Drunkle got one of the better explanations. It is easier to think of it with fewer objects than 1000 though. Simply the following.
We have goat1, goat2 and car. Goat1 and Goat2 are effectively equivalent.
Odds on first pick of picking either Goat1, Goat2 or Car are 33%.
1)If on first pick, you picked Goat1, the announcer will reveal Goat2 making the Car the other unselected pick. Odds of being in this state is 33% (see above). Switch has odds of 100% of success in this state (unless you need a goat)
2)If on first pick, you picked Goat2, the announcer will reveal Goat1 making the Car the other unselected pick. Odds of being in this state is 33% (see 2nd above). Switch has odds of 100% success in this state.
3)If on first pick, you picked Car, the announcer will reveal Goat1 or Goat2 randomly since they are both unselected. Odds of being in this state are 33% (see 3rd above). Switch has odds of 0% success in this state.
Therefore, there is a 66% chance of being in a state where switching choices will benefit and 33% chance where it won’t. It is important to note that the announcers actions are not random when in state 1 and 2.
It does seem counterintuitive. The key is that the announcer’s actions are not random in 66% of the cases.
