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April 10, 2018

RPG Math: Calculating Chance to Hit

RPG Math

This time we will be system-specific and look into a valuable formula for D&D 5e, chance to hit. This formula will allow you to make an educated guess for how likely you are to hit your target with your spell or weapon attack. The best part is that the math is even easier than our last post which was calculating the average dice roll! Let’s get to it.

Definitions

Probability

The chance for a specific outcome to happen. When you see the word “chance” in mathematics, it’s safe to assume probability is involved.

Chance to Hit

The likelihood of landing a successful attack on your target. This is calculated as a percentage.

Chance to Hit Formula

Chance to Hit = ( ( 21 – ( Target’s AC – Player’s Attack Bonus ) / 20 ) * 100

This is an extremely simple formula as we merely have to add all the numbers in the numerator together and then divide by 20. Next, we will multiply the decimal value by 100 to put it into the proper percentage format.

Why do we have a 21 in the numerator when we only have a 20-sided die? You will always have at least a 5% chance to hit any target in D&D 5e as rolling a 20 is an automatic hit. Conversely, you will never have a greater chance than 95% to hit as a 1 is an automatic miss.

Examples

Fighter vs. Dragon

For our first example we will be a fighter with a +9 to hit. Its target is an Adult Bronze Dragon with an AC of 19. This is going to be a tough fight, what’re our chances of dealing some damage?

1) Chance to Hit = ( ( 21 – ( 19 – 9) / 20 ) * 100
2) Chance to Hit = ( ( 21 – 10) / 20 ) * 100
3) Chance to Hit = ( 11 / 20 ) * 100
4) Chance to Hit = 55%

55% is certainly not the best odds, but worth a shot. This is a situation where you have found yourself swinging above your weight-class.

Sorcerer vs. Ogre

In our second example we are a sorcerer with a +7 to hit with our spells. Our target is an ogre with an AC of 11, which our front line is keeping at bay for us to unleash our scorching ray.

1) Chance to Hit = ( ( 21 – ( 11 – 7 ) / 20 ) * 100
2) Chance to Hit = ( ( 21 – 4 ) / 20 ) * 100
3) Chance to Hit = ( 17 / 20 ) * 100
4) Chance to Hit = 85%

85% are some great odds! However, this is only the chance for a single ray to hit. What if we wanted to know what the likelihood of all 3 rays hitting was?

While it’s a bit more tricky, all you’ll have to do is take the value of your chance to hit and raise it to the power of attack rolls you are making. In our case, we have 3 rays so we will take 0.85³ which is 0.614125. This is roughly 61.4%. We have an above-average chance at hitting with all 3 of our rays which is pretty rare.

Use Cases

The obvious use case for this formula is to calculate our chance to hit an enemy. However, we can also use it to determine our likelihood of success on ability checks and saving throws in D&D, given that we know the DC. Simply change the formula to the following:

Chance to Succeed = ( ( 20 – ( DC of the check – Ability Score – Proficiency) / 20 ) * 100

Did you notice that we are using 20 instead of 21? Although it is a popular house rule, a natural 20 is not an automatic success on saving throws and ability checks. Likewise, a natural 1 is not an automatic failure. Therefore, our maximum chance to hit with this formula is 100% and our minimum chance to hit is 0%.

Knowing how to use this formula can grant you many advantages in D&D. There are many situations in which missing an attack or failing a check is worse than attempting the action. Having the ability to know when to avoid these checks can be the difference between life and death.

Conclusions

My intentions for this semi-regular series is not to have a lot of D&D 5e-specific formulas. However, I felt that this formula is a great introduction into calculating basic probabilities. Having an understanding of probability will be helpful for the more complex formulas in RPGs! I’ve written a different post on calculating probability before as well.

It is possible to calculate your chance to hit with advantage and disadvantage. However, I will save that for another post where I can go into much more detail on the math as it is much more complicated. I also intend to write an entire post on advantage and disadvantage beforehand.

Depending on the system you are playing this formula could be tweaked based on how your dice rolls, bonuses, and enemy armor works in another system. So long as you have all 3 of these components, you can make a chance to hit formula for that system.

Announcements

I saw on Twitter today that there is an annual contest for creating a One Page Dungeon. For those of you that don’t know, a One Page Dungeon is an entire dungeon that fits on a single sheet of paper. I’ve never made a One Page Dungeon before, but it sounds like a fun idea and I have made plenty of dungeons in my time so I’m going to try my hand at it!

The submissions must be in by May 1st, so I have decided that because I have 3 weeks to work on this I will make a post on my design processes every Friday. If this is something that interests you be sure to follow me on Twitter or sign up for our email alerts using that form on the right-hand side of the site!

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