Loaded Dice Probability Calculator (Bias-Weighted Odds)
🎲 Loaded Dice Probability Calculator
Bias-weighted face odds, exact totals, and target chances for custom loaded rolls.
⚖ Bias-Weighted Formulas
Face probability
Each face gets a weight, then the weights are normalized into a real probability mass.
p_i = w_i / sum(w)
Loaded die average
The loaded expected value is the weighted sum of each face probability multiplied by its face number.
E[X] = sum(i x p_i)
Exact total odds
For multiple dice, the final total comes from convolving the same loaded face distribution across every roll.
P(S = T) = coeff of (sum(p_i x^i))^n
🧮 Loaded Die Component Grid
0.00Fair EV
Baseline mean for the current die size.
0.00Loaded EV
Weighted mean after the bias profile.
+0.00Bias Lift
Change versus a fair die.
0Peak Total
Most likely sum in the current pool.
📌 Presets
⚙ Calculator Inputs
Exact target0.00%match chance
At least target0.00%minimum total
Loaded EV0.00expected total
Any high face0.00%per pool
📊 Full Breakdown
📑 Formula Trail
ProfileBalanced
Weight sum0.000
Single high face0.00%
Fair EV0.00
Loaded EV0.00
Bias lift+0.00
Peak total0
Peak chance0.00%
Weighted faces are normalized after weighting, then the pool is convolved to produce exact totals and cumulative odds.
| Total | Exact | At least | Note |
|---|
📋 Reference Tables
| Profile | Curve | EV Trend | Best Use |
|---|---|---|---|
| Balanced | 1:1 | Zero shift | Fair comparison |
| High face drift | Rises up | Positive | Top-heavy rolls |
| Low face drift | Falls down | Negative | Bottom-heavy rolls |
| Center crown | Middle peak | Near fair | Mid faces matter |
| Edge spike | Outer peak | Wide swing | Wild extremes |
| Custom focus | Focus face | User set | One-face loads |
| Face | Weight | Chance | Bias |
|---|
| Scenario | Profile | Dice | Goal |
|---|---|---|---|
| Fair d6 baseline | Balanced | 2d6 | 7 total |
| High drift test | High face | 3d6 | 10 total |
| Low drift test | Low face | 3d6 | 9 total |
| Center crown | Center | 2d8 | 10 total |
| Edge spike | Edge | 4d10 | 22 total |
| Custom focus | Custom | 1d12 | 12 total |
| Target | Exact | At least | Note |
|---|
💬 Tips
Check the lift, not just the total
A loaded die is easiest to compare when you watch the EV shift against the fair baseline.
Use custom focus for one-face loads
The custom profile is best when a single face needs to spike without flattening the rest.
Loaded dice are those where not all sides have equal chance to appear. Two main physical reasons cause that. One of them is the entropy so the number of sides on the cube.
The second relates to asymmetric dispersed weight that affects the gravitational potential energy. You can modify dice in different ways. For instance burdening one side you make the six side more probable.
What Are Loaded Dice and How They Change Chances
Some use shavings or capping for that. That means to remove a bit of material or add a layer for subtle change of weight. Also rounding of edges can affect that the cube favors certain sides.
In some games with dice you use loaded dice to control the game. They allow you for example to get three one-cost units with 75 % probability on the third level. They help also for vertical setups as Dragonmancer if you want triple units.
Sometimes the loaded dice have same probabilities as the shop. When cube simply favors one number as the five every roll has equal chance of success.
For count probabilities of dice rolls you use special calculators. They help to estimate chances for a set of cubes. You can calculate precise number or values less than something or amounts.
This operates for many kinds also D4 (tetrahedron) D6 (cube) and D20 (icosahedron). Two D6 give 36 possible combinations. Probability of a certain number comes of dividing the right cases by means of 36.
That results in a fraction from 36.
Different forms of dice give different distributions. One cube has flat distribution with equal chances. Two cubes give a pyramid.
Three or more form natural bell with rolls around the center. If producer makes loaded dice it can advertise that they fall on a sum of 7 in a third part of time. For control whether cubes are fair you do tests and use clear criteria.
