A GUIDE TO PROBABILITIES

Each roll of a die can seal a character’s fate.

Hedge your bets by understanding probabilities.
A REFERENCE FOR UNDERSTANDING DIE ROLL PROBABILITIES
IN 5TH EDITION DUNGEONS & DRAGONS

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J M Gunnarsson
 JAMJIE’S BOOK OF ODDS
A GUIDE TO PROBABILITIES

BY J M GUNNARSSON

Revision: 1

Template created by William Tian

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K
NOWLEDGE IS POWER. WHETHER YOU ARE A DM CAREFULLY PLANNING AN ENCOUNTER OR A PLAYER
trying to get smarter about the risks you take, it pays to know the odds. This module provides a
wealth of references to understand not just normal d20 odds, but also more complex situations
like advantage and disadvantage, falls from great heights, and effects of the bless, guidance, and resistance
cantrips.

While we all have a good general sense of the odds of succeeding on a DC15 Strength (Athletics) check
with a +3 modifier (45%), most of us find our brains a bit more stretched when trying to figure the odds
that a skeleton with a full 13 hit points will survive after falling 30 feet. (Keep in mind that skeletons are
vulnerable to bludgeoning damage—and by the way, if you’re ever fighting skeletons near a cliff, try
giving them a push—its odds of survival are a meager 9%.)

Perhaps the most useful part of this reference is the section on random ability scores. These rolls are
perhaps the most important of an entire campaign, as they steer the trajectory of a character’s fate. Many
players and even DMs are in a sense afraid to leave something so important up to chance. But I think this
section will help not only ease your concerns but even stoke your sense of possibility for achieving more
heroically high ability scores.

For the mathematically inclined, the Appendix contains a primer on basic probability and how the
information in this reference is derived.

I hope this reference of probabilities will help you plan more precise and balanced encounters (or as a
player, help you survive those that are perhaps less fairly balanced). Please reach out to me on Reddit
(/u/fifthstringdm) or by e-mail (jeffrey.gunnarsson@gmail.com) with any feedback, thoughts,
comments, or requests for changes and additions.

— J M Gunnarsson
June 2019

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Introduction ..................................3 Ability Scores .............................10
Table of Contents ........................4 Single score odds .....................10
The d20 ..........................................5 Odds across all scores .............11
Overview of d20 odds...............5 Appendix .....................................13
Guidance and resistance ...........5 The d20 ......................................13
Dice sums ......................................7 Dice sums..................................14

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 are 11/20 = 55%. In general, we can calculate
these odds:

F
ATES OF MANY GREAT HEROES HAVE BEEN Odds that a d20 roll
sealed by the humble d20. As the result yields n or = (21 – n) / 20
executors of these fates, Dungeon Masters greater
must transform overwhelmingly complex
situations into a single number: the Difficulty
Class (DC). This translation of the complex to Cumulative odds are straightforward for a
the definite is a solemn responsibility, one that single roll, but trickier for advantage and
behooves DMs to understand the true odds that disadvantage.
they are putting their players up against.
DISADVANTAGE
This chapter explains d20 odds in various Rolling with disadvantage means rolling twice
situations, providing a formula for each case. and taking the lower result. The odds of such a
Graphs are used to summarize the odds as a roll (or pair of rolls) are harder to calculate. The
quick reference to cut out and hang on your DM formula below gives the odds that a roll with
screen. advantage is greater than or equal to a value:

OVERVIEW OF D20 ODDS Odds that a d20

1.1025 –
check with
There are 20 sides on a d20 die. For every roll of = 0.105*n +
disadvantage
the die, each result has a 1 in 20 chance of 0.0025*n2
yields n or greater
occurring, or 5%. This part is simple, but the
numbers get trickier as we consider the
following questions: A simpler way to get the value is to take the
straight odds for a single d20 roll of n or greater
• What are the odds that a roll is greater and square it. For example, the odds of getting
than or equal to a certain value? 16 or greater are 25%. If the roll is made with
• How do the odds change when disadvantage, the odds of getting 16 or more
applying advantage or disadvantage? diminish to 0.25*0.25 = 0.0625, or about 6%.
• How much does a resistance or guidance
cantrip help a roll’s odds? ADVANTAGE
The answers to these questions are in the When rolling with advantage, a player rolls
twice and takes the higher result. In this case,
following sections.
the odds are given by the formula below.
CUMULATIVE D20 ODDS
Odds that a d20 check 0.9975 +
When a player rolls a d20, they are trying to
with advantage = 0.005*n –
achieve a certain value or higher. A goblin’s
Armor Class (AC) of 15 means that a fighter
yields n or greater 0.0025*n2
with a +5 attack bonus needs to roll at least a 10
to hit—but an 11, 12, 13, or 20 are okay too.
GUIDANCE AND RESISTANCE
The probability that we care about, then, is the
cumulative probability of several outcomes: 10, The guidance cantrip allows its caster to add a d4
11, 12, 13… 19, 20. There are 11 possible d20 to any ability check. Likewise, the resistance
outcomes in all that will yield a hit (10 and cantrip allows a d4 adder to any saving throw.
above), and 9 that will not (9 and below). In this The odds get pretty tricky in these cases, so refer
case, the odds of the fighter hitting the goblin to the graphs on the next page for the odds.

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 tedious, especially as the type and count of the
rolled dice increase. The short story is that it is
most useful to let a computer do the heavy
lifting. The graphs on the following pages are

T
HE MIGHTY FIREBALL SPELL CAN EMBLAZE
generated from numerical results, not a formula.
targets with a scorching 8d6 fire damage.
We know the average roll result would be READING DICE SUM GRAPHS
28 damage (8 rolls times the average d6 value of The graphs show the probability (on the
3.5). So what are the odds that an intrepid young vertical axis) that the total of a particular roll is
fighter with only 22 hit points can survive the greater than or equal to a given value (on the
magical conflagration? And on the other horizontal axis). For example, suppose a wizard
extreme, how likely is the grim circumstance of directs all three of his magic missile darts at a
the fighter taking the 44 hit points of damage goblin with 7 hit points. Each missile does 1d4+1
necessary to kill her outright? force damage; the +1 modifier already gives the
The answer is that she has just shy of a 10% wizard 3 damage. To slay the goblin altogether,
chance of withstanding the blast—grim odds, to the player’s 3d4 roll must be 4 or more. In the
be sure. But on the blazing bright side, her odds graph below, we can see that the odds of this
of a one-hit instant death are only about 1 in occurring are about 98%. Knowing this high
3000. likelihood, the wizard may be wiser to direct
only two of his darts at the goblin and steer the
So where do these odds come from? The third at another foe. With only two darts, his
answer is convoluted. In fact, the answer is odds of killing the goblin are still 60%—good
convolution. The Appendix gives a rough odds for any clever young wizard.
outline of what this operation entails, walking
through an example for the sum of a 2d4 roll.
However, the operation is extraordinarily

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 SINGLE SCORE ODDS
Let’s recap how ability scores are randomly
generated. A player rolls 4d6, then ignores the

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O DUNGEONS & DRAGONS TRADITION
lowest result and sums the others to get a score.
immerses players in the stakes of
This process is repeated until the player has 6
randomness quite like the ritual of rolling scores, which they can then assign to Strength,
for ability scores. Indeed, many DMs and
Dexterity, and so on.
players bristle at the idea of leaving something
as important as ability scores to chance, opting The graph below shows the odds that a single
instead for the static scores or point-buy options score is greater than or equal to a given value.
provided in the Player’s Handbook. But random
ability scores yield characters whose very stories WHAT THE ODDS MEAN
are bound up in their unique combination of From the “Single Ability Score” graph, we can
talents and shortcomings. already answer a few questions:

Still, the least a DM can do is help the players • The odds of getting a score less than 8
understand the odds. This section can answer are very low—only about 6%. A score of
many questions players might have: less than 6 almost never occurs.
• The highest default score of 15 is not
• What are the odds that my scores will be
that unlikely. The odds of getting 15 or
better than the defaults or point-buy?
more are about 22%.
• What if all my scores are low?
• Perhaps disappointingly, players are
• Do I have a shot at getting one really very unlikely to hit it big with a
high score? or a really low one? maximum score of 18, the odds of which
The good news is that, when comparing to the are only about 2%.
default scores, the odds are (slightly) in the
players’ favor when rolling for random scores.

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 the odds for getting a totally lame (or totally
ODDS ACROSS ALL SCORES heroic) character really are.
Few players will mind getting one or two low
scores. What really worries them is that they WHAT THE ODDS MEAN
will end up with many or most of their scores Viewing the odds in this way is a bit obtuse, so
being abysmally low. (After all, we humans are let’s think through an example. If a player is
far more afraid of loss than we are hopeful of worried that none of his scores will be 15 or
gain.) So rather than looking at the odds of just higher, his DM can remind him that the odds of
one score, let’s consider the odds across all six. that occurring are only 20%. On the other hand,
the DM might be worried that his players end
This graph shows two types of odds: First, the up overpowered: What if a player rolls all high
probability that all six scores are less than a scores? Again, the odds are reassuring: The
given value; second, the converse: the odds that probability of a player rolling all 6 scores of 12
all six scores are greater than a given value. or higher are only about 5%.
These odds give us a high-level picture of what

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COMPARING TO STATIC SCORES (horizontal axis), and includes the examples we
The most pertinent question a player or DM have given here for reference:
might have regarding random ability scores is • Default: The odds of rolling a set of
how the odds compare to the default scores and scores whose average is greater than or
the point buy system. equal to the default scores are 62%.
To answer this question, we can look at the • Point-Buy: A lucky 39% of players will
average value of all ability scores. For the end up with an average they never
default values, this total is equal to could have attained with the point-buy
(15+14+13+12+10+8)/6 = 12. For a commoner, a system. On the other hand, an unlucky
generic NPC stat block in the Monster Manual 24% won’t even end up with the lowest
whose ability scores are all 10, the average is of point-buy average.
course 10. For a player rolling for random ability • Commoner: As you might expect for a
scores, the maximum possible average is 18 heroic adventurer, the odds of ending
(though they are far more likely to be struck by up worse off than a lowly commoner are
lightning—twice—than to get scores so high). miniscule: less than 3%.

The point-buy system is a bit more Not surprisingly, even that tiny 3% chance of
complicated than the default scores, but it’s easy winding up worse than a commoner is enough
enough to work out the range of average scores: to scare off some risk-averse players. But
(13+13+13+12+12+12)/6 = 12.5 is the maximum, consider the other extreme: The probability of
while (15+15+15+8+8+8)/6 = 11.5 is the getting average scores of 13 or more is almost
minimum (unless a player chooses to buy less 30%.
for some strange narrative reason).
What courageous adventurer wouldn’t act on
The graph below shows the probability those odds?
(vertical axis) of a given average of ability scores

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 another way, the player must roll two failures in
order to fail the action altogether.

The probability of a single d20 failure is simply

This appendix provides more detailed 100% (or 1) minus the probability of a success:
mathematical derivations of the formulas given
:::::::::
𝑃*+ (𝑛) = 1 − 𝑃*+ (𝑛)
throughout this module.
In order to fail a roll with advantage, a player
THE D20 must fail twice. The odds of failing twice are
simply the squared odds of failing once:
When rolling a die with N sides, the probability
of rolling a particular value greater than or ::::::::: * *
;𝑃 *+ (𝑛)< = ;1 − 𝑃*+ (𝑛)<
equal to n is:
*
𝑛−1 ::::::::: * 𝑛−1
𝑃" (𝑛) = 1 − ;𝑃 *+ (𝑛)< = =1 − 51 − 6>
𝑁 20

For example, on a d20, the probability of * 𝑛−1 *

:::::::::
;𝑃 *+ (𝑛)< = 5 6
rolling 16 or higher is 20
16 − 1 15 The probability of succeeding with advantage is
𝑃*+ (16) = 1 − =1− = 25%
20 20 100% (or 1) minus the probability of failing that
we just derived:
If a character has an attack bonus of +3 and is
attacking a creature with an AC of 19, they need :::::::::
𝑃*+,?2@ (𝑛) = 1 − ;𝑃
*
*+ (𝑛)<
to roll a 16 on their attack, meaning their odds of
hitting are 25%. 𝑛−1 *
𝑃*+,?2@ (𝑛) = 1 − 5 6
20
DISADVANTAGE
When a player has disadvantage, then they This formula can be expanded to yield
need to roll this result twice to achieve a success. 𝑛* − 2𝑛 + 1
The probability of obtaining a result of n or 𝑃*+,?2@ (𝑛) = 1 +
400
higher on a d20 roll with disadvantage is simply
the squared probability of a normal roll: If we use decimals instead of fractions,

𝑛−1 * 𝑃*+,?2@ (𝑛) = 1.0025 − 0.005𝑛 + 0.0025𝑛*

𝑃*+,234 (𝑛) = 51 − 6
20 Again, this expression is the form given in the
This formula can be expanded to yield main section.

𝑛* − 42𝑛 + 41
𝑃*+,234 (𝑛) = 1 +
400
If we use decimals instead of fractions,

𝑃*+,234 (𝑛) = 1.1025 − 0.105𝑛 + 0.0025𝑛*

This is the expression given in the main

section.

ADVANTAGE
With advantage, only one successful roll out of
two is needed for the action to be successful. Put

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 1 2 3
DICE SUMS 2 2 4
For the roll of a single die with N sides, the 3 2 5
probability of getting any single value is just 4 2 6
1/N. For example, on a d4, the probability of 1 3 4
2 3 5
rolling a 2 is 1/4. Similarly, the odds of rolling a
3 3 6
1 are 1/4 as well. We can write out these odds in
4 3 7
a table: 1 4 5
d4 result Probability 2 4 6
1 0.25 3 4 7
2 0.25 4 4 8
3 0.25
4 0.25 Now, we can just count up how many times
each sum result occurred and divide by the total
This probability distribution is called a number of possibilities, 16, to get the odds:
uniform probability distribution, because each
2d4 result Occurrences Probability
possible value is equally likely. 2 1 1/16 = 6.25%
What if we roll 2 dice? We can list the possible 3 2 2/16 = 12.5%
4 3 3/16 = 18.75%
sum results—for example, a 3 is possible, but a 1
5 4 4/16 = 25%
is not, and the maximum value is 8—but what
6 3 3/16 = 18.75%
are the odds of each value? 7 2 2/16 = 12.5%
2d4 result Probability 8 1 1/16 = 6.25%
1 Not possible
2 ? Do you see the pattern? The odds start low,
3 ?
ramp up to a peak, then ramp back down for the
4 ?
high values. The distribution is no longer
5 ?
6 ? uniform as it was for a single roll.
7 ? Obviously, this was a tedious calculation. If we
8 ?
want to repeat this for a third d4, there are now
9 or more Not possible
43 = 64 possibilities. On the extreme end, a large
roll like 8d6 has 68 = 1,679,616 possible
We can also intuitively figure out that some outcomes. For larger numbers of dice, the
values are more likely than others: 5, for probability distribution tends toward a
example, can result from several pairs of rolls: 1 Gaussian (a bell curve).
and 4, 2 and 3, 3 and 2, or 4 and 1. On the other
The general process we just went through is
hand, a result of 2 can only happen if both dice
called convolution. It is an ugly, time-
yield a result of 1.
consuming mathematical operation that is better
One way to figure out the exact odds of each left to computers, especially when looking at
result is to write out every possible combination larger die types and larger numbers of dice.
of individual d4 die rolls:
The graphs given in this module for the sums
First d4 Second d4 Sum of various dice rolls are calculated with a
1 1 2 computer using this method of convolution.
2 1 3
3 1 4
4 1 5

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