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The Pool — any-one trigger component
R17 · Intermediate · Scoped component
Question and scope
The Pool uses any natural 1 as its success trigger. Given an already assembled pool of two d6, how often does at least one 1 appear? Predict why adding a die helps less when the pool is already large.
Source verification pending. The explicit mathematical procedure below is executable but must not be treated as certified game rules.
Inputs and model limits
Supplied integer pool size 1–5 for the displayed table; default question uses two d6. Pool assembly and gambling/resource choices are excluded.
Complete runnable model
.p_any(1) asks whether one or more faces match. .p_none(1) asks whether no face matches. Both return probabilities, not counts. Pool assembly, gambling, rewards, and narration decisions are outside this trigger component.
pool = dice_pool(2, 6)
output("Any natural one", pool.p_any(1))
output("No natural ones", pool.p_none(1))
rows = []
for count in range(1, 6):
rows.append((str(count) + " dice", dice_pool(count, 6).p_any(1)))
output("Success trigger by supplied pool size", prob_table(rows))
Any natural one · Prob
| outcome | % | frac | X |
|---|---|---|---|
| Any natural one | 30.6 | 11/36 | 0.306 |
No natural ones · Prob
| outcome | % | frac | X |
|---|---|---|---|
| No natural ones | 69.4 | 25/36 | 0.694 |
Success trigger by supplied pool size · Table
| outcome | % | frac | X |
|---|---|---|---|
| 1 dice | 16.7 | 1/6 | 0.167 |
| 2 dice | 30.6 | 11/36 | 0.306 |
| 3 dice | 42.1 | 91/216 | 0.421 |
| 4 dice | 51.8 | 423/817 | 0.518 |
| 5 dice | 59.8 | 637/1065 | 0.598 |
Reading the report and independent checks
No ones on two dice means 5 × 5 of 36 pairs: 25/36. Its complement is 11/36. For n dice, any one is 1 − (5/6)^n. Each added die rescues only some previously unsuccessful rolls, so the gains diminish.
Safe changes
Use one die: any one is 1/6 and none is 5/6. No probability can grow above one.
Composition and boundary checks
Ask for any six instead. Answer: the probability is unchanged for fair d6, but the event is different. For a biased custom die you must check face weights. “Any” does not tell you how many matches occurred.
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
The Pool trigger component; author-rules source review remains pending. The fully specified mathematical question here is at least one 1 on n independent fair d6, not a certified complete game procedure.
Related learning
Lesson L14 explains the modelling idea step by step. Cookbook index offers both game and mechanic views of these same recipes.
Content ID: R17 · Review status: source pending
Prerequisites: L14