Conditional probability you can feel: what a positive test really means (Bayes' theorem), a 300-game Monty Hall simulation, and the prosecutor's fallacy — with a 10,000-person grid painted live by you.
Three doors, one prize. You pick. The host — who knows where the prize is — always opens an empty door you didn't choose. Stay, or switch? Your gut says 50/50. The maths says switch wins 2/3 of the time. Watch it happen.
Why. Sticking wins only if your first guess was right — one door in three. Switching wins in the other two cases, every time. The host's reveal isn't information for you; it's a gift to switchers.
A forensic test says only one person in a million would match. In a city of ten million, that means about ten people match — and one of them is on trial. Each dot below is a match; the hot one is the defendant.
The probability of the evidence given innocence (one in a million) is not the probability of innocence given the evidence. Bayes' theorem is the bridge between the two — and it always asks: how many of *those* are the guilty one? Exhibit A shows the same trap in medicine.
Numbers are exact arithmetic (Bayes' theorem) on the chosen inputs; presets use commonly cited figures. "Test accuracy" is deliberately split into its two honest halves: sensitivity (catches the ill) and specificity (spares the healthy). All of this is classical probability — no coin is being bent.