Great. Let's do that. You will notice that the "contestant chooses door randomly and then switches" is both a different answer to your answer, and a different question to your question, making different assumptions. I don't want to debate which one is a better question, better assumptions, and better result. But this is an approach which is reliably sourced and which both has an informal version which is good to know and a more sophisticated version which all the smart guys (financial engineers, business school, economists) know about. Take fate into your own hands by throwing your own dice. Finesses all possible complications. You can't think when you're under the lights in the last round of the game show, that's why no-one ever switched, except for the occasional pigeon. You must think this through in advance. You must think out of the box and realize that you have TWO not one decision moments and the first is just as important as the second.