Dice Math | Running the Game

everybody Matt Koval here we just did two videos on undead so let's use undead as our running example in this video let's imagine I had a group of first level player characters I wanted to throw some undead at them I choose ghouls because I think ghouls are an interesting low level bad guy and I can throw a few of them at the players ghouls have an armor class of 12 how likely are my players to hit these ghouls that's what we're gonna talk about in this video we're gonna talk about the math of Dungeons and Dragons and we're only good talk it's gonna be it's gonna be a short video because I only want to give you enough information I don't know overwhelm a brand new player this is gonna be super obvious stuff for most of you but if you are a brand new player if you're new dungeon master and you're looking at this and you feel overwhelmed by the math I can't completely solve that problem I mean if you have math phobia there's only so much I can do about that my powers are limited but I can at least demystify the math a little bit and give you a little bit more confidence so that you feel comfortable making changes on the fly so how likely are my players to hit these ghouls well there's two ways to go about figuring this out one is to just ask my players what their bonuses to hit is and then do the math manually but that's a pain in the butt would take a long time so instead I'm just gonna do it my head and I know I can tell in my head without having to write anything down that my players have about seventy percent chance to hit one of these ghouls how do I know that I know they have a 70% chance to hit one of these schools because I know how they generated their stats they either use the 46 take the highest three method and then they placed them in the order they want or they took the list of numbers from The Player's Handbook and put them in the order they want but in either instance I can count on them having a 15 let's say a 15 in their highest stat in other words we can assume the fighter is attacking with their strengths the rogue is attacking with their dexterity the wizard if they are going to make an attack role for some reason is going to be attacking with their intelligence and if we assume that they picked a mathematically advantageous species that means they're gonna put a +2 in that stat which means they're gonna go from a 15 let's use the fighter as an example the fighter is gonna put their 50 in our high stat in their strengths they're gonna pick a species that give some +2 to their strengths but that's gonna get them to 17 a 17 is a +3 they're also proficient and whatever it is they're trying to do that's another plus 2 that's the mysterious plus 2 where does it come from and all this math equations it comes you were doing something you know how to do you get your proficiency bonus which starts off at +2 it does go up over time but we are assuming this is a first level party three plus two is five which means when the players are rolling to hit they're gonna roll it 20 sided die and this is this is important this is the engine that drives all the math in this game they're gonna roll a 20 sided die and they're gonna add five to it and they have to beat a 12 because that's the armor class of a ghoul which means they have to roll a seven they have to roll a seven to hit a ghoul seven or higher on a 20 sided die is a 70% chance to hit which is quite high by the way so you can see out of the gate I did a little bit of math in my head and came up with an accurate result how likely are the players did these ghouls actually they are very likely did them already I have used a lot of terminology and a lot of math that I don't want to assume you know so for a lot of people this is gonna be super obvious and they're gonna want they're gonna check out of this video already because they're gonna feel like well I know all this stuff that's fantastic this video is not for you it's for new Dungeon Master's who really know nothing about math so when I say my players have about a 70% chance to hit this ghoul where am I getting that number from well it starts with the d20 the d20 has 20 sides on it which means any given side let's imagine you picked a specific number seven what are the odds of rolling exactly seven on a 20 sided die 5% how did I arrive at that number let's figure out well because percent means per hundred if I rolled this die a hundred times I can I would be it would be reasonable to bet that the die would come up exactly seven about five times everybody likes rolling or crit a natural 20 we call it we mean natural as opposed to unmodified right and that something happens all the time when we play someone will say the number they rolled when you need to know their their result and sometimes someone will say the result and you think they just told you the number they rolled that's why we use the term natural this is a natural 20 what are the odds of rolling a natural 25 percent one in twenty that means if I roll the die a hundred times about five times I will roll exactly 20 we players ascribe a lot of meaning to this number and this number but the die doesn't know the difference between them technically there is no friends between as far as the dye is concerned there's no difference between rolling this as opposed to rolling this rolling this as far as the dyes concerned they're all the same they each are equally likely to come up they each have a 5% chance of coming up but the math in D&D is based on target numbers if I tell you that this ghoul has a 12 armor class and you have you know plus 5 to hit for instance and you figure out you do the math your head you're like I need to roll a 7 to beat it it's technically 7 or higher that means you have this whole spread of numbers from 7 all the way over to 20 that will result in you hitting so we're counting all of these each of those is 5% so if you did the math in your head you'd go 5 10 15 20 25 and you'd get to 70% that's how I know my players have about a 70% chance to hit this school let's imagine I want to make this school harder than it I could give it armor for instance I could raise its dexterity or I could just say it has thicker than normal natural hide and I could using those methods I could get its armor from 12 to whatever number I wanted well what number do I want to get it to that is this is how I think as a dungeon master I don't just think well I'm gonna I want this ghoul to be harder to hit I will give it plus one to its armor class that would make it harder to hit that is technically true but how how big an impact is that one difference going to make let's talk about that if you've been paying attention you know that if I increase the ghouls armor class from 12 to 13 I have made it 5% less likely that my players will hit how much of a difference is that 5% it's not a lot but I think you can count on a first level party in one battle against your AC 13 super rule it will come up maybe once one guy roll one attack in the entire battle how did I arrive at that conclusion well it given battle lasts between 5 & 6 rounds and if you have about 4 or 5 players that means you're looking at about roughly 20 attack rolls per battle obviously that number is gonna vary wildly between one battle and another one group and another it makes a lot of assumptions but if we have adjusted the armor class by one if we've made it 5% harder to hit and there's twenty five percents on this dye that means you'd have to roll this die 20 times in order for there to be one instance where I would have hit the ac-12 ghoul but I missed the AC 13 ghoul because we're dealing with a 20-sided die we are constantly dealing in five percents and that's I think useful for us because dealing in fives multiplying by 5s is relatively easy if your players are only fighting one ghoul then they're not gonna get 20 opportunities to attack that thing and so you won't see that you shouldn't expect to see I will say you shouldn't expect to see that one that one point difference in armor class really come up but once you've got four or five goals in the battle now you've made it so that your players are probably going to attack about 20 times over the course of this encounter and in that process there will be on average one die roll that would have hit your ac-12 goal but misses your AC 13 ghouls so me as a dungeon master if I want my ghouls to be tougher I'm not going to give them just plus one AC because I don't think it's going to come up that often giving them plus two to their armor class taking it from 12 to 14 and now instead of it being of 70% likely that the players are going to hit now it is 60% likely that they're going to hit and now we're getting close to 50% 50% is hitting half the time one out of every 10 attacks that would have hit my AC 12 ghoul are going to miss my AC 14 ghoul and now at that point I think we're definitely going to see that come up in the course of one given battle so as far as ghouls are concerned increasing their armor class from 12 to 13 not really gonna make a difference 12 to 14 now we're gonna start to see a difference but that's all predicated on a ghouls armor class I googled only has our class of 12 what about something else what about something like a bugbear chieftain a bugbear chieftain has an armor class of 17 what are the odds of you hitting this a c-17 bugbear well let's imagine you had no bonus to hit let's imagine you had a 10 strength and you were using weapon that you were not proficient with you have to roll it twice sided die and have to it in a c-17 there's a little bit of a misleading I think there's a tendency when we first start playing to do the math wrong in our heads because we naturally want to subtract 17 from 20 and that leaves us with 3 3 times 5 5% chance for each number is 15 that would mean a 15% chance to hit our bugbear no because we win if we equal 17 so we have to count the 17 we're doing the math so we don't subtract 17 from 20 we start at 17 in kind of 17 18 19 20 there's actually a 5 10 15 20 a 20 percent chance to hit our bugbear with an unmodified roll your players probably are going to be using unmodified rolls they're probably going to be +5 to hit +5 to hit is a + 25 percent chance which means now they've got a 45 percent chance to hit right I haven't even really figured out what the target number is in my head I've just taken the 20% that they need from an unmodified roll 17 18 19 20 5 10 15 20 % and I've added their plus 5 to hit on their die roll 5 10 15 20 25 25 and 20 is 45% they have a 45 percent chance to hit 45 percent chance to hit is less than 50 percent which means they're going to be midlet here's what that means I just want to help you internalize this even though it may seem like a 45 percent chance to hit is still pretty beefy still pretty meaty and not that far away from a 60 percent chance to hit 45 percent chest hit does mean your players are going to be missing most of the time a 45 percent chance of hitting means a 55 percent chance of missing and 55 percent is more than half so that is something to keep in mind when you're dealing with translating the math of D&D into player psychology players don't like to miss it would be better to have many easy to hit monsters than few hard to hit monster because even though the math might work out the same your players experience is going to be very different in one scenario many easy to hit monsters the players are going to be hitting all the time and they're gonna feel heroic the other scenario a few hard to hit monsters the players are going to be missing all the time it's gonna be really frustrating especially if you've got five or six or seven players sitting around the table and the players have to wait maybe as much as twenty minutes between taking turns and they roll one twenty sided die and they miss and then they're done and their entire turn took 30 seconds and now they're really frustrated so that's the danger zone that's the that's the danger zone is between I would say a fifty five percent chance to hit and a 45 percent chance that you're in that threshold of I'm now going to be missing an app or more of the time and there is a psychology here taking your AC's 17 bugbear and saying gosh 45 percent chance of hitting to me still sounds like a lot let's be from up to 18 armor class now I only have a 40% chance of hitting and that 5% difference happening between 45 and 40 is gonna make the players feel even more frustrated mathematically it is linear but psychologically it is not this is something you learn as a game designer especially in video games where your game is live and people can complain about it you better you better patch it quick is that even if your telemetry is telling you these these characters are all equally useful if the players conclude that that is not the case you don't get to ship a copy of Matt Koval with every game - except there on the couch next to the player and explain how clever you are and how dumb they are and then actually they're all equally likely you have to deal with the players psychology not your math that's why I am far less likely to give for instance our bugbear chieftain an 18 armor class as opposed to as normal 17 then I am a ghoul a ghoul because he's got a 12 armor class because the players are 70% likely to hit him going from 70 percent to 65 percent to 60% 60% is still most of the time but 55 percent 50% now we're getting into player getting frustrated territory hopefully this is useful to new players just starting out we are trying to figure out how likely are my players to hit and what kind of difference will it make if I change this monsters armor class you can rely on your players having a +5 to hit they may have more and they will eventually get more as their stats go up as a proficiency bonus goes up as you give as your award magic items but +5 to hit against any given armor class is the distance from that armor class to 20 including that armor class so for ghoul right with an armor class of 12 y/o 12 13 14 15 16 17 18 19 20 that's 9 the range of 9 9 times 5 is 45 plus the plus 5 the players get to there die roll which is 25 25 and 45 is how you get to a 70% chance to hit and I guarantee you there are people watching this right first of all there are people watching this right now they're like duh but I guarantee you there are people watching this right now we're like okay you lost me and this is the kind of thing we're doing it over and over again actually will result in understanding and you'll start to live your life behind the screen in five percent increments like anything else in life when we sit down and start to do it it will be tricky and it'll take a little bit of work and thought but once you start thinking in terms of the distance on the number line which is how we should be living our lives the distance between the armor class of 20 including the armor class that's one percentage and then we add it to our players to hit bonus and that's how we know what are the odds of them actually hitting obviously another way of doing it is starting with the armor class armor class of 12 they're plus five to hit so subtract five from that you get to seven seven to twenty is a 70% chance inning so there's a couple of different ways to skin this cat I don't know fluffy cat why does it weigh what skin you why would anyone skin my tiny cat no I love done pretty cool those are the two ways to calculate how likely your players are to hit you start with their attack bonus which you can assume is going to be plus five at first level and you can either subtract that from monsters armor class and then measure the difference between the result and 20 multiply that by five that's your percentage to hit or you can just add up percentages and these are things I do I can do on the fly because I've done this a million times so there's a the beginning of this process is some understanding and then a lot of practice so what does it mean to give your players a +1 magic item well it means they're 5% more likely to hit it really doesn't matter what the target number is your bug bear with a 17 armor class or your ghoul with a 12 armor class in both instances and also it doesn't matter what the what level your players are or what their attack bonus is maybe some of them are plus 5 to hit maybe some of them are plus 7 in both instances giving your players a plus-one long sword means they're all anyone who's using it is plus 5 percent more likely to it which again is not going to come up that often you it maybe will come up 5% plus 1 will maybe come up once per night the problem is that your players have lots of ways to get bonuses and they all add up so that's one of the reasons the game does not recommend you give your players a plus-one weapon until they're about fifth level for instance one of the easiest ways for my players to get bonus to hit and in fact this is something probably change is I play with the optional rule in Dungeon Master's guide that flanking awards advantage I play with a highly casual game but I think flanking granting advantage is too too much of a bonus and so I'm going to switch it from advantage to +2 to hit and let's talk about why advantage as you probably know means you're gonna roll to 20 sided dice and you're gonna take the highest one and use that as your attack roll but what does that mean mathematically well you may be surprised to learn that mathematically rolling to 20 sided dice and taking the highest instead one means on average you're plus 5 more to hit if we assume your players are attacking with the stat that is their highest stat and that they're proficient with the weapon and they've got plus 5 coming out of the gate and then you get them advantage that's another plus 5 there now Plus tend to hit basically which means they would be hitting an armor class of twenty fifty percent of the time that's that's crazy right it's essentially I look at it this way giving your players advantage on the attack role guarantees they're going to hit and I say that even though they'll be people who dooblydoo he'll be like oh it's not a guarantee when they're there they're there correct it's not but I think of it as being guaranteed because armor class does not come in every flavor your players are not going to be fighting creatures in the armor class of three or armor class of six they're going to be fighting characters that bad guys that have mostly armor classes between 13 and 17 and at that point 13 to 17 starting off a +5 to hit giving them advantage which is another +5 that's like plus 10 to it if the armor class is 50 and they only need to roll 5 they're gonna be hitting 75% of the time Haidee now an 80% chance to hit is not the same as a hundred percent chance to hit this is the lesson everyone paying attention to 538 nate silver's forecast of the last election learned is that saying that this candidate has an 80% chance of winning or in fact a 70% chance of winning means they still have a 30% chance of losing that's not how we think we don't think that way when we hear that this cannon has a 70% chance of winning we think oh it's in the bag for our purposes though as Dungeon Master's granting your players advantage is guaranteeing asterisk not a guarantee that they're going to hit and there's something else with advantage that I discovered kind of bugs me a little bit is that because you get to do you know in dungeon on the die this number is no different than this number or this number but in D&D this number is totally different than the others because if I if this five percent chance again which is the same five percent chance as this or this if this five percent chance comes up your players get to roll their damage twice and if they have advantage there's double the odds that they're going to get a twenty instead of it being five percent is 10 percent chance no if you consider the fact that they're getting advantage from flanking which means there's two characters who are on opposite sides of the bad guy which prevents the bad guy from being able to be in two places at once and defend against both of them now you've got instead of two dice each trying to get a 5% chance we've got four dice each trying to get a 5% chance it is now much you've doubled it is now much more likely that your players are gonna start getting critical hits once they start flanking if you grant them advantage from flanking which is why I think in my next game at least when we start we're gonna begin with flanking just giving you plus two which is how it used to work in I think third and fourth edition dandy plus two is just plus 10% and that is a meaningful difference enough for the players to seek out and want to get into dangerous situations just so they can flank and that makes me happy that makes the game more tactical it makes strategy more important but it's not such an overwhelming bonus like +5 from advantage alternatively another thing you could do and I might try this is say advantage gives you an extra d4 on top of your d 20 because mathematically that's gonna come out at about ten percent on average so thinking sitting you're thinking about the idea of allowing my players if they have advantage to it's their role in twenty twenty just add a d4 to their die I want to know how is that going to change the math well I can figure that out and this happens a lot in D&D if I figure out what is the average result on any given die what is the average result on a d4 it's two and a half which again if you are not a math person it may not be obvious why that should be but let's figure it out using a d4 in our example is useful to us because we can easily plot it here in this video we can put a 1 over here we can put a 4 over here and we put the other numbers in between and we want to know what is the average result on a d4 well what is the number in the middle and again there will be people in the dooblydoo talking about mean and median and mode and that's fine if you want to go read that stuff and look into it more I encourage you to do so but for the just for the purposes of the new player that knows nothing about math who just wants to know what is the average result on a d-10 you're looking for the number in the middle and all the diets all have an even number of facings which means there is no number in the middle if instead of a d4 we had a d5 which some people make for games that I don't play suddenly now we do have a number in the middle the 3 is in the middle we have 2 numbers to the left of the 3 and 2 numbers to the right of the 3 so we can confidently say that the average result on a five-sided is 3 we don't have a five-sided dog we have a four-sided die and there is no number in the middle so we're just gonna split the number line in half and put a number here that number would be 2 and a half obviously I think using the number line to understand this is more intuitive but also you could just add up the facings and divide by the number of facings which is how they teach you how to do at school 4 + 3 + 2 + 1 adding up all the facings gets us 10 divided by the number of facings which is 4 10 divided by 4 is 2 & a half so the average result on a d4 is 2 an ASP which translates to 12 and a half percent on average if we're giving our players an extra d4 on top of their day 20 to hit because of advantage of flanking which is really just it's an example I used in order to talk about how to figure out averages on dice but same thing the same process works for a d-10 what's the average result on a d-10 it's 5 and 1/2 what's the average result this is important on D 20 it's 10 and 1/2 it's always half the value so 20 divided by 2 is 10 plus 1/2 a percent 10 and a half this is important to us because for instance in D&D we have to make death saving throws in the target number for death saving throw is and what are the odds of us being at 10 without even calculating percentages you know that you are likely to make it because the average result on a 20 sided die is 10 and a half and the number you're trying to beat is a 10 so without even adding up all the 5% says you know that you are likely to make not by much you are likely to make your death savings know in any given round it's a little better and 50% if you do the math it turns out to be 55% because you're counting all the numbers from 10 to 20 including the tenth intuitively if you're doing the math and you're counting five percents you're always gonna be well I've got a 50% chance tip because twenty from ten is ten times five is fifty but then you'll go back and go wait I am including the number ten because I I still win if I had a ten so technically there's 11 numbers and eleven times five is 55% again I guarantee you I've already it may be very this may be very obvious to many of you but there was a time when it was not obvious to me and playing D&D helped make it obvious so I guarantee you there are some people watching this that I have already lost I lost him a long time ago but maybe you watch the video more than once I'm sure there are other videos out there on this subject but let's talk about what we learned because I actually think we covered quite a lot in this video we now know how to figure out what what is the average result on any given die we talked about what does granting advantage do mathematically we talked about what does giving a player a +1 weapon do mathematically we talked about how to figure out how likely you are to hit without actually going and figuring out what your players stats are and how much does changing a monster's armor class change the likelihood to hit what is the difference in a plus one what's the difference between being you know 55% likelihood and 45% likely did so to my way of thinking we've covered a lot of ground I'm not a math teacher I have no idea whether or not this lesson was useful and it is my hope that if you got lost along the way I encourage you to comment in the doobly-doo and we will all do our best to Shepherd you along because playing D&D is one of the things that makes you better at math you start learning what is what does it really mean to only have a 15% chance of hitting it some it feels very unlikely sometimes but then you will discover it does come up let's all start rolling and there are real-world applications here you will discover when everyone else is acting like something that has a 70% likelihood of succeeding is a sure thing you will be focused on that 30% and you'll have a very real understanding humans do not come built-in with an understanding of odds we are incredibly poor when it comes to judging odds and risk that is not something that our evolutionary heritage has gifted us with but if you play a lot of D&D you will velop a sense for that stuff and it will stand you in good stead throughout your entire life so get started get started doing the math in your head get started figuring these things out and get started working out for yourself what does it mean to give your ghoul bonus to their dexterity so that now instead of needing an armored class being an armed class of 12 they have to be in an armored class of 14 or maybe even 15 you will start noticing significant differences there as opposed to going from 12 to 13 which you probably won't notice that's the video on dice math folks I hope you liked it I hope it was useful for some of you please comment in the doobly-doo below let me know what you think next video we're gonna talk about comics in general and specifically a comic that I wrote and because that is not a D&D related subject I'm gonna make sure that when I upload this video I'm going to be immediately the same time upload another video that is D&D related and also sort of related tangentially to the subject of this comic no spoilers I want to encourage all new Dungeon Master's to come by adventure look up calm I'll put a link in the doobly-doo and also I am still working on the stronghold rules Kickstarter a lot doing a lot of research working on getting an LLC and tax ID and stuff like that and that stuff all takes time but if you want to be alerted when the stronghold Kickstarter goes live there is a link in the doobly you follow it you give us your email and we promise we will only email you once and it's to let you know when the stronghold Kickstarter goes live finally as you know I I do not run ads in front of my videos I'm an independent fantasy author if you want to help support the channel come by my amazon page there's a link to that in the doobly-do and there are lots of links to the doobly-doo it seems like it grows all the time like me wait what thanks for sticking around through the dice math video folks until next time peace out

Channel: Matthew Colville

Views: 290,384

Keywords:

Id: YDjD0Gjtgik

Length: 25min 58sec (1558 seconds)

Published: Mon Oct 02 2017

Reddit Comments

[u/IamFootfungus]

Reading this title, I realized I am a giant nerd... "Dice math, oohhh exciting!"

[u/LoL-Guru]

"This is going to be a short video"

Check video length....just under 26 minutes.

EDIT: As an aside, advantage is not exactly like adding +5 to hit

For instance, if you have a 90% chance to hit and then advantage you do not have a 115% chance to hit (what +5 would give you) You have a 99% chance to hit.

Advantage just takes whatever your odds of missing are and multiplies it on itself because essentially you have to miss twice.

90% hit chance is a 10% miss chance, 10% of 10% is 1%.

If you had a 30% chance to hit and attack at advantage you now have a 51% hit chance (not +25% from +5).

Essentially the worse your odds of hitting the worse advantage becomes

[u/Zaenille]

Surprisingly informative! This changes how I view ACs now.

[u/timnitro]

What about the reverse? How likely is this monster to hit a PC?

Off the top of my head, the lowest AC I can think of is 11 while the highest is 20. So the average is 15.5.

Meaning, a monster with no attack bonus will hit, on average 25%- 30% chance of hitting a PC. But even Kobolds and Goblins have +4 to hit, so actually most creatures are going to hit 45%-50% of the time. Fairly often.

[u/[deleted]]

[removed]

[u/ajholman]

The stuff about advantage being like a +5 is really interesting. I couldn't see how it could be true at first, and when I worked it out it averaged a little over +3. But that ignores the fact that target numbers are much more likely in the middle of the range: your DM probably won't ask you to roll if you only need 2 or more on the dice and needing a natural 20 is rare. If your target is between 8 and 14 advantage does indeed give you a 23% to 25% better chance.

[u/powerfamiliar]

This is why I dislike dicepools (specially with opposed rolls or variable TNs), or variable dice (savage worlds like). It's incredibly hard for a DM to gauge the players probability of success on any given test on the fly.

[u/fulllotusyoga]

So I crunched the numbers on advantage and got the following:

ROLL % ADV%

20+ 5 9.75

19+ 10 19

18+ 15 27.75

17+ 20 36

16+ 25 43.75

15+ 30 51

14+ 35 57.75

13+ 40 64

12+ 45 69.75

11+ 50 75

10+ 55 79.75

9+ 60 84

8+ 65 87.75

7+ 70 91

6+ 75 93.75

5+ 80 96

4+ 85 97.75

3+ 90 99

2+ 95 99.75

1+ 100 100%

AVG% 52.5% 69.125%

AVG# 10.5 13.825

Giving an average advantage boost of +3.325 - about the same as adding a D6 to the D20, though that wouldn't boost the odds of a crit.

Any corrections? I've heard advantage is equivalent to +4 and Colville said +5 but this math says otherwise.

[u/bokodasu]

I thought this was a great video - I remember finally learning all this stuff sometime in the 3.5 era, which meant probably 20 years of picking random numbers for monster stats and having no idea of what they'd do. And yeah, I learned it because I had a DM friend who explained it almost exactly like this. Statistics was the only course I dropped in college because I just could not wrap my head around it - I credit D&D for making me finally understand all that stuff.