Making the Monty Hall problem weirder but obvious
> It’s important that Monty looked behind the doors before choosing which to open.<p>This is often not explicitly stated when the problem is given. It is even not a 100% clear from the statement above. Monty <i>always chooses a door with a goat</i>. So:<p>1. You choose a door.<p>2. Prob that there is a car behind it: 1/3<p>3. Prob that the car is behind the two other doors: 2/3<p>4. If the car was behind the two other doors (which, remember, has p=2/3), Monty <i>will choose the door without a car for you</i>, and the door with a car will remain closed. In this case you are <i>guaranteed</i> to have the car if you switched.<p>So with switching, the overall probability is 2/3. Without, its the original 1/3.<p>If you did not understand that Monty always chooses a goat door, but the person giving you the problem does, or vice versa, then what usually happens is that both of you try to explain why your intuition is correct. Because most people don't talk formal probabilities, your explanations will be so vague that the other person will not realize your different understanding. You will discuss forever, you will both be right, and you will part ways with the strange feeling that maybe the other person <i>was</i> right, when all along you were talking about different problems. This is why this problem is so notorious.