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Probability

This article was first broadcast in Episode Twenty-One on 25th April 2018.

Ryu: Are you sure about this?
Lennon: I think we have to
Ryu: I’m getting the hat
Lennon: No, no… Let’s just let him have this one.
Ryu: You’re REALLY sure? I mean, last time some people didn’t wake up for days.
Lennon: Well, we can dress it up as a public service. Insomnia is a real issue for a lot of people. Now, breathe in… and out… ok, go get em!
Ryu: Okay… Hey Ostron? What’cha been working on in the Gnomish Workshop?

 

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With all of this talk of Mordenkainen’s Mayhem, and the Mike Mearls Happy Fun Hour, it got us thinking about designing creatures, and then how much chance the players had to defeat them. You see, there’s a quote about Las Vegas that comes from comedian Penn Jillette: Las Vegas is a city built by tourists with bad math skills.

This concept was usually described in the show he and Teller gave in the city itself. Despite being tongue in cheek, it’s not wrong. All of the games in Las Vegas have odds that are heavily skewed toward the casinos, but people keep gambling. Usually there’s one of three reasons: they enjoy the gambling and don’t care about winning, they’re suffering from gambling addiction and need to seek treatment, or they believe the odds don’t apply to them.

All of that is wonderful, but we’re here to discuss D&D, not gambling, so who cares? Well, here’s the reality: a lot of the frustration people have in D&D comes from the same problem: players or DMs who lack an understanding of the math. Fortunately, Ostron is known to haunt people’s dreams with his dice math, so we thought we’d take a moment to clear the mists for you.

Because D&D involves dice, it involves probability. Fortunately, unlike Vegas, the game is not inherently designed to screw you over (note: if your DM is actively trying to screw you over, understanding probability is irrelevant. We suggest cookies, or possibly pizza).

Every die is, at heart, a probability equation. On a d6, you have a one in six chance of rolling any given result on the die. On a d20, it’s a one in twenty chance, which means theoretically for every twenty attempts, it will succeed an average of once. Simple!

Now here’s where many, many people get in serious trouble. The key word in that last sentence is “theoretically”. Nowhere does it say that if you roll a d20 twenty times, you are guaranteed to roll a 20 at least once. If nothing else, you are or you know someone for whom this is demonstrably not the case. Think of how lucky Critical Roll’s Taliesin is vs. how unlucky Wil Wheaton is. The average we were talking about is the average of every d20 roll everywhere in the multiverse across the entire history of time. It is not true when you zoom that view down to an individual person.

The principle that governs all of this is called “The Law of Large Numbers.” It has nothing to do with the average damage an ancient gold dragon can output and everything to do with the idea that if you repeat a task that has an expected average over and over again, the results will eventually trend toward the average. If you want to test this out for yourself, keep track of the result of every d20 roll made at your gaming table. For a year. Or you can find a computer that will do it for you. That “For a year” is the key bit. If you only consider one game session, for example, you might conclude that certain players have no luck at all, or that d20s never roll 11s or something like that. After an entire year of gathering data, you should see that each value comes up about as much as any other, but there’s no pattern to when that happens.

Failure to understand that leads to something called the gambler’s fallacy — the idea that because you’ve rolled a string of low numbers, that probability dictates you have a higher chance of rolling a higher number next right? Afraid not. To use a D&D example, many Paladins hold off on using smite because they want to wait for a critical, where their smite damage rolls will also be doubled along with their regular weapon damage. If they haven’t rolled a critical yet, they will hold off because they’re sure one is coming. It’s been so long that one has to be coming up soon, right?

Nope.

There is nowhere in the laws of D&D or the universe that guarantees a critical roll will appear (except on Thursdays, of course). Further, statistics and probability say there are a non-zero number of players who will never roll a critical in their entire lives. There are even more people for whom critical rolls will be a true rarity, showing up once every year or so. These people deserve our pity.

So what’s the point, other than to depress our pessimistic listeners? The point is that you should not base any of your playing decisions on the idea that your past failures means that you’re due to succeed. Players often believe after a string of low skill check rolls, their chances of rolling higher increase. This is also incorrect. So what is the solution? As a DM, what can we do to ensure players don’t feel frustrated and cheated? Easy — change the math.

Modifiers

We previously covered a lot of different dice modifiers way back in Episode Four when Ryu first found that santa-hat with the words “Killer DM” on it, but to summarise, in general modifiers can be lumped into two major categories: modifiers to the roll and additional dice. Modifiers are the easy part. If you have a higher modifier to the roll, your chances of success improve. If you’re making an unmodified attack against a creature with an AC of 16, you have a 25% chance of hitting. Now if you’re wondering where that 25% number comes from, let’s break it down. Each value on a d20 is worth 5% — after all there are 20x 5s in 100, and 20 sides on a d20, so each side is worth 5%. To make a successful attack against a creature with an AC of 16, you have to meet-or-beat, so you need to roll a 16, 17, 18, 19 or 20 — five distinct values that would win. As there are 5 values, and each value is worth 5%, that means you have a 25% chance of hitting the creature. Now, if you had a +5 modifier to the attack, you would only need to roll an 11 or higher (as 11+5 = 16), meaning there are 10 possible winning values (11 through 20), and so you have a 50% chance instead. Similarly, the Fighter Champion’s much derided bonus to critical chance, where they can crit on a 19 or a 20 is mathematically significant. The 5% chance of critting now becomes a 10% chance as there are two values that can cause a crit.

Additional Dice

Additional dice is where things get complicated, and it only matters if you’re adding the dice together. Some people think that the only thing that matters with multiple dice is the totals. This is why a lot of people assume rolling 2d6 is roughly equivalent to rolling a d12, because the maximum value is still 12. But statistically that’s not true. With 2d6 there are multiple ways to get certain values. With a d12 you have a 1 in 12 chance of rolling a 7 — there is only a single 7 on that die. But with 2d6 you have multiple chances of rolling 7 — 6+1, 5+2, 4+3, and then of course 3+4, 2+5 and 6+1. This actually gives you 6 chances of rolling a total of 7. And what’s more, you’re actually less likely to roll a 12 with 2d6 than with a d12, because you have to roll 6s on both dice, not just on one. Oh, and there’s no possible way to roll a 1 on 2d6.

If you add dice to a roll, you usually improve your chances of rolling middle values and reduce your chances of rolling a maximum. This is why the “Bless” spell has a deceptively significant effect. For those unfamiliar with the spell, Bless targets up to three creatures, and then any attack rolls or saving throws that creature makes before the spell ends can roll an additional d4. The d4 means the most likely totals of your roll will add up to 5 through 21, but also doesn’t decrease your chances of rolling a critical, because critical rolls only depend on the d20. To relate that back to the tabletop, if you have a +5 to your attack roll, and you’re Blessed, you have an 85% chance of hitting most of the creatures Wizards has released so far.

Advantage and disadvantage are more complicated mathematically. There is a link in the show notes that explains the probability calculations and the values for those who are interested., but the short summary is that advantage gives you an 84% chance of rolling a 9 or better, and a nearly 45% chance of rolling a 16 or better. Disadvantage is exactly opposite (because that’s how math works). The biggest thing with advantage is that it provides almost a 10% chance of rolling a 20. That means any character making an attack with advantage effectively gets the Champion Fighter’s class ability for rolling criticals.

Common Damage Rolls

If you’re sitting behind the screen, these probability principles are definitely a lot easier to see, simply because you’re rolling dice more often. If you look in the Monster Manual, the monsters all have a quick damage value to use if you don’t want to roll the dice. Those values represent the most common damage rolls, and the reason they can be used without unbalancing the game is the Law of Large numbers mentioned earlier: as more and more attacks hit from the same creatures, the average damage done will trend toward the quick damage value. That also means if you want to save time by not rolling damage but you don’t want to do the same damage each time, you can use a value one or two above or below the given value without upsetting the balance much.

However, the Gambler’s Fallacy can crop up too. If you’re still rolling the dice and assuming it will be near the quick damage value every time, you could end up in trouble. A creature that does 2d10 damage every hit is still capable of rolling 2 or 3. If you’re relying on the creature doing heavy damage to make a statement at the start of combat, you may want to just use the average value to begin with, then go back to rolling later so the characters can feel like they got lucky when the creature rolls low. Conversely, if you want to give the player’s barbarian a nice tap on the shoulder to make them use their healing potion and actually roll the damage, you could end up rolling high values and killing them instead.

 

 

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Links: Advantage and Disadvantage Probability

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