{"query": "E[Binom(10, ½)] = n·p = 5 · P(X=2) = C(10,2)·½¹⁰ = 45/1024 ≈", "count": 3, "results": [{"id": "card_works_probability", "title": "Ten fair coins: the average and a particular outcome", "shelf": "the-works", "surface": "secular", "snippet": "Flip a fair coin ten times. On average five come up heads; the chance of exactly two heads is C(10,2)/2¹⁰. The engine confirms both the expectation and the exact probability.  Worked & sealed by the e", "authority_tier": "verified", "source": "The Works — worked & sealed", "readable": false, "generated": false}, {"id": "card_v_e25edb860b45", "title": "E[Binom(10, ½)] = n·p = 5 · P(X=2) = C(10,2)·½¹⁰ = 45/1024 ≈ 0.04395", "shelf": "probability", "surface": "secular", "snippet": "Verdict: HOLDS.\n• probability.binomial_mean: E[Binom(10,0.5)] = 5.0\n• probability.binomial: P(X=2 | n=10, p=0.5) = 0.0439453\nSealed and independently re-checkable: https://narrowhighway.com/s/e50261aa", "authority_tier": "engine", "source": "Verified by the engine — probability", "readable": false, "generated": false}, {"id": "card_seal_afafa0c5f8a818947458016ce6026b37bfe86fe7265786e2c8b8f32891303797", "title": "Receipt afafa0c5f8a8… — sealed: record", "shelf": "seals", "surface": "secular", "snippet": "{\n \"anchors\": [],\n \"axis_coords\": {\n  \"axis\": \"probability\",\n  \"dimensions\": [\n   \"conservation_balance\",\n   \"discreteness\",\n   \"reasoning\"\n  ]\n },\n \"gate_results\": [\n  {\n   \"details\": {\n    \"broken_a", "authority_tier": "engine_derived", "source": "Narrow Highway engine — sealed record", "readable": false, "generated": false}], "house": {"door": "FIND", "kind": "cards", "trail": "results", "seal": null, "next_step": {"do": "open the top card", "door": "FIND", "tool": "card_get", "params": {"id": "card_works_probability"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}