Risk 3v2 Attack Odds Lookup

Enumerate at most 7776 independent fair-d6 Risk battle rolls, sorting each side and resolving ties for defender; one engagement only, not full territory invasion probability.

Description

Risk compares sorted attacking and defending dice, with ties favoring the defender.

Risk compares sorted attacking and defending dice, with ties favoring the defender. Exhaustive enumeration gives an exact fair-dice casualty model.^[1]

Enter attacker dice 1–3 and defender dice 1–2. Enumerate all 6^(attacker+defender) ordered rolls. Return a distribution indexed by attacker losses and both expected casualty counts.

When to use Risk 3v2 Attack Odds Lookup

Use Risk 3v2 Attack Odds Lookup when you need to evaluate the supplied records under the calculation and evidence rules below. No complete battle, retreat policy or biased dice are modeled.

Prepare the supplied game state, observations and explicit variant settings. Follow the card, board, coordinate and state conventions in the calculation rules.

How to use Risk 3v2 Attack Odds Lookup and understand its results

Prepare the supplied game state, observations and explicit variant settings. Follow the card, board, coordinate and state conventions in the calculation rules.
Enter attackerDice, defenderDice using the field table and calculation rules, then run the tool.
Read attackerLossDistribution, expectedAttackerLosses, expectedDefenderLosses, totalEnumeratedRolls. The explanation below defines the units, signs, ordering and any flags or placeholder values.
If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.

Input field	Meaning	Value type; see rules below	Requirement
attackerDice	Attacker dice	number	Required
defenderDice	Defender dice	number	Required

Enter attacker dice 1–3 and defender dice 1–2. Enumerate all 6^(attacker+defender) ordered rolls. Return a distribution indexed by attacker losses and both expected casualty counts.

compare highest pairs; attacker loses when attack≤defense

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
attackerLossDistribution	Attacker loss distribution	list of number
expectedAttackerLosses	Expected attacker losses	number
expectedDefenderLosses	Expected defender losses	number
totalEnumeratedRolls	Total enumerated rolls	number

json

{
  "input": {
    "attackerDice": 3,
    "defenderDice": 2
  },
  "output": {
    "attackerLossDistribution": [
      0.37165637860082307,
      0.3357767489711934,
      0.2925668724279835
    ],
    "expectedAttackerLosses": 0.9209104938271604,
    "expectedDefenderLosses": 1.0790895061728396,
    "totalEnumeratedRolls": 7776
  }
}

Limitations and assumptions

No complete battle, retreat policy or biased dice are modeled.

Numeric inputs, where present, must be finite. Follow the calculation-specific rules for overflow, underflow and exceptional cases. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.

Alternative or Complementary methods

Use 2d6 Sum Probability Table: Two fair six-sided dice have 36 equally likely ordered outcomes. Different sums have different counts. For a complementary task, use Nd6 Dice Sum Mean & Variance: Loaded-die weights define a probability model only after normalization. Independent dice add expectations and variances.

References

Risk 3v2 Attack Odds Lookup — tool.io

Similar or alternative tools

2d6 Sum Probability Table
Two fair six-sided dice have 36 equally likely ordered outcomes. Different sums have different counts.
Nd6 Dice Sum Mean & Variance
Loaded-die weights define a probability model only after normalization. Independent dice add expectations and variances.