RPG Math: Calculating Probability

There are a plethora of tabletop RPGs out there. If there’s a genre of story, there’s an extremely high chance that there’s at least one tabletop RPG system made to play it. As you can imagine, this means that there are a ton of different games out there, each with their unique ruleset. It’s difficult to pick one, much less play it regularly!

However, there’s at least one thing that all tabletop RPGs have in common with each other; they all use probability in their ruleset.

Most RPGs use dice. Some use coins or other tools to generate random numbers, but regardless, these are all used to generate outcomes for specific mechanics within an RPG. We as players and GMs are at the mercy of these random numbers.

Though, we don’t have to be. If you have an understanding of how to calculate probability, you can give yourself an enormous leg-up when it comes to making tough decisions based on random outcomes. While it requires a bit of math to do so, it’s not difficult at all.

Let’s get started!

Definitions

Random Number Generator (RNG) – A tool, device, etc. that randomly selects a number from a predetermined set of numbers.

Probability – The likelihood of a specific random event or events occurring.

Independent Event – A random event that is not impacted by another event happening or not happening.

Dependent Event – A random event whose probability can be influenced by other events occurring or not occurring.

Probability Formula

Probability = Desired Event / Total # of Possible Outcomes

I wasn’t lying when I said this wasn’t difficult!

The real challenge of calculating the probability of an event is determining what you’d like to calculate. Honestly, even that usually isn’t difficult either.

Probability Formula That Includes Multiple Events

Multi-Event Probability = Probability of Event 1 * Probability of Event 2 * … Probability of Event n

These calculations do get more complex as more variables are added to the calculation. However, it’s still just multiplication. Though I will concede that fractions do make everything annoying!

Some events will simply never happen. For example, rolling an 11 on a d10 is impossible, so therefore the probability of that happening will always be 0.

Examples

Rolling Maximum Damage With a 3rd-Level Fireball

Fireball is a 3rd-level spell in D&D 5e that deals 8d6 fire damage on a failed save. That’s an average of 28 fire damage per cast! It’s also an AoE so it’s an exceptional AoE damage spell.

28 fire damage is nice and all, but we’re not average. We don’t settle for just that. We want it all! What’s the likelihood of us rolling all 6’s on our 8d6?

1) Probability = 1/6 * 1/6 * 1/6 * 1/6 * 1/6 * 1/6 * 1/6 * 1/6
2) Probability = 1/1,679,616

Let’s just say that it’s not likely you’re going to roll maximum damage. You’re more likely to be struck by lightning this year than your Fireball dealing 48 damage.

Each die roll is an independent event. They do not impact each other in any way, shape or form. This means that it’s pretty easy to calculate as it’s just a bit of multiplication, but the probability of the event becomes less likely as new events are added to the calculation.

For example, if you cast Fireball using a higher-level spell slot, you would be even less likely to roll maximum damage for it.

It’s also possible to determine how likely you are to succeed on the fireball’s saving throw. I’ve written an article on this previously for those interested, but it’s not a relevant event with regards to the possibility of the spellcaster rolling maximum damage on the spell.

anydice graph of fireball's damage output probability
And here’s fireball’s damage output probability in graph form thanks to Anydice!

A Beholder Using Three Specific Eye Rays in a Single Turn

Beholders in D&D 5e have 10 different Eye Rays that they may use as an Attack action. The catch, however, is that the beholder cannot choose which Eye Rays to shoot at the target. You have to roll 1d10’s to determine which rays will be used for this action, and you must reroll any duplicate values.

This is a dependent event since you cannot, for example, roll two 10’s and use two Death Rays in the same turn. So yes, while you can roll two 10’s you’ll be rerolling your d10 until you get a unique value.

As the DM you may want three specific rays to be rolled because some of their rays are better than others. But what are the chances of this?

1) Probability = 3/10 * 2/9 * 1/8
2) Probability = 6/90 * 1/8
3) Probability = 1/15 * 1/8
4) Probability = 1/120

Quick sidebar, but whenever you are working with fractions look for ways to simplify them as I did in step 3 of this example. Your calculations will become considerably less complex when you do so.

Understandably your chances of getting exactly three of the 10 rays on a single roll are very low. Specifically, you only have a 0.83% chance of using three specific eye rays on a single turn!

It makes logical sense that this event has such a low probability. You want three very specific things to happen at a specific point in time. The more restrictive your constraints are, the less likely the event will take place.

Use Cases

There are a ton of use cases for calculating probability in RPGs! Seriously, everyone at the table can benefit from being able to do basic probability calculations before they make decisions.

One of the most obvious benefits is that players can determine what their chances are for a favorable outcome of a decision they’re about to make. Keep in mind that I’m not suggesting that you all break out the calculators before every decision, but a bit of quick mental math goes a long way!

Probability calculators are also super useful for DMs, GMs, etc. anyone who is going to implement a homebrew mechanic, rule, creature, etc. into a game. Being able to quickly crunch the numbers will give you a good idea as to how likely a favorable or unfavorable outcome is and you can balance item in question as desired.

Game designers for RPG systems are immersed in probability. Each mechanic or rule they create needs to be weighted accordingly to ensure that it has the appropriate outcome(s) that they envisioned it to have.

One specific game mechanic use case is the D&D 5e recharge mechanic. I’ve written an article on how it uses probability to balance an action with respect to the likelihood of an action recharging each turn.

Conclusions

Probability is fairly simple to calculate, but it gets increasingly more difficult as you add more events into a single calculation. Both of my examples were both too complex to do in your head quickly at the table, but something like calculating the odds of rolling a 15+ on a d20 isn’t all that difficult.

For the record, there’s a 6/20 or 3/10 chance of that happening.

It’s also important to be able to distinguish the events you wish to run your calculations on. Your calculations will change slightly depending on if you are using dependent or independent events as we saw in the examples above.

Probability is all around us, and being able to do some basic calculations gives you a chance to look before you leap into a decision. That’s a valuable edge to have for sure!

If you enjoyed what you read be sure to check out my ongoing review for all of the official D&D 5e books!

Sign up to get e-mail updates for new articles on Dungeon Solvers using the form below!

2 Comments

  1. The chance of roll a 15+ on a d20 is not 1/4. There are 6 results, 15,16,17,18,19,& 20 that satisfy the condition of “15 or better”. Thus the probabillity is 6/20 or 3/10, which is 30%.

Leave a Reply