A game show has 3 doors — one hides a prize, two are empty

Why

Ayush's instinct — two doors left, so 50-50 — feels obvious, but it quietly ignores something important: the host did not open a door randomly; he ALWAYS knows where the prize is and deliberately avoids it. Your first pick had exactly a 1/3 chance of being right, and nothing the host does afterward changes that original 1/3 — it stays 1/3 forever. That means the other two doors, together, held a 2/3 chance between them; once the host reveals one of THOSE two as empty (which he can always do, since he knows the prize's location), that entire 2/3 chance does not vanish — it all collapses onto the one remaining unopened door. Test it: imagine playing this game 300 times, always switching — since your first pick is wrong about 200 times out of 300, and switching from a wrong first pick always lands on the prize (the host removed the only other empty door), switching wins roughly 200 times, while staying only wins the roughly 100 times your first guess was already correct. His teacher is right: switching gives a 2/3 chance of winning versus 1/3 for staying — the host's guaranteed, non-random choice of which door to open is exactly what breaks the '50-50' assumption.