Savage Worlds Dice Probabilities

This application provides an overview of the probabilities for rolling specific results in the Savage Worlds role-playing game system.

You can create or edit dice configurations and adjust modifiers to see how the statistics change in real time.

Probabilities are displayed in two views:

• Cumulative – shows the probability of rolling at least a certain value.
• Distribution – shows the probability of rolling exactly that value.

Dice System Overview

In Savage Worlds, when a player wants to take an action, they roll two dice:

• Trait Die: ranges from a D4 to a D12, depending on the character’s skill.
• Wild Die: usually a D6. The higher of the two rolls is the result that is taken.

Exploding Dice

When the result of a die matches its highest possible value (e.g., a 6 on a D6), the die "explodes" and is rolled again.
The new value is added to the previous one. This can happen multiple times on a single roll.

Critical Failure

Rolling a 1 on both the Trait Die and the Wild Die is considered a Critical Failure.

Modifiers

Certain modifiers may be applied to the final result of the roll, either increasing or decreasing it.
Characters can have modifiers that apply to specific skills (e.g., +2 to every Persuasion roll) or global modifiers (e.g., –1 to all rolls because the character is wounded).

Probability Calculation

Definitions

$d$ = generic dice value
$t$ = Trait Die value
$w$ = Wild Die value
$R_t$ = wanted result
$m_d$ = configuration specific modifiers
$m_g$ = global modifiers
$R$ = result to be achieved
$E$ = explosions needed to reach $R$
$r$ = result needed after esplosions
$p_{c}(d_1,d_2)$ = probability of getting $1$ on two dice
$p_{=}(R,d)$ = probability of getting exactly $R$ for a die $d$
$p_{\geq}(R,d)$ = probability of getting $R$ or more for a die $d$
$p_{<}(R,d)$ = probability of getting less than $R$ for a die $d$

Relations

$R = R_t - m_g - m_d$
$E = \left\lfloor{\frac{R}{d}}\right\rfloor$
$r = R\mod d$

Critical Failure

A special case where both dice roll a $1$.
This is displayed in both Cumulative and Distribution mode.

$$p_{c}(t,w)=\frac{1}{t} \cdot \frac{1}{w}$$

Cumulative Probabilities

Probability of getting at least the expected result with one die:

If the expected result is less than or equal to $1$, the result is always achieved ($p = 1$).
If the expected result is a multiple of the dice value, the result can be achived by rolling the dice max value ($p = 1/d$) for $E$ times in a row.
In other cases, you need to roll the required number of explosions and than roll at least the remaining value.
So you need to roll an exact value of the dice ($p = 1/d$) for $E$ times in a row, and a value greater than or equal to the remaining value on the last roll ($p = (d - r + 1) / d$).

$$p_{\geq}(R,d)= \begin{cases} 1 & \text{if } R \leq 1,\\\ \left(\frac{1}{d}\right)^{E} = \left(\frac{1}{d}\right)^{\left\lfloor{\frac{R}{d}}\right\rfloor} & \text{if } (R\mod d) = 0,\\\ \left(\frac{1}{d}\right)^{E} \cdot \left(\frac{d - r + 1}{d}\right) = \left(\frac{1}{d}\right)^{\left\lfloor{\frac{R}{d}}\right\rfloor} \cdot \left[\frac{d - (R\mod d) + 1}{d}\right] & \text{otherwise.} \end{cases}$$

Combining probability of both dice:

$A$ = rolling at least $R$ with Trait Die
$B$ = rolling at least $R$ with Wild Die

$$\begin{align*} P(A \cup B) & = P(A) + P(B) - P(A \cap B) \\\ & = P(A) + P(B) - (P(A) \cdot P(B)) \\\ & = p_{\geq}(R,t) + p_{\geq}(R,w) - (p_{\geq}(R,t) \cdot p_{\geq}(R,w)) \end{align*}$$

Distribution Probabilities

Probability of getting exactly the expected result with one die:

Due to the nature of exploding dice, if the expected result is a multiple of the dice value, it's impossible to get that exact number ($p = 0$).
In other cases, you need to roll the required number of explosions and than roll the exact remaining value.
So you need to roll an exact value of the dice ($p = 1/d$) for $E + 1$ times in a row.

$$p_{=}(R,d)= \begin{cases} 0 & \text{if } (R\mod d) = 0,\\\ \left(\frac{1}{d}\right)^{E+1} = \left(\frac{1}{d}\right)^{\left\lfloor{\frac{R}{d}}\right\rfloor + 1} = \left(\frac{1}{d}\right)^{\left\lceil{\frac{R}{d}}\right\rceil} & \text{otherwise.} \end{cases}$$

Combining probability of both dice:

$A$ = rolling exactly $R$ with Trait Die and less than $R$ on Wild Die
$B$ = rolling exactly $R$ with Wild Die and less than $R$ on Trait Die
$C$ = rolling exactly $R$ with both dice

$$\begin{align*} P(A \cup B \cup C) & = P(A) + P(B) + P(C) \\\ & = (p_{=}(R,t) \cdot p_{<}(R,w)) + (p_{<}(R,t) \cdot p_{=}(R,w)) + (p_{=}(R,t) \cdot p_{=}(R,w)) \\\ & = [p_{=}(R,t) \cdot (1-p_{\geq}(R,w))] + [(1-p_{\geq}(R,t)) \cdot p_{=}(R,w)] + (p_{=}(R,t) \cdot p_{=}(R,w)) \end{align*}$$