One Die or a Curve? How Dice Shape Predictability

Dice do not make a game predictable or chaotic by themselves. The distribution created by the roll determines how often the middle appears, how expensive extreme results are, and how much a modifier changes the chance of crossing a target.

A d20, 2d6, four Fate dice, and a pool of d6s can all resolve “Does this work?” Yet the same +1 or extra die does not mean the same thing across them. The useful comparison is not the objects in your hand. It is the shape of their possible results and the rule used to read that shape.

Start with three questions

When reading a resolution mechanic, ask:

1. Where is probability concentrated? Are all totals equally likely, or do results cluster?
2. How is success read? Against one target, through outcome bands, or from the single highest die?
3. What does improvement change? The total, the number of dice, a reroll, or the interpretation of a result?

These questions prevent a common shortcut: calling one system “swingy” and another “reliable” without identifying whose reliability, at which target, and under which special rules.

A single d20 is flat

On a fair d20, every face has a 5% chance. For a generic roll-over test with no automatic success or failure faces, moving the final result by +1 converts one previously failing face into a success. At an interior target, that is a fixed increase of five percentage points.

Target on the die	Success chance	With +1	Change
6+	75%	80%	+5 points
11+	50%	55%	+5 points
16+	25%	30%	+5 points

The regularity is easy to explain: +1 always buys one face. But its relative value changes. Going from 25% to 30% is a 20% increase relative to the original chance; going from 75% to 80% is only about a 6.7% relative increase.

Named games can alter this generic picture. D&D’s current Basic Rules, for example, add special natural-face behavior to attack rolls and use roll-two-take-higher or roll-two-take-lower for advantage and disadvantage. Those rules reshape outcomes without changing the d20 itself.

Summed dice create a center

With 2d6, totals are not equally likely. There are six ordered ways to make 7, but only one way to make 2 and one way to make 12.

Total	Ways out of 36	Probability
2 or 12	1 each	2.8% each
3 or 11	2 each	5.6% each
4 or 10	3 each	8.3% each
5 or 9	4 each	11.1% each
6 or 8	5 each	13.9% each
7	6	16.7%

This curve changes what +1 buys. A modifier shifts the threshold across a part of the distribution; the number of outcomes crossed depends on where that threshold sits.

Generic 2d6 target	Base chance	Chance with +1	Change
7+	58.3%	72.2%	+13.9 points
10+	16.7%	27.8%	+11.1 points

At 7+, the +1 captures all five ways to roll a natural 6. At 10+, it captures the four ways to roll 9. Unlike the d20 example, a +1 does not have a constant effect.

This is why a summed-dice system can make small bonuses especially influential near its common middle. The curve suppresses extremes, but it can amplify the practical value of moving a central threshold.

Centered dice make competence louder than noise

Fate rolls four dice whose faces each contribute -1, 0, or +1. The official Fate SRD gives results from -4 to +4 and notes that they occur most often between -2 and +2.

Exhaustive enumeration shows the center more sharply:

Dice result	Ways out of 81	Probability
-4 or +4	1 each	1.2% each
-3 or +3	4 each	4.9% each
-2 or +2	10 each	12.3% each
-1 or +1	16 each	19.8% each
0	19	23.5%

The roll lands from -1 through +1 in 51 of 81 outcomes, or about 63%. Because the dice are added to a skill rating, the character’s rating usually remains visible in the final total. The dice frequently adjust competence rather than replace it with a result from an unrelated part of a wide range.

That does not make every action safe. Difficulty, opposed rolls, invokes, costs, and outcome rules still determine what the final total does. The distribution only tells us how strongly the random component clusters around zero.

Highest-die pools change confidence through redundancy

Blades in the Dark provides a clear example of a different structure: roll several d6s and read only the highest. Its core rules interpret a highest 6 as full success, 4–5 as partial success, and 1–3 as a bad outcome.

For the broad question “Did the pool produce at least a 4?”, the probability is the complement of every die showing 1–3:

Dice in pool	At least one 4+	At least one 6
1	50.0%	16.7%
2	75.0%	30.6%
3	87.5%	42.1%
4	93.8%	51.8%

The first extra die raises the chance of avoiding the lowest band by 25 percentage points. The next adds 12.5, then 6.25. This is diminishing return: more dice still help, but each new die has less unresolved failure probability left to remove.

The chance of at least one 6 grows more gradually. That lets a pool make “something useful happens” reliable before it makes the cleanest outcome routine. A specific game may add criticals, zero-dice rules, stress, position, effect, or consequences, so the table is a lens on the pool—not a summary of the whole system.

The same bonus language hides different operations

“You get +1” sounds portable, but it can mean:

• add one to a flat roll, converting one face at a threshold;
• add one to a summed roll, crossing a variable amount of central probability;
• add one to a skill while centered dice perturb it;
• add one die to a pool, improving the chance that at least one die reaches a band;
• reroll, keep the highest, or change the outcome after seeing it.

Compare bonuses by the probability and decisions they change, not by the printed numeral. A +2 in one system is not inherently “bigger” than an extra die in another.

Choose a shape by the behavior you want

Use a flat single die when you want transparent odds, constant percentage-point modifiers at ordinary thresholds, and a broad spread of equally likely raw results.

Use summed dice when you want common results to gather near a center and want modifiers to interact strongly with thresholds near that center.

Use centered additive dice when you want a rating to anchor the result while randomness usually nudges it and only occasionally pushes far away.

Use a highest-die pool when you want improvement to feel like added confidence, outcome bands to carry consequences, and gains to taper as the pool grows.

These are tendencies, not verdicts. Target numbers, automatic face rules, rerolls, resources, opposition, consequence design, and roll frequency can overwhelm the raw distribution. A game that rolls rarely at decisive moments can feel more volatile than one with wider dice that rolls constantly and offers many recovery tools.

Probability answers one important question: what outcomes does this procedure make common? Design begins when those frequencies are connected to the decisions players make before and after the roll.