I'm working on a simple 3d6 homebrew system, and I'm looking for probability help. Here's my idea: Players have a small pool of Action Points. You have two options: Spend 1 AP before you make a skill roll, and you can roll 4d6 (instead of 3d6) and take the best 3. Or you can spend 1 AP after the roll to reroll the 3d6.

The goal is to make the first option better than the second, but I don't know enough about probability to know if it works. Any math experts out there?

Let me suggest www.anydice.com.  It will do the probabilities of most dice rolls for you.  He's the results for your question.

The table below lists the probability that the roll will add up to at least the number value on the left.  So:

"rolling 4d6 once",
#,%
3,100
4,99.9228395061728
5,99.6141975308638
6,98.8425925925918
7,97.2222222222218
8,94.2901234567918
9,89.5061728395118
10,82.4845679012418
11,73.0709876543318
12,61.6512345679318
13,48.7654320988318
14,35.4938271605318
15,23.1481481482318
16,13.0401234568318
17,5.7870370370818005
18,1.6203703704118002

"rolling 3d6 twice and taking the best roll",
#,%
3,100
4,99.99785665294925
5,99.96570644718796
6,99.78566529492495
7,99.14266117969895
8,97.37439986282895
9,93.27846364883895
10,85.93750000000895
11,75.00000000000895
12,60.93750000000895
13,45.13031550070895
14,29.78180727020895
15,17.66117969820895
16,9.044924554178952
17,3.669410150888951
18,0.9237825788689511

So as you can see, you end up with about a 2% higher chance of rolling above 11 by rolling 4d6 than 3d6 twice.  Don't know if that is enough of an effect for you.  Here's the setup I used.  Try it yourself and play around until you get the results you want:

output [highest 3 of 4d6] named "rolling 4d6 once"
output [highest of 3d6 and 3d6] named "rolling 3d6 twice and taking the best roll"

Thanks, much appreciated. My proposed design isn't meeting the goal. Need to rethink it.

I used the formula provided by Eirikrautha but did it as a graph. I also changed the 4D6 to 5D6 for comparison.

I think the prospect of rolling TWO additional dice might appeal enough to the players that they favor that option.

Quote from: Godsmonkey on January 12, 2021, 01:01:40 PM
I used the formula provided by Eirikrautha but did it as a graph. I also changed the 4D6 to 5D6 for comparison.

I think the prospect of rolling TWO additional dice might appeal enough to the players that they favor that option.

Excellent idea! Thanks.

In case you're interested, The Action! System by Gold Rush Publishing uses a 3d6 task resolution and Action Points. http://www.darkshire.net/jhkim/rpg/srd/srd_action/index.html

Quote from: BoxCrayonTales on February 06, 2021, 01:40:37 PM
In case you're interested, The Action! System by Gold Rush Publishing uses a 3d6 task resolution and Action Points. http://www.darkshire.net/jhkim/rpg/srd/srd_action/index.html

Thanks! Already know about Action System. It's a great 3d6 rollover system, until you get to magic, and then you're out of luck. it doesn't do quite what I'm looking for in terms of action points, which is

Option 1: Spend one before the roll, for a good bonus
Option 2: Spend one after a roll, for possibly that same bonus but maybe not

Quote from: Aglondir on February 09, 2021, 02:41:59 AM
it doesn't do quite what I'm looking for in terms of action points, which is

Option 1: Spend one before the roll, for a good bonus
Option 2: Spend one after a roll, for possibly that same bonus but maybe not

Why not invent your own rules for it? Kobold Press has several variants: https://koboldpress.com/5-action-point-variants/

Quote from: Aglondir on February 09, 2021, 02:41:59 AM
until you get to magic, and then you're out of luck.

Why not invent your own magic rules for it? There's no shortage of mechanics in other games that you can take inspiration from.

For example: Add a new Magic Group for Attributes, similar to the suggested Psi Group. Add the suggested Mana derived attribute too. Add a new set of Magic Skills to represent magical powers, such as Alchemy, Necromancy, Divination, etc. Adapt Traits to represent magical phenomena not covered by the attributes and skills. Does that work?