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Xyk
2020-05-04, 02:35 PM
I'm toying with a new idea for a dice pool mechanic for a modular adventure rpg system, but I don't know enough about statistics to really understand the odds of different outcomes.

The idea is that a player will roll a number of d6s equal to their ability+skill and count the number of matching pairs for degrees of success (comparing them to either a contested roll or set number of pairs needed). 6's are wild.

The idea appeals to me because:

It makes some level of success pretty likely with a relatively low skill.
Makes at least one matching pair inevitable for pools greater than 6 dice.
The wild 6 definitely increases the likelihood of matching pairs, and multiple matching pairs, but I don't know for sure by how much.
Only one roll is required to get both "does it succeed" and "by what degree does it succeed or fail"
No matter how many dice you roll, the maximum number of pairs you can get is number of dice/2, so levels of success don't escalate too quickly, and the bounds are clearly defined.
I intend to use the specific numbers that are paired up as like an evens vs odds system for the magic/superscience parts of the system, where odd pairs cause unwanted side effects, so the wild 6 can give the player additional agency in those situations, giving some choice between higher degrees of success or fewer side effects.
It's fun to roll a handful of dice, and almost no math is required to count pairs.
Most people have at least some d6s in their home somewhere, and the d6 is what people think of when they think of dice.


Based on the chart of possible outcomes I made, it seems like there's a 4/9 chance of getting at least one matching pair with 2 dice rolled, and a 2/3 chance of getting at least one pair with 3 dice rolled, and I don't really understand the stats to do more than that or check my work effectively.

Any amount of help figuring this out would be super helpful, and if someone knows how to express it as a function of how many dice rolled to chance of numbers of pairs, I would be very interested in learning that.

Thanks y'all

Elves
2020-05-04, 04:04 PM
Seems like it would be a pain unless the different faces of the dice were color-coded. But that gives you some unique merch to sell I guess, lol.

For the math portion maybe post on a math or stats forum?


I could see some cool possibilities in mixing dice of different types in a single throw (3d6 and 2d8 or whatever). Then you get different benefits for matching odds - evens and for matching digits.

aimlessPolymath
2020-05-04, 05:12 PM
I ran a Monte Carlo simulation to test this for higher dice numbers. I didn't spend enough time to test even/odd sets with wilds. It looks like at higher numbers of dice, there are enough wilds going around that practically all dice end up matched- for example, if you have 12 dice, you have 2 wilds on average, which means that unless there's an odd number of at least three of 1's, 2's, 3's, 4's, and 5's, etc, you probably can make your whole die pool into matches.
Running 1000000 trials of 2 dice
Number of 0 pairs: 555000, probability 55.50000000000001%
Number of 1 pairs: 445000, probability 44.5%
Running 1000000 trials of 3 dice
Number of 0 pairs: 277852, probability 27.7852%
Number of 1 pairs: 722148, probability 72.2148%
Running 1000000 trials of 4 dice
Number of 0 pairs: 92135, probability 9.2135%
Number of 1 pairs: 524454, probability 52.4454%
Number of 2 pairs: 383411, probability 38.3411%
Running 1000000 trials of 5 dice
Number of 0 pairs: 15567, probability 1.5567%
Number of 1 pairs: 308448, probability 30.8448%
Number of 2 pairs: 675985, probability 67.5985%
Running 1000000 trials of 6 dice
Number of 0 pairs: 0, probability 0.0%
Number of 1 pairs: 105854, probability 10.5854%
Number of 2 pairs: 489880, probability 48.988%
Number of 3 pairs: 404266, probability 40.4266%
Running 1000000 trials of 7 dice
Number of 0 pairs: 0, probability 0.0%
Number of 1 pairs: 14929, probability 1.4929%
Number of 2 pairs: 283067, probability 28.306700000000003%
Number of 3 pairs: 702004, probability 70.2004%
Running 1000000 trials of 8 dice
Number of 0 pairs: 0, probability 0.0%
Number of 1 pairs: 0, probability 0.0%
Number of 2 pairs: 88925, probability 8.8925%
Number of 3 pairs: 448164, probability 44.8164%
Number of 4 pairs: 462911, probability 46.2911%
Running 1000000 trials of 9 dice
Number of 0 pairs: 0, probability 0.0%
Number of 1 pairs: 0, probability 0.0%
Number of 2 pairs: 11459, probability 1.1459000000000001%
Number of 3 pairs: 238582, probability 23.8582%
Number of 4 pairs: 749959, probability 74.9959%
Running 1000000 trials of 10 dice
Number of 0 pairs: 0, probability 0.0%
Number of 1 pairs: 0, probability 0.0%
Number of 2 pairs: 0, probability 0.0%
Number of 3 pairs: 68897, probability 6.8897%
Number of 4 pairs: 395498, probability 39.549800000000005%
Number of 5 pairs: 535605, probability 53.5605%
Running 1000000 trials of 11 dice
Number of 0 pairs: 0, probability 0.0%
Number of 1 pairs: 0, probability 0.0%
Number of 2 pairs: 0, probability 0.0%
Number of 3 pairs: 8187, probability 0.8187%
Number of 4 pairs: 192246, probability 19.2246%
Number of 5 pairs: 799567, probability 79.9567%
Running 1000000 trials of 12 dice
Number of 0 pairs: 0, probability 0.0%
Number of 1 pairs: 0, probability 0.0%
Number of 2 pairs: 0, probability 0.0%
Number of 3 pairs: 0, probability 0.0%
Number of 4 pairs: 51198, probability 5.1198%
Number of 5 pairs: 340885, probability 34.088499999999996%
Number of 6 pairs: 607917, probability 60.791700000000006%

Segev
2020-05-04, 05:29 PM
I'm toying with a new idea for a dice pool mechanic for a modular adventure rpg system, but I don't know enough about statistics to really understand the odds of different outcomes.

The idea is that a player will roll a number of d6s equal to their ability+skill and count the number of matching pairs for degrees of success (comparing them to either a contested roll or set number of pairs needed). 6's are wild.

The idea appeals to me because:

It makes some level of success pretty likely with a relatively low skill.
Makes at least one matching pair inevitable for pools greater than 6 dice.
The wild 6 definitely increases the likelihood of matching pairs, and multiple matching pairs, but I don't know for sure by how much.
Only one roll is required to get both "does it succeed" and "by what degree does it succeed or fail"
No matter how many dice you roll, the maximum number of pairs you can get is number of dice/2, so levels of success don't escalate too quickly, and the bounds are clearly defined.
I intend to use the specific numbers that are paired up as like an evens vs odds system for the magic/superscience parts of the system, where odd pairs cause unwanted side effects, so the wild 6 can give the player additional agency in those situations, giving some choice between higher degrees of success or fewer side effects.
It's fun to roll a handful of dice, and almost no math is required to count pairs.
Most people have at least some d6s in their home somewhere, and the d6 is what people think of when they think of dice.


Based on the chart of possible outcomes I made, it seems like there's a 4/9 chance of getting at least one matching pair with 2 dice rolled, and a 2/3 chance of getting at least one pair with 3 dice rolled, and I don't really understand the stats to do more than that or check my work effectively.

Any amount of help figuring this out would be super helpful, and if someone knows how to express it as a function of how many dice rolled to chance of numbers of pairs, I would be very interested in learning that.

Thanks y'all

Clarification question: other than whether it's even/odd, does it matter WHAT the matches are, or merely how many dice are in the "Set?"

And is it one success per two dice you can match (so max success is half the dice rolled), or is it your success has a rating equal to the number of dice in a set (so if you rolled 2 5s and 4 1s, you'd take the 1s for a success rating of 4)?

Xyk
2020-05-05, 04:55 PM
I ran a Monte Carlo simulation to test this for higher dice numbers. I didn't spend enough time to test even/odd sets with wilds. It looks like at higher numbers of dice, there are enough wilds going around that practically all dice end up matched- for example, if you have 12 dice, you have 2 wilds on average, which means that unless there's an odd number of at least three of 1's, 2's, 3's, 4's, and 5's, etc, you probably can make your whole die pool into matches.
Running 1000000 trials of 2 dice
Number of 0 pairs: 555000, probability 55.50000000000001%
Number of 1 pairs: 445000, probability 44.5%
Running 1000000 trials of 3 dice
Number of 0 pairs: 277852, probability 27.7852%
Number of 1 pairs: 722148, probability 72.2148%
Running 1000000 trials of 4 dice
Number of 0 pairs: 92135, probability 9.2135%
Number of 1 pairs: 524454, probability 52.4454%
Number of 2 pairs: 383411, probability 38.3411%
Running 1000000 trials of 5 dice
Number of 0 pairs: 15567, probability 1.5567%
Number of 1 pairs: 308448, probability 30.8448%
Number of 2 pairs: 675985, probability 67.5985%
Running 1000000 trials of 6 dice
Number of 0 pairs: 0, probability 0.0%
Number of 1 pairs: 105854, probability 10.5854%
Number of 2 pairs: 489880, probability 48.988%
Number of 3 pairs: 404266, probability 40.4266%
Running 1000000 trials of 7 dice
Number of 0 pairs: 0, probability 0.0%
Number of 1 pairs: 14929, probability 1.4929%
Number of 2 pairs: 283067, probability 28.306700000000003%
Number of 3 pairs: 702004, probability 70.2004%
Running 1000000 trials of 8 dice
Number of 0 pairs: 0, probability 0.0%
Number of 1 pairs: 0, probability 0.0%
Number of 2 pairs: 88925, probability 8.8925%
Number of 3 pairs: 448164, probability 44.8164%
Number of 4 pairs: 462911, probability 46.2911%
Running 1000000 trials of 9 dice
Number of 0 pairs: 0, probability 0.0%
Number of 1 pairs: 0, probability 0.0%
Number of 2 pairs: 11459, probability 1.1459000000000001%
Number of 3 pairs: 238582, probability 23.8582%
Number of 4 pairs: 749959, probability 74.9959%
Running 1000000 trials of 10 dice
Number of 0 pairs: 0, probability 0.0%
Number of 1 pairs: 0, probability 0.0%
Number of 2 pairs: 0, probability 0.0%
Number of 3 pairs: 68897, probability 6.8897%
Number of 4 pairs: 395498, probability 39.549800000000005%
Number of 5 pairs: 535605, probability 53.5605%
Running 1000000 trials of 11 dice
Number of 0 pairs: 0, probability 0.0%
Number of 1 pairs: 0, probability 0.0%
Number of 2 pairs: 0, probability 0.0%
Number of 3 pairs: 8187, probability 0.8187%
Number of 4 pairs: 192246, probability 19.2246%
Number of 5 pairs: 799567, probability 79.9567%
Running 1000000 trials of 12 dice
Number of 0 pairs: 0, probability 0.0%
Number of 1 pairs: 0, probability 0.0%
Number of 2 pairs: 0, probability 0.0%
Number of 3 pairs: 0, probability 0.0%
Number of 4 pairs: 51198, probability 5.1198%
Number of 5 pairs: 340885, probability 34.088499999999996%
Number of 6 pairs: 607917, probability 60.791700000000006%

Okay that is really helpful, thank you! Seems like the system breaks down pretty quickly and didn't work all that well from the start, you would want to see much more even distributions than those. I could see dice with more sides helping that some, or getting rid of the wild 6, or maybe just forgetting the whole "matching pairs" idea.

Trying it out with just a dice rolling simulator, (I don't know how to program one to roll a million and count pairs), seems like doing it with d20s evens out that curve significantly. Rolling 10d20s (with wild 20s), I'm rarely getting 4 pairs and usually getting 1 or 2, and rarely getting 0.

To clarify what I meant by "matching pairs" also, here's an example roll using 8d6s:
(6,6,5,4,4,1,1,1) would result in a score of 4 successes (match a 6 to a 1 and a 6 to a 5, to get 4 distinct pairs), which does illustrate how d6s don't have enough variance.

With 8d20s, a sample roll could be:
(20,17,14,13,10,6,4,4), which would be 2 successes (match the 20 to any number but 4).

I'm still gonna mess around with it, and see if I can get someone to teach me enough javascript or whatever to run simulations.

aimlessPolymath
2020-05-05, 05:10 PM
I reran without the "wild 6's" aspect. Results: Somewhat increased spread; odd behavior at larger dice counts where odd vs. even numbers of dice results in a varying spread size.

Running 1000000 trials of 2 dice
Number of 0 pairs: 832991, probability 83.29910000000001%
Number of 1 pairs: 167009, probability 16.7009%
Running 1000000 trials of 3 dice
Number of 0 pairs: 556132, probability 55.6132%
Number of 1 pairs: 443868, probability 44.3868%
Running 1000000 trials of 4 dice
Number of 0 pairs: 277551, probability 27.7551%
Number of 1 pairs: 648031, probability 64.8031%
Number of 2 pairs: 74418, probability 7.4418%
Running 1000000 trials of 5 dice
Number of 0 pairs: 92539, probability 9.2539%
Number of 1 pairs: 617512, probability 61.7512%
Number of 2 pairs: 289949, probability 28.9949%
Running 1000000 trials of 6 dice
Number of 0 pairs: 15510, probability 1.551%
Number of 1 pairs: 385489, probability 38.5489%
Number of 2 pairs: 550927, probability 55.092699999999994%
Number of 3 pairs: 48074, probability 4.8073999999999995%
Running 1000000 trials of 7 dice
Number of 0 pairs: 0, probability 0.0%
Number of 1 pairs: 143633, probability 14.3633%
Number of 2 pairs: 624910, probability 62.491%
Number of 3 pairs: 231457, probability 23.145699999999998%
Running 1000000 trials of 8 dice
Number of 0 pairs: 0, probability 0.0%
Number of 1 pairs: 23975, probability 2.3975%
Number of 2 pairs: 432690, probability 43.269000000000005%
Number of 3 pairs: 504556, probability 50.455600000000004%
Number of 4 pairs: 38779, probability 3.8779%
Running 1000000 trials of 9 dice
Number of 0 pairs: 0, probability 0.0%
Number of 1 pairs: 0, probability 0.0%
Number of 2 pairs: 167754, probability 16.775399999999998%
Number of 3 pairs: 624605, probability 62.460499999999996%
Number of 4 pairs: 207641, probability 20.7641%
Running 1000000 trials of 10 dice
Number of 0 pairs: 0, probability 0.0%
Number of 1 pairs: 0, probability 0.0%
Number of 2 pairs: 28003, probability 2.8003%
Number of 3 pairs: 451668, probability 45.1668%
Number of 4 pairs: 485749, probability 48.5749%
Number of 5 pairs: 34580, probability 3.458%
Running 1000000 trials of 11 dice
Number of 0 pairs: 0, probability 0.0%
Number of 1 pairs: 0, probability 0.0%
Number of 2 pairs: 0, probability 0.0%
Number of 3 pairs: 178781, probability 17.8781%
Number of 4 pairs: 625046, probability 62.504599999999996%
Number of 5 pairs: 196173, probability 19.6173%
Running 1000000 trials of 12 dice
Number of 0 pairs: 0, probability 0.0%
Number of 1 pairs: 0, probability 0.0%
Number of 2 pairs: 0, probability 0.0%
Number of 3 pairs: 29652, probability 2.9652000000000003%
Number of 4 pairs: 461459, probability 46.1459%
Number of 5 pairs: 476056, probability 47.605599999999995%
Number of 6 pairs: 32833, probability 3.2833%




public class dieRoller {

static int trials = 1000000;
static int sides = 6;
int[] n = new int[sides];
int[] pairsFromTrials;
public static void main(String[] args){
dieRoller roller = new dieRoller();
for(int i=2; i<=12; i++){
roller.trials(trials, i, false);
}

}

public void output(){
for(int i=1; i<=sides; i++){
System.out.println("Value: "+ i + "Result: "+ n[i-1]);
}
}

public void trials(int howMany, int howBig, boolean wild){
pairsFromTrials = new int[howBig/2+1];
for(int i=0; i< howMany; i++){
resetN();
rollN(howBig);
pairsFromTrials[numPairs(wild)]++;
}
System.out.println("Running "+ howMany + " trials of " + howBig + " dice");
for(int i=0; i<pairsFromTrials.length; i++){
System.out.println(
"Number of " + i +
" pairs: " + pairsFromTrials[i] +
", probability " + 100*divide(pairsFromTrials[i], howMany) + "%");

}
}

public double divide(int top, int bottom){
return ((double) top)/((double) bottom);
}

public int numPairs(boolean wildMax){
int spares = 0;
int pairs = 0;
if(wildMax){
for(int i=0;i<(sides-1);i++){ //i<5 if you want 6's wild
pairs += n[i]/2;
spares += n[i]%2;
}
int wilds = n[sides-1];
pairs += Math.min(wilds, spares);
wilds -= Math.min(wilds, spares);
pairs += wilds/2;
}else {
for(int i=0;i<sides;i++){
pairs += n[i]/2;
spares += n[i]%2;
}
}
return pairs;
}

public void rollN(int num){
for(int i=0;i<num;i++){
rollAdd();
}
}


public void resetN(){
n = new int[sides];
}

public void rollAdd(){
n[roll()-1]++;
}

//returns a number between 1 and sides
public int roll(){
return (int) (Math.random()*sides) +1;
}
}

Segev
2020-05-05, 05:14 PM
Okay that is really helpful, thank you! Seems like the system breaks down pretty quickly and didn't work all that well from the start, you would want to see much more even distributions than those. I could see dice with more sides helping that some, or getting rid of the wild 6, or maybe just forgetting the whole "matching pairs" idea.

Trying it out with just a dice rolling simulator, (I don't know how to program one to roll a million and count pairs), seems like doing it with d20s evens out that curve significantly. Rolling 10d20s (with wild 20s), I'm rarely getting 4 pairs and usually getting 1 or 2, and rarely getting 0.

To clarify what I meant by "matching pairs" also, here's an example roll using 8d6s:
(6,6,5,4,4,1,1,1) would result in a score of 4 successes (match a 6 to a 1 and a 6 to a 5, to get 4 distinct pairs), which does illustrate how d6s don't have enough variance.

With 8d20s, a sample roll could be:
(20,17,14,13,10,6,4,4), which would be 2 successes (match the 20 to any number but 4).

I'm still gonna mess around with it, and see if I can get someone to teach me enough javascript or whatever to run simulations.

The Godlike game system used die pools of d10s, but had success be based on groups of matched dice. The MORE dice that matched, the "broader" the success, which indicated speed (primarily). The HIGHER the value of the match, the "taller" the success, which indicated quality (primarily).

So you roll a pool of dice, then pick any group of matched dice you like, aiming for speed or quality as you desire.

Xyk
2020-05-05, 06:32 PM
The Godlike game system used die pools of d10s, but had success be based on groups of matched dice. The MORE dice that matched, the "broader" the success, which indicated speed (primarily). The HIGHER the value of the match, the "taller" the success, which indicated quality (primarily).

So you roll a pool of dice, then pick any group of matched dice you like, aiming for speed or quality as you desire.

Wow, that system is practically exactly what I was trying to piece together, I will be looking into this system, apparently called One Roll Engine (ORE) with many different settings including Godlike.

Segev
2020-05-05, 07:47 PM
Wow, that system is practically exactly what I was trying to piece together, I will be looking into this system, apparently called One Roll Engine (ORE) with many different settings including Godlike.

I only saw it for a month or so in college, so you may more more about it than I very shortly. Good luck and have fun!