Let’s just start by saying it’s ok for gamers to have superstitions about dice. It’s ok to retire dice that misbehave, or even “dice shame” them. It’s ok to “cleanse” dice with sage or a salt water bath, which was popular when I first started playing a few decades ago. Dice are our weapons and our tools, our characters live and die by them. It’s ok to add a little ritual to their use. That said, we often give dice qualities they don’t actually have. Let’s examine some of the odd quirks of randomness and maybe give you a slightly better edge then setting all your dice on the table with their highest values showing. What’s that supposed to do anyway?
I wrote this piece with the helpful guidance of Glen Whitman. Glen is a writer and an economist who’s capable of doing the statistical math behind what happens when you roll dice. He’s also a genre television writer, having written on The Strain and FRINGE. To top it off, he is a gamer and knows all too well the lies we tell ourselves about dice. So, thanks to Glen in advance for double-checking my work.
I HAVEN’T ROLLED A CRITICAL HIT IN A WHILE… SO I’M DUE
Probably the most common myth or misconception about dice is the idea that a number that hasn’t come up in a while is somehow more likely to show up in subsequent roles. This is called the “Gambler’s Fallacy,” and it amounts to the idea that if you’ve rolled your 20-sided die a bunch of times and never gotten above a 10, the next roll has some sort of statistical (or fated) reason to be higher.
The fact is the die (provided it’s fairly balanced) has as much chance of rolling any number as any other number every time you roll it. On a 20-sided die there is a 5% chance of rolling a 20, a 5% chance of rolling a 2, a 5% chance of rolling a 16.That probability doesn’t change between rolls. The die doesn’t have a memory, and its probability doesn’t get altered by its history, or for that matter, how badly you need to critically hit that orc.
Of course that die could roll a 1. It could even roll a 1 six times in a row. If you’re wondering, the chances of that are 0.0000015625%, or 1 in 64 Million. This is where things get confusing. You can calculate the probability of rolling a die multiple times and getting the same result every time, and it’s going to be a very very low number. However, that math only holds until you roll the dice. Let’s say you just rolled your fifth natural 1 in a row on your D20. This is not your day. But you probably don’t have to worry about rolling a 1 on your next turn because, after all, the probability of rolling six 1’s in a row is 1/64,000,000 right? RIGHT? Sadly, no. The chances of rolling a 1 on your next roll are still 1 in 20 (5%), just like any other roll. The previous rolls simply don’t matter.
So while you can calculate the probability of any number coming up six or sixty times in a row, that’s only a statement about the future. Once you’ve rolled the actual dice, anything that’s already happened is in the past. Previous rolls have no effect on the future behavior of fair dice. Neither does cursing at them or throwing them across the room, in case you were wondering.
JUST HOW ADVANTAGEOUS IS ADVANTAGE?
If you’re not yet playing D&D 5th edition, “advantage” describes a situation where a player can roll two dice instead of one and take the higher result. “Disadvantage,” by contrast, describes a situation where a player must roll two dice and take the lower result. What effect do advantage and disadvantage have on your odds of success?
Let’s take a practical example. Say you’re a second level rogue with a strength score of 14. You’ve just picked the lock on your jail cell and stolen a weapon you’re not familiar with from the sleeping guard. Now he’s woken up and you’re trying to hit an enemy dressed in plate armor (AC18). You’re rolling at +2 ( your ability score bonus, because you can’t add proficiency with the Maul you took in your escape) so you need to roll a 16. Okay, that was a lot of detail, but the math we’re about to do will work for any example where you need to roll a 16.
Let’s start with just one die. While the chance of rolling a specific number on a twenty-sided die is always the same (5%), you can calculate the chance of hitting a target number or higher with a little easy math. Just multiply 5% by the number of sides that will do the trick. For our example, since you need to hit a 16, there are five sides with 16 or better, so that’s a 25% (5 x 5%) chance. (If your target were 12, there are nine sides with 12 or better, so that would be a 45% chance.)
So how does a second die affect the probabilities? It’s actually easier to start by thinking about disadvantage. Let’s say the guard you’re now attacking with his own weapon chooses to take the “dodge” action for a round while he gets his wits about him. With disadvantage you have to hit a 16 or better not once, but twice. Now imagine you throw the dice in sequence. You throw the first die. There’s a 75% chance you lose immediately, and a 25% chance you’re still “in it.” Suppose the latter happens. You still have to throw another die, with a 25% chance of winning. So one quarter of the time, you have a one quarter chance of actually winning. One quarter of one quarter is one sixteenth (1/4 x 1/4) or 0.0625 (a little over 6%). That’s not a very good chance. Extending the logic, you can always square your one-roll probability to find your probability of doing it twice. Thus, if you need to hit a 12 with disadvantage, you need a 12 or better twice, or 0.45 squared, which is 0.2025 (about 20%).
Now let’s use the same example–your 2nd-level rogue–but this time assume he has the advantage. Maybe you’ve been saving up a point of inspiration to get you out of a jam. The fight seems like a great time to use it. With disadvantage, you had to win twice. With advantage, it means you have to fail twice. So the math is easier if you start with your chance of failure. The chance of hitting a 16 once is 25%, which means the chance of missing it is 75%. For the odds of missing twice, we square 0.75 to get 0.5625 (about 56%). That means your chance of not missing twice–that is, your chance of hitting with advantage–is 1 minus 0.5625, which equals 0.4375 (about 44%). Extending the logic, you can always square your one-roll probability of failing to find your probability of failing twice, and then subtract that from 1 to get your probability of hitting with advantage.
You’ve probably already guessed that the bonus probability you get from advantage is less for higher target numbers. Rolling at advantage to hit the number 19 only adds a little under a 10% bonus to your chances, whereas when rolling for 16 the bonus is almost 19%. So if your DM has awarded you “inspiration” which you can trade in for advantage, you actually get the most practical effect from using it when you have a target number in the middle of the range. Or you can use it to try and hit a natural 20 which is only about a 4% added chance, but hey, dream big.
I’d include a whole table of the advantage math here but a little searching online turned up a place that’s already done that. A presumably very smart guy named Bob Carpenter did a great write up on all things advantageous and disadvantageous here. Check it out if you want to know more.
GREATAXE v. GREATSWORD: DAWN OF PROBABILITY
If your character uses heavy weapons, you might be trying to decide between wielding a Greataxe or a Greatsword. The axe rolls a 12-sided die while the sword rolls two 6-sided dice. Just how much mathematical effect does this have? If you’re like me when I was 10 years old, you might think, “The great sword is clearly better because it has no chance of hitting a 1.” But that’s not the right way to think about it.
It turns out that statistically the greataxe has a greater chance of hitting extremes. That is to say, it produces both high and low numbers more often than the sword. The axe has a 1 in 12 chance (about 8.3%) of any result, including a 12. The greatsword, which has to come up as a six on both of its two dice to hit the number 12, has only a 1 in 36 (about 2.7%) chance. On the other hand, the greatsword has a much larger chance of rolling numbers in the middle of the range such as 7. Anyone who has played Settlers of Catan knows why this works. There are more combinations of two dice faces that add up to 7 then anything else (1-6, 2-5, 3-4, and the reverse of all of these). So the greataxe will more often do the most damage, but also more often do the least damage. The greatsword brings in more consistent results.
So off the top you can choose the weapon that most fits your playstyle. If you take risks but want big rewards, you can take the greataxe. If you want to chip away at enemies and know you’ll get there in the end, choose the sword. Or you could carry both and draw whichever one suits the situation. If you’re hanging onto life by a string but you expect your opponent has 11 hit points left, choose the greataxe. It has a higher chance of delivering the 11 or 12 damage you need to kill your opponent before you take a hit that will put you down. If you have hit points to burn, then choose the greatsword because you’re much more likely to do at least 11 damage over two hits. Of course as we learned from the “gambler’s fallacy” above, all this math could mean nothing and you could roll nothing but 1’s forever–but it’s still better to know the odds.
If you want to keep theorycrafting, you can start figuring in the optional fighting style for some classes called “Great Weapon Fighter.” That would let you re-roll any 1s or 2s on the damage dice, but you have to keep the results. Or take a look at the effects of critical hits, which double the damage dice you roll. These abilities and features ramp up the damage in a big way, but in the end, you’re still looking at more risk with the greataxe and a safer, more consistent greatsword. I guess what I’m saying is that four greataxes would make a much more interesting reenactment of the movie “The Hangover.” Wait? Why am I saying that? If you want to really mess with the math, look at Half-Orcs and Barbarians which both have features that add one die to critical hits. Yes, that’s just one die which means the consistency of the greatsword just got a little less impressive.
You can also extend the basic lesson here to other multiple dice rolls. With a single die, all outcomes are equally likely, but that’s not true with multiple dice. The more dice you throw, the more likely you’ll get a middle of the road result (on average). For example, if you’re a 3rd-level wizard casting a Magic Missile with a second level spell slot, you’ll be firing four missiles at 1d4+1 damage each. Obviously, four hits is better than one. But with the larger number of missiles, don’t assume that 20 damage (a roll of 16 + 4) is very likely. The most likely result from the dice is actually 14 (a roll of 10 + 4).
If you want to check out the math on these things I can get you started with this link to a very informative reddit post that does the hard math on Great Weapon Fighter (and much more).
Between starting and finishing this article, I actually had my regular weekly Dungeons and Dragons game. I hadn’t intended to be watching the dice as I played, but after a while I caught myself using one of my D20s for skill checks and another to attack. I remembered that weeks earlier I had decided that one particular die, a black D20 with white text, seemed to perform well on skill checks but not when attacking. Even after a week of doing research on how dice worked, that still made more sense in my head then anything else. When the next perception or investigation check came around, I picked up that die and used it. Why? Well I might know it’s not based on facts or fate, but if it works, it works. So choose the dice you like. It’s a game after all. Heck it’s a game with magic in it so you might as well enjoy the rituals that you and everyone else brings to the table.
Thanks again to Glen Whitman for all the help. On top of checking out his episodes of FRINGE you can also pick up his book that describes economics principles with the useful tool of zombies. Its called Economics of the Undead and it’s on Amazon.
What dice rituals do you have? Have you ever done the math on an attack or a spell before you decided to cast it? Seriously tho aren’t you due for a 20? Tell us in the comments.
Header Image Credit: Bryan Forrest, Used with Permission.