A portal · the ground becomes a network
The Condition You Weren't Told
Switching doors wins two-thirds of the time — but only if the host knew where the car was. A 99%-accurate test says you're sick — but only if the disease is common. Six famous results are true conditionals whose "if" the popular version quietly deletes. Operate the dropped condition and watch each one flip, vanish, or finally become real.
Every one of the six below has the same shape: “if A, then B.” The result is B. The condition is A — a prior, a null, a rule, a scale, a protocol — the baseline the result is measured against. Folklore repeats B as a free-standing fact and drops A on the floor. And the move is always the same, because a result only ever means something relative to what it's compared to:
The same machine, twice
Two of the six are literally one calculation — Bayes' theorem — and you can watch the deleted condition be the only input that changes the answer. The host's protocol and the disease's base rate are the same kind of thing: the baseline that tells you what the evidence is worth.
Monty HallYou picked a door. The host opened another, showing a goat. Switch?
66.7% switching wins — because his choice of door was evidence.
simulated: —
The positive testA 99%-accurate test just came back positive. Are you sick?
50.0% chance you're actually sick
of everyone who tests positive, the red are truly sick
Same theorem. Both answers are P(cause | evidence) = (how often the evidence shows up when the cause is real) ÷ (how often it shows up at all). The host's protocol decides whether his opened door is evidence; the base rate decides how much a positive test is worth. Delete either baseline and you've thrown away half the denominator — and the famous number was never a fact about doors or tests, but about what you were comparing against.
Six results, six deleted conditions
The same move across probability, statistics, economics, and physics — which is the point: it isn't a quirk of any one field, it's a habit of reading. Each card links to the full instrument, where you can operate the condition yourself.
The check
Every number on this page is re-derived from exact arithmetic and a seeded Monte Carlo in research/the-condition-you-werent-told/derive.mjs — both sides of each toggle:
- Monty Hall: knowing host → switching wins exactly 2/3; blind host, conditioned on a goat being revealed → exactly 1/2. Enumerated and Monte-Carlo-confirmed (200k games).
- Positive test: a 99%/99% test at 1% prevalence → 50.0%; at 50% → 99.0%; the Casscells 1978 case (1/1000, 5% false-positive, ~100% sensitive) → ~2%, against the doctors' median guess of 95%.
- p-value: 60 heads in 100 fair flips → two-sided p = 0.0569 (exact binomial, no normal approximation). That is P(data | fair), not P(fair | data).
- Dunning–Kruger: a seeded sim with skill–confidence correlation built to ≈ 0 still draws the scissors — the bottom quartile lands near the 12th actual / 62nd perceived percentile, from no real effect.
- Commons: open-access efficiency 4n/(n+1)² → 100% (sole owner), 88.9% (n=2), 33.1% (n=10), → 0; a stint restores 100%.
- Drain: Rossby number Ro = U/(fL) → ≈1 for a hurricane (rotation governs), ≈10⁴ for a sink (rotation negligible by four orders of magnitude).
Reproduce: node research/the-condition-you-werent-told/derive.mjs — 26/26 pass.