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Reroll a bad face once

C03 · Advanced · Scoped component

Question and scope

A house rule says “if your first d6 is a 1, reroll it once and keep the second face, even another 1.” Is rolling a 1 now impossible? Predict before examining the report.

Source sections are linked under Rules below; primary-source review is recorded in the release dossier. Independent human and browser/usability review are separate gates.

Inputs and model limits

First d6 and one independent potential replacement. Replace a first 1 once; must keep the second face even if worse. No resource cost.

Complete runnable model

Use a two-die pool for the first and potential replacement faces. Index 0 is first, index 1 is second. The callback decides using only the first face, then returns either that face or the mandatory replacement. pool_map returns the numeric distribution.

def reroll_once(faces):
    first = faces[0]
    replacement = faces[1]
    if first == 1:
        return replacement
    return first
result = pool_map(dice_pool(2, 6), reroll_once)
output("Reroll one once", result)
output("A one remains possible", result.pmf(1))

Reroll one once · DieRoll · mean 3.917

A one remains possible · Prob

Reading the report and independent checks

Only pair (1,1) ends at 1, so its probability is 1/36. Each other face gets its original 1/6 plus replacement probability 1/36 = 7/36. Mean is 47/12. The potential replacement is ignored when the first roll is kept.

Safe changes

Reroll 1 or 2 instead by testing first <= 2. Each of 1 and 2 then has probability 2/36; each other face has 8/36.

Composition and boundary checks

Why is always rolling two dice and keeping the higher a different rule? Answer: a first 2 and potential second 6 would become 6, while our once-only policy keeps the 2 and ignores the potential second face. For the guided variant that rerolls 1 or 2, a first 2 and replacement 1 must become 1—not retain the 2 after inspecting the replacement. A named game ability needs its own edition-specific timing check.

Do not combine separate marginal reports and assume they preserve the same rolled faces. When an event depends on several properties of one roll, keep those properties together inside a single classifier or pool callback.

Rules

Explicit generic house rule. Two independent d6, at most one replacement, no resource cost. Do not silently substitute it for a named game’s reroll timing or optional keep rule.

Related learning

Lesson L26 explains the modelling idea step by step. Cookbook index offers both game and mechanic views of these same recipes.

Content ID: C03 · Review status: pilot

Prerequisites: L26