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Rerolling once is a policy, not filtering

Part G · L26 · Prerequisite: L25 · One modelling idea

Situation, question, and prediction

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.

Build

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

Run, read, and check

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.

Change one thing

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.

Try it yourself

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.

Rules and model notes

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.

What you now know / where next

Rerolling once is a policy, not filtering is the reusable idea. Follow the generated previous/next links below, or return to the course index. For a complete self-contained application, see reroll-once.

Content ID: L26 · Review status: pilot

Prerequisites: L25