Analysis of roll values (New difficulty chart inside)

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Rawle Nyanzi
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Analysis of roll values (New difficulty chart inside)

Post by Rawle Nyanzi »

In my quest to be a better Game Master, one thing I give a lot of thought to is difficulty. Unsatisfied with selecting difficulty numbers arbitrarily, I devised a way to choose difficulty that brought some of that arbitrariness under control: using playing cards to indicate how many dice I should roll, then using the result of those dice as the difficulty number. But as I did this in a recent game, I noticed that certain numbers occurred far more often than others. I wondered if my method was, in effect, selecting a difficulty number in a more roundabout way.

Looking for answers, I fired up my computer and got to work. I would measure how often specific numbers came up whenever I rolled certain numbers of dice. The Law of Large Numbers would ensure that whatever results I got would be roughly the same if someone else repeated this investigation; to this end, I rolled two through twenty-four dice 50,000 times each and recorded how often each value came up.

Here is a sample of the results:
Image

Looking at this data alone, we see one of the earliest things I noticed: the most frequently occurring value is usually a multiple of six. This is because six is the largest value the dice can give you, meaning that it takes very few sixes to exceed the other matched sets.

When we look at larger numbers of dice, we see another phenomenon as well:
Image

These charts exclude values that didn't appear in the results even if it was mathematically possible for that value to appear. But note which type of value is extremely unlikely to appear: a prime number. Prime numbers have only two factors -- itself and 1. In OVA, this means that the only way to get a prime number result that isn't already on the dice (2, 3, and 5) is to get a result where a set of ones decides the roll. Ones are the lowest number the dice can give you, so it takes an awful lot of ones to exceed the other matched sets. Even if enough ones appeared to make a prime number, it would likely get exceeded by sets of the other numbers and thus won't decide the roll's value. Furthermore, a prime number not already on the dice and larger than the number of dice being rolled cannot appear at all.

Another interesting case is semiprimes, or numbers with exactly two prime factors. Excluding six and nine, semiprimes in which one of the factors is either two or three are very uncommon, since two and three not only need the correct number of occurrences in a roll, but also the largest matched set (semiprimes with five, on the other hand, occur frequently since five's larger size allows it to decide the roll more easily.)

Which brings me to my next point. As I looked at the frequencies, I noticed a very large flaw in my old difficulty chart: it used rarely-occurring values as difficulty benchmarks.

The values 14 and 22 stand out. Both are semiprimes, with 14 factoring out to 2*7 and 22 factoring out to 2*11. Not only must these values be arrived at with twos, but the set of twos must exceed all the other sets. As a result, 14 occurs very infrequently and 22 almost never appears at all. This explains why the probabilities for 22 and 24 are nearly identical -- beating 22 is not easier than beating 24 since 22 is so rare that using that value won't improve your chances above beating 24. When rolling less than eleven dice, getting 22 isn't even mathematically possible, so the chances of beating 22 and 24 are absolutely identical when rolling four through ten dice.

Noticing how uncommon some of the benchmarks were, I came up with new difficulty numbers that reflect far more commonly occurring values (all percentages are rounded to the nearest whole number.)
Image

With the exception of 9 and 35, the values that occur between the difficulty numbers are very uncommon; you can estimate the probability of difficulties in between by checking the probability of the larger neighbor (for example, if a 21 turns up, you can estimate its probability by checking the chances of beating 24.) If you are simply declaring difficulties per the rules, these are my recommended numbers to use. You'll also find that these numbers turn up a lot when you roll the dice.

Dissecting OVA's dice-rolling mechanics has helped me become a better Game Master, since it allowed me to see probabilities much more clearly and take the right actions at the table to keep the game interesting. Below, I have provided a link to the Excel spreadsheet where I recorded the frequencies as well as the Python source code of the program I used to find them and derive the new difficulty chart. I've also included the spreadsheet for the difficulty charts. My hope is that my work will also help other OVA Game Masters.

Roll Frequency Data
Difficulty and opposed roll charts
Python source code of program that was used to make all of the tables
Joe_Mello
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Re: Analysis of roll values (New difficulty chart inside)

Post by Joe_Mello »

I find this interesting, but lacking in utility. If a player is rolling more than 10 dice, I have to think something has gone incredibly wrong in your game.

However, it is interesting how good 6 is as a target number. If you want to favor characters with a certain ability without punishing those that don't, six seems a reasonable bar to clear.
Joe_Mello: Could you make a common sense roll, please, Ryu?
Ryushikaze: With Smart?
Joe_Mello: Sure
*Ryushikaze rolls*
Joe_Mello: SHE'S DEAD!
sniffycrab
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Re: Analysis of roll values (New difficulty chart inside)

Post by sniffycrab »

Begone with your sinister math and nerdy charts. I literally feel the fun being sucked out of the room by this.
Clay
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Re: Analysis of roll values (New difficulty chart inside)

Post by Clay »

OVA, as written, was designed to pretty much cap out at 12 dice. That requires TWO +5 Abilities, which is really unlikely. That said, if you’re using a digital dice roller, there’s no real reason you can’t chuck fistfuls of virtual dice. So if that’s what works for a group, great! I think a chart like this certainly makes that more doable, since, as I said, OVA really only accounts for a smaller number.

Joe: Yeah, 6 is supposed to be the highest “untrained” difficulty. There’s a sharp drop after that, making them solely the territory of more skilled characters. I wish there were a good way to emphasize this in the text itself, but hopefully the fact that it doesn’t require doubles and isn’t described as “difficult” imply it enough.

Sniffycrab: While it’s well and good if you don’t like “sinister math,” it would be more polite of you to simply not read this particular thread than put down what others may find fun or interesting. I actually enjoy seeing the math behind things, and doing calculations like this is ESSENTIAL when designing an RPG. You have to build the bouncy-castle and make sure it’s structurally sound before you can jump on it.
Joe_Mello
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Re: Analysis of roll values (New difficulty chart inside)

Post by Joe_Mello »

Clay wrote:Joe: Yeah, 6 is supposed to be the highest “untrained” difficulty. There’s a sharp drop after that, making them solely the territory of more skilled characters. I wish there were a good way to emphasize this in the text itself, but hopefully the fact that it doesn’t require doubles and isn’t described as “difficult” imply it enough.
A couple sentences in the GM section regarding setting difficulty? Like "If you want your players to successfully complete a task, but don't want to make it an automatic success, make the target 4 or 6. Anything higher means that likely only those with training in the proper abilities will be able to pass."
Joe_Mello: Could you make a common sense roll, please, Ryu?
Ryushikaze: With Smart?
Joe_Mello: Sure
*Ryushikaze rolls*
Joe_Mello: SHE'S DEAD!
Clay
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Re: Analysis of roll values (New difficulty chart inside)

Post by Clay »

If you take a look at the GM’s section, it’s pretty packed as it is. :) But looking at it again, the drop isn’t THAT exorbitant. I think the odds of 4 (not shown here) is like 80%, 6 40%, and 8 close to 10%. So it’s actually a smoother progression for unskilled characters. It’s just that difficult remains at least a little difficult for increasing amounts of dice.
Rawle Nyanzi
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Re: Analysis of roll values (New difficulty chart inside)

Post by Rawle Nyanzi »

Joe_Mello wrote:If a player is rolling more than 10 dice, I have to think something has gone incredibly wrong in your game.
Ridiculous numbers of dice are the norm in my games.
Joe_Mello wrote:However, it is interesting how good 6 is as a target number. If you want to favor characters with a certain ability without punishing those that don't, six seems a reasonable bar to clear.
Agreed. It's the most frequently occurring number when rolling two dice, but lower numbers are more likely to occur. The ease of beating it with higher numbers of dice indicates greater training by that character.
Rawle Nyanzi
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Re: Analysis of roll values (New difficulty chart inside)

Post by Rawle Nyanzi »

Clay wrote:OVA, as written, was designed to pretty much cap out at 12 dice. That requires TWO +5 Abilities, which is really unlikely. That said, if you’re using a digital dice roller, there’s no real reason you can’t chuck fistfuls of virtual dice. So if that’s what works for a group, great! I think a chart like this certainly makes that more doable, since, as I said, OVA really only accounts for a smaller number.
I got the sense that high numbers of dice weren't typical. Still, I wanted to see how it played out at larger dice numbers due to the trajectory my games usually take.

However, my players strongly prefer physical dice. I actually have an absurd number of d6 dice, so this is extremely feasible.

Once again, thanks for designing this game.
S'drolion
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Re: Analysis of roll values (New difficulty chart inside)

Post by S'drolion »

I do have one character in my game that has (unless I've drastically misinterpreted things) a reliable 12 dice for casting Magic, Witchcraft spells (Level 5 Magic, Witchcraft + Level 5 Smart), but that's also while he's in a Transformed state, and the character's basically been set up specifically to fulfill a debuff role.
Rawle Nyanzi
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Re: Analysis of roll values (New difficulty chart inside)

Post by Rawle Nyanzi »

S'drolion wrote:I do have one character in my game that has (unless I've drastically misinterpreted things) a reliable 12 dice for casting Magic, Witchcraft spells (Level 5 Magic, Witchcraft + Level 5 Smart), but that's also while he's in a Transformed state, and the character's basically been set up specifically to fulfill a debuff role.
Sounds like fun times. These charts should prove very helpful, then.
bushido11
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Re: Analysis of roll values (New difficulty chart inside)

Post by bushido11 »

Here's a similar probability list to the one listed above starting at a -4 dice pool (taking only the lowest number) up to a 14 dice pool with their corresponding chances of succeeding against a difficulty of Easy (2) up to 1st Edition's difficulty of Nigh Impossible (15).

Dice Pool Probabilities (in %) for Succeeding against DN 2/4/6/8/10/12/15
-4 33.5/1.6/0.002/0 for the rest
-3 40.2/3.1/0.013/0 for the rest
-2 48.2/6.3/0.077/0 for the rest
1 83.3/50/16.7/0 for the rest
2 97.2/80.5/38.9/8.3/5.5/2.8/0
3 100/94/60.6/22.7/15.3/8.3/1
4 100/98.4/77.9/39.9/27.6/16.5/3.3
5 100/100/89.1/57/41.1/26.8/7.4
6 100/100/95.4/71.7/54.3/38.4/13.4
7 100/100/100/83/66.4/50.4/20.1
8 100/100/100/90.8/76.6/61.8/30
9 100/100/100/100/84.6/72/39.2
10 100/100/100/100/90.4/80.4/49.2
11 100/100/100/100/100/87/58.9
12 100/100/100/100/100/91.8/67.7
13 100/100/100/100/100/100/75.6
14 100/100/100/100/100/100/82.2

Note that values of 100% are not actually 100%, but just very close. As far as rolling all 1s is concerned (or rolling at least one 1 on a negative dice pool), page 99 dictates that rolling all 1s generates the worst failure possible, though, unlike 1st Edition, you can still add up 1s (though there’s still technically not a 100% guarantee of success, as you can still roll all 1s). I don't have an image of the chart, hence the typing out of the information in the format above.
Rawle Nyanzi
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Re: Analysis of roll values (New difficulty chart inside)

Post by Rawle Nyanzi »

bushido11 wrote:Here's a similar probability list to the one listed above starting at a -4 dice pool (taking only the lowest number) up to a 14 dice pool with their corresponding chances of succeeding against a difficulty of Easy (2) up to 1st Edition's difficulty of Nigh Impossible (15).

Dice Pool Probabilities (in %) for Succeeding against DN 2/4/6/8/10/12/15
-4 33.5/1.6/0.002/0 for the rest
-3 40.2/3.1/0.013/0 for the rest
-2 48.2/6.3/0.077/0 for the rest
1 83.3/50/16.7/0 for the rest
2 97.2/80.5/38.9/8.3/5.5/2.8/0
3 100/94/60.6/22.7/15.3/8.3/1
4 100/98.4/77.9/39.9/27.6/16.5/3.3
5 100/100/89.1/57/41.1/26.8/7.4
6 100/100/95.4/71.7/54.3/38.4/13.4
7 100/100/100/83/66.4/50.4/20.1
8 100/100/100/90.8/76.6/61.8/30
9 100/100/100/100/84.6/72/39.2
10 100/100/100/100/90.4/80.4/49.2
11 100/100/100/100/100/87/58.9
12 100/100/100/100/100/91.8/67.7
13 100/100/100/100/100/100/75.6
14 100/100/100/100/100/100/82.2

Note that values of 100% are not actually 100%, but just very close. As far as rolling all 1s is concerned (or rolling at least one 1 on a negative dice pool), page 99 dictates that rolling all 1s generates the worst failure possible, though, unlike 1st Edition, you can still add up 1s (though there’s still technically not a 100% guarantee of success, as you can still roll all 1s). I don't have an image of the chart, hence the typing out of the information in the format above.
Nice. It's very similar to my own chart; I generally avoid decimals, though, because it would be too difficult for average readers to visualize fractional percents.
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Re: Analysis of roll values (New difficulty chart inside)

Post by Lucas Yew »

After getting my hands in the paperback copy of OVA in a lucky encounter in Boston last summer, I finally decided to ask the Internet about which exact probabilities and number of cases does the OVA dice rules produce. And then I found that somebody already posted a rounded, cleared up version already in the official forums, so I decided to enroll in the forums to compliment his great work, and yet still ask for what I am searching for.

1. What is the average value for each number of dices rolled and executed in OVA rules?
2. What is the least possible value for each (...) ?

Apparently, I only had spare time to figure out that the unmodified 2d6 OVA roll results in the hypothetical average value of "5 + 5/90", lower compared to the common result of 7 and just short of the usual Challenging difficulty. No result of 1, so the minimum value is 2, automatically bypassing the Easy difficulty.

Can someone with time and brainpower to spend freely find out what are the answers for other number of dices, please? Like, up to 22, which is apparently the maximum bonus (+5 times 3+1) you can add up before attacking/defending plus advantageous Scale... (And probably suitable for the "world's most powerful organism" type character; but maybe TOO strong for a PC material...?)

----

P.S. This is the exact results for all 36 combinations that happens in a 2d6 OVA roll, summed up and then divided for the exact average value.
(1*0+2*3+3*4+4*7+5*8+6*11+8*1+10*1+12*1)/36 = 5 + 5/90
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