We've all been there – someone wants to do something weird with a spell, and you want it to be unlikely but not 'nat 20' unlikely. As someone who enjoys the crunchier aspects of the game, I've used a few statistical tricks over my time DMing, and thought I'd share. These are totally optional and system agnostic, they're just a fun way of mixing stuff up when a player asks you if they can do a backflip as part of their attack or whatever. So I'll list the statistic tools/tricks I've used over my time DM'ing, what the math behind them is, and if you have more, feel free to comment below!

## Basics

5e rests on a few fundamental probability and statistic things, and I felt the need to briefly review those. Feel free to skip this.

% Chance – On a d20 there is a 1/20 (5%) chance in getting any particular number, including 1 or 20. This holds true that any die of size x has a 1/x chance of rolling that number. For DCs, you also add in the probability of rolling the numbers higher than it. For a DC of 15, a player can roll 5 numbers (15-20) and succeed, which is a 5/20 or 25% chance.

Advantage/disadvantage – Rolling two dice and taking the higher/lower skews this distribution, so that the higher/lower numbers are more likely. With advantage, it makes rolling a 20 nearly twice as likely and rolling a 1 nearly impossible (1/400 chance). The average roll becomes about 13 with advantage and 7 with disadvantage. But remember, what advantage/disadvantage really adds is consistency, because the very lowest/highest values become the least likely.

Average dice value = (1+x)/2 – This is a small thing, but a lot of people forget that there's no zero on a die, so the average is actually not just half the maximum (d20 average ≠ 10). The average value of rolling, for example, a d6 is actually 3.5, and therefore rolling 2d6 will give an average of 7. Pop quiz – if a fireball does 8d6 damage, on average this is 28.

## Dice Math Tools

Random number generator – Roll 3d4 rather than 1d12 – In the same way that advantage adds consistency, adding multiple dice also provides that. Say you need to give your party a number of potions, but you want to leave it up to chance. You're thinking around 6 or 7 because you need them to be well-stocked for an upcoming fight. You tell them to roll a d12, knowing the average is 6.5. You groan as they, of course, get that 1/12 chance and roll the 1, leaving them with a single potion. The party dies. You weep. Instead, if you want them to have 6 potions, roll multiple smaller die. The more die, the more consistent. In this case, 2d6 or 3d4 would work fine. Here's what the distributions look like: 1d12, 2d6, 3d4, 6d2.

Plus, people just love rolling a bunch of dice. I wanted to give a random amount of gold once, so I had them roll 10d10 once, and they enjoyed the dice rolling more than the fantasy money.

Using the damage die for effect – We're already rolling damage die, why not use them? A player once wanted to target a specific body part on a bad guy. This was a minor ask, more flavor than anything, so I set the DC with the damage dice rather than a separate roll in order to keep it all moving. It's also RP-friendly, as it would take a bigger hit to do what the player is likely asking for. For something with multiple die like a greatsword or fireball, it's not as straightfoward as a single roll, but that's fine! If you know the average result of the roll, you can base a DC off of that, keeping in mind the width of the distribution is based on number of dice. You could even say they need to have a certain number of dice hit their maximum value. For 2 dice, the odds of this are basically 2x the odds of rolling it normally, so getting at least one 10 on 2d10 is ~20%. This rule becomes less accurate for bigger sets, or for setting a DC of at least two max damage dice. This is a calculator for that if you are interested, but if this doesn't click with you, you definitely shouldn't slow down your combat to implement this.

Magic item bonuses beyond a +1 to attack/damage – There are other posts dedicated entirely to this, but I wanted to repeat a few of the math-based principles here. The first is that anything that that happens on a critical hit will have its potency reduced by a factor of 20 (barring other effects like consistent advantage), because they happen so rarely. There's a few features in the game such as as Champion Fighter's critting on 19 & 20 or the Barbarian's Savage Critical (3x dice rather than 2x) that, on a magic item, are statistically worse than a +1 weapon but feel more impactful. In that same vein, "Exploding" die are when you roll max value on a die (typically for damage die), you can roll it again and add to the original value (and it can chain explode ad infinitum). This generally adds (less than +1 damage per die), but in an exciting way.

AC in 5e – Effects that increase AC give damage resistance, which has a non-linear effect on how much damage they actually take on average. Basically, each point of resistance is more valuable than the previous, since 100% means full immunity and effectively infinite health (because of crits, 5e is 'capped' at 95% resistance). 95% damage resistance is twice as good as 90% for example. When designing your encounters, a good rule of thumb is damage resistance above 50% is very impactful. In other words, when your party has a +5 to hit, each point above 15 AC is going to start becoming extremely noticeable, and you effectively double the health of the bad guy by going from 15 to 20 AC.

## The End

I hope this didn't get too boring with all the math, and that it either helped someone or was at least interesting to read! I'm not at all suggesting you whip out a calculator mid-game, but if you grok this you might be able to use math to inform your DM'ing a little bit more. Feel free to suggest your own dice tricks or math quirks in the comments, I'm always happy to hear more!

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