Comment by tgb

This is what I thought at first but simulating it out, it’s not true. The strategy of switching doors works 2/3rds of the time but only if you allow picking the opened door (when it has the prize). That’s easy to see since it works whenever the prize isn’t behind your first choice. Conditioning on the opened door revealing a goat gives that strategy only 50% chance. After all, the opposite situation happens 1/3rd of the time and has a 100% success rate (since you can see the car), so this other case (opened door has a goat) has to have a 50% chance to add up to the unconditioned success rate. It’s still unintuitive to me though.