the obvious final result is likely failure. Oct 21 '14 at 13:54
• Edit it into the post! Comments are ephemeral: their primary purpose is to help figure out how questions and answers can get edited to be better.
– BESW
Oct 21 '14 at 13:56
• @BESW, he's home-brewed, so what general ruleset would I reference? I was simply trying to come up with something simple and useful. Oct 21 '14 at 19:11
• @Ellesedil, not necessarily. If he beat his stonemasonry but failed his artistry the block would be ugly but functional. The other way around it looks great but crumbles when he sets it in place. Oct 21 '14 at 19:13

I would use this idea: You use just one die (and one skill - player choice which one, he would probably choose the higher skill no doubt) then the second skill I'd devise a use for as a "support influence". So if you got Stone Cutting 50 and Artistry 35 then let's say for each full tens in the support skill (here, Artistry) you get +5 to the roll. So in this instance you'd get 50+15 (as the last 5 points of Artistry doesn't convey any support) = 65% chance. That way the secondary "support skill" is useful anyway. But I'd set the limit of support skills to just one. Or, it could be another character doing this "supporting" with his skill.

I would suggest perhaps having one roll per skill, but make note of the amount by which the roll either exceeds or falls short of the requirement. Tally up the first five points above what was required, and half of all points beyond, for each roll that exceeded requirements; then tally up the first five points below what was required and double all points beyond, for each roll that fell short. If the first value was at least as great as the second, the action succeeds. One may need to tweak the thresholds and scaling ratios to see what values make the game enjoyable.

For example, if someone has a 60 skill at woodcutting (a roll of 40 or above counts as "success"), a 30 skill at clay-forming (roll 70), and a 65 skill at painting (roll 35), then they would role one die for the result of each skill. If the dice come up (42, 71, 38) then all rolls would be above their requirements so the action would succeed. If they'd been (42, 71, 33) then the last roll would be two short, but the first two rolls would have had a total of three extra points available to make up the shortfall so the action would still succeed. If (63, 71, 25) the first roll would have 23 extra points, but only the first five would count full value and the other 18 would count half (for a total of 5+9=14). The second roll would have one extra point. The third roll would be short 10; of that shortfall, all but the first five units would count double (yielding 15). So the excess would barely cover the shortfall.

Another approach that might be combined with this is to have the consequences for failure vary depending upon the skill for which an insufficient value was rolled and the amount of the shortfall (if some die-points are transferable between skills, allow the player to distribute them for the least undesirable consequences). If two skills, but with a 50% success factor, are required to perform some task, a player would only have a 25% chance of getting through without any penalty, but would also have only a 25% chance of getting two penalties and no success.

• Your first paragraph could use an example to illustrate the method you're suggesting. Oct 21 '14 at 22:28
• @Miniman: Is that better? If excess points or shortfall could freely transfer, it wouldn't matter what die roll was associated with which ability; I think it's better to have die rolls "mean something", though the exact formulations could probably use adjustment. Oct 21 '14 at 22:42
• Yep, that makes more sense. Oct 21 '14 at 22:47