Dragon Age stunt economy — when P(stunt) ≠ P(success) × P(doubles)

Independence works at exactly one TN

3d6 is symmetric around 10.5 — the joint distribution of (a, b, c) is invariant under (a, b, c) → (7 − a, 7 − b, 7 − c), which maps doubles to doubles and sums to 21 − sum. Success at TN=11 means sum ≥ 11; the symmetry maps that to sum ≤ 10, the failure half. Every doubles outcome has exactly one mirrored partner across the threshold, so the doubles mass splits exactly evenly:

P(success ∧ doubles) = 1/2 · P(doubles) = 1/2 · 4/9 = 2/9.

And P(success) · P(doubles) = 1/2 · 4/9 = 2/9 too. Independence holds — but as a coincidence of two separate facts (P(success) = 1/2, mirror symmetry of the doubles set), not because the events are actually independent.

Move off TN=11 in either direction and the trick stops working.

Where the correlation shows up

Three exact-rational results, no modifiers, computed by direct enumeration of the 216 outcomes of (a, b, c):

The deviation is small in absolute terms — about 1.4 percentage points — but the sign carries information. Below TN=11 the success cutoff is in the lower half of the doubles distribution and excludes more of it; above TN=11 the cutoff sits in the upper half and includes proportionally more. The modal-sum clustering of doubles is the entire effect.

The 39 success-doubles triples at TN=12 enumerated explicitly: (3,3,6), (4,4,5), (4,4,6), (5,5,2), (5,5,3), (5,5,4), (5,5,6), (6,6,1), (6,6,2), (6,6,3), (6,6,4), (6,6,5) (3 permutations each = 36) plus (4,4,4), (5,5,5), (6,6,6) (1 each) = 39. None of (1,1,b) or (2,2,b) qualifies — the cutoff has eaten the low-double half of the doubles distribution.

The conditional SP distribution isn't uniform either

Even after a stunt fires, the stunt die's value is not uniformly distributed. The same clustering effect that biased P(stunt) upward at high TN biases the stunt-die value upward whenever a stunt fires at all. At TN=11 with no modifiers, the 48 success-doubles outcomes bucket like:

The expected value works out to exactly:

E[SP | stunt at TN=11]
  = (1·2 + 2·2 + 3·7 + 4·9 + 5·14 + 6·14) / 48
  = 217 / 48
  ≈ 4.5208.

Substantially above the unconditional stunt-die mean of 3.5. Climb to TN=12 and the cutoff selects more of the (5,5,·) and (6,6,·) families: E[SP | stunt at TN=12] = 184/39 ≈ 4.72. Harder tests don't just trigger less often — when they do trigger, the stunts are more expensive on average.

What you can afford on a triggered stunt

Most published Dragon Age stunts cost between 1 and 5 SP. The reachability table at TN=11, no modifiers — given that a stunt fires:

Combine with the P(stunt) = 2/9 headline number for the unconditional reach:

P(SP ≥ 4 across all rolls)
  = P(stunt) · P(SP ≥ 4 | stunt)
  = 2/9 · 37/48
  = 37/216
  ≈ 17.13%.

About one in six tests at TN=11 lets you afford a 4 SP stunt. One in three triggered tests is a 6 SP haul. The economy is more generous than the 2/9 trigger rate suggests because the conditional distribution skews high.

How modifiers reshape the picture

ability + focus shifts the effective TN linearly: ability +2 and focus +2 against TN=11 is mathematically identical to +0/+0 against TN=7. That moves the success cutoff into the lower-doubles half of the distribution — well below the symmetry centre — and the correlation flips negative.

For a trained PC (focus +2, ability +3) facing a typical TN=11 obstacle, the effective TN is 6, which means success ≈ 95% and the stunt-trigger rate climbs to around 40% of all rolls. But the conditional E[SP | stunt] drops back toward 3.5 — the easier the test, the closer the stunt-die conditional gets to the unconditional uniform. Hard tests trigger stunts less often but pay better when they do.

The engine surface that backs every number on this page is one function call, exposed via NIF and cached in ETS:

{:ok, r} = Diceplots.Engine.analyze_3d6_test(ability_mod, focus_mod, tn)
r.p_stunt_exact            #=> "13 / 72"  (at TN=12, +0)
r.expected_sp_given_stunt  #=> 4.5208... (at TN=11, +0)

Trigger defaults to :success_and_doubles (standard AGE). :any_doubles is the variant where doubles fire a stunt regardless of success — useful for house-rule analyses and for confirming the trigger probability is exactly 4/9 uniformly across modifiers.

Where this matters in practice

Stunt-economy reasoning shows up in three places worth modelling with the exact-rational substrate rather than the product-law approximation.

Encounter design. Dropping a TN from 13 to 11 doesn't just make the test easier — it changes the shape of the stunt economy because the success cutoff moves through the doubles cluster. Tests that sit on either side of the TN=11 symmetry centre have meaningfully different stunt-frequency vs. stunt-richness tradeoffs. Boss fights at high TN trigger fewer stunts but, conditional on a trigger, are more likely to deliver a Lethal Blow.

Focus value. Picking up a focus is usually framed as +2 on the test — but it also moves the effective TN, which moves the success-doubles correlation sign. Focus on a TN=13 test pulls you toward the symmetry centre and trades a small amount of conditional stunt richness for reliability. Focus on a TN=9 test pushes you below centre and trades the other way. The reliable-vs-nuke tradeoff applies here at the stunt-economy level.

Cross-system: any "success-AND-trigger" mechanic. The product-law assumption fails the same way for any system where a secondary trigger condition correlates with the success threshold — Pendragon's critical-success rules, FATE's bonus effects on success-with-style, Cthulhu's hard/extreme success bands (covered in Call of Cthulhu's success bands). The symmetry-at-the-modal-sum trick is specific to 3d6, but the "naive product overstates by some fraction of a percentage point" pattern is general.