Monty Hall Switch

Consider the probability your initial door was correct before any doors were opened.

The host's action never reveals the prize, so it doesn't change your original door's probability of being correct.

1. 1. Initial probability

   Your first choice has probability $1/3$ of being the prize door and $2/3$ of being wrong.

2. 2. Host's reveal doesn't change the initial split

   The host always opens a losing door, so this action carries no new information about your original door; the $2/3$ probability of 'prize is elsewhere' becomes concentrated entirely on the remaining unopened door.

3. 3. Conclude

   Switching wins with probability $2/3$.

Answer: 2/3