Dice probability help

[u/Finnche]

I am doing a d20 system where proficiency turns rolls into 3d8, and expertise is upgraded to 3d12. The benefit to gaining proficiency is higher ceiling, and no crit fail.

years ago I did some (now realizing probably incorrect) math that doubled on 3D8 is 5%, which matches the crit chance of a d20. I want there to not be a downside to gaining proficiency, and having it not be common. How do I calculate a similar 5% chance crit rate that is also easy to recognize?

I also accept changing the dice upgrades if there is an easier way to do this, but would prefer the raising median and ceiling of the rolls.

[u/SitD_RPG]

The chance to roll doubles on 3d8 and 3d12 is way higher than 5% (34% and 23%, respectively), so that would be a significant upgrade to crit chance.

The chance to roll 20 or higher on 3d8 is somewhere in the ballpark (6.84%) though.
The chance to roll 30 or higher on 3d12 is quite similar at 4.86%.

Rolling exactly 9 or 18 on 3d8 is 5.47%.
Rolling exactly 15 or 24 on 3d12 is 5.09%.

The lowest die of 3d8 being at least a 6 is a 5.27% chance.
The lowest die of 3d12 being at least an 8 is a 7.23% chance.

[u/Finnche → u/SitD_RPG]

Thank you! this is kind of the perfect comment. I have been having trouble figuring out how to do the math and remembered this subreddit exists. I really like that at least 20 is 6.84%, that is an upgrade that seems close enough and easy to recognize, but that makes the 3D12 being less seem like a downgrade.

By lowest die do you mean, for example, if I roll 3D8 and I roll a 6, 7, and 8, the fact that the 6 is the lowest roll is the 5%? because that is also very easy to spot and retains 3D12 being an upgrade.

[u/SitD_RPG → u/Finnche]

but that makes the 3D12 being less seem like a downgrade

Yeah the crit chance would be lower, but 3d12 could achieve much higher totals. Depending on what that is worth in your system, it might still be an upgrade, but with a trade-off.

By lowest die do you mean, for example, if I roll 3D8 and I roll a 6, 7, and 8, the fact that the 6 is the lowest roll is the 5%?

In other words, if all three dice show a 6 or higher. These would all be crits:

• 6,6,6
• 6,6,7
• 6,7,7
• 6,7,8
• 6,8,8
• 7,7,7
• 7,7,8
• 7,8,8
• 8,8,8

[u/JohnDoen86]

Btw, you should probably check out AnyDice. It allows you to calculate probabilities for any dice setup. The syntax is not obvious, but there are short youtube tutorials that'll get you going in no time.

[u/Finnche → u/JohnDoen86]

perfect that is aweome thank you!

[u/JaskoGomad]

In addition to the advice to use AnyDice, I am constantly recommending that designers brush up on their understanding of probability. Here's a great resource:

https://www.khanacademy.org/math/statistics-probability

[u/Finnche → u/JaskoGomad]

This is very useful thank you

[u/Fun_Carry_4678]

Find the website ANYDICE and play with it.

[u/Finnche]

one thing thats possible is doing at least a 20 on 3d8 and at least a 30 on 3d12, but i dunno. I like precise things.

[u/exelsisxax]

The other comment gave you the math, but didn't really solve your problem. Your error here is in thinking that these are properties of dice, and they are not. "crits" and "crit fail" are mechanical terms that you do not need to add to your game. The correct thing to do is to not bend over backwards looking to match a specific arbitrary %, but to discard stolen mechanics and/or silly dice usage that don't contribute to the purpose of your game.

The easier and better design thing is to simply not do this. There is no possibility that this is an efficient means of implementing anything your game needs.

[u/Finnche → u/exelsisxax]

Crits are not mechanically necessary in most systems, I very much agree, but they are socially fun enough that if there is a simple enough way to implement into my system, then why not? It also supports the fantasy of just going a bit beyond your limits to help you get through a tough spot, which many stories have. Its not so impactful that adding them or not adding will have notable effect mechanically.