# [RPG] How to approach the probability of a d100, using 2 d100’s

dicegame-designstatistics

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.

## 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.)