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The Monty Hall Problem

Suppose I've made it to the final round of a game show hosted by Monty Hall where I'm now given the choice of 3 closed doors to choose from, and I get to keep whatever is behind the door that I choose.

The good news is that behind one of the doors is a brand new car! However, hidden behind each of the other doors is a goat. And since the car and goats have been randomly placed behind the doors, I must choose my door without any helpful information to guide me.

"Which door would you like to select?" asks Monty, to which I hesitantly reply, "I'll take Door Number 1". Monty continues, "Thank you. But before we open up Door Number 1, I'd like to make things more interesting and let you in on the fact that not only do I already know what is behind each of the doors, but I'm now required to open up one of the two unselected doors and reveal what's behind it. However, I'm not allowed to open a door that would reveal the car, and if both doors have goats behind them I must randomly pick one to open."

Monty then motions to his assistant, who opens Door Number 2 and reveals a goat. Now there are only two closed doors remaining, Door Number 1 (our selection) and Door Number 3. One of the doors has a brand new car waiting behind it, and the other hides the second goat. "Additionally, I would now like to give you the option of either staying with your first choice of Door Number 1 OR switching your choice to Door Number 3." After a few moments of thoughtful deliberation, I lean forward and declare my decision into the microphone.

"Monty, I'd like to switch to Door Number 3. In fact, it's always to my advantage to switch based on the facts you've presented." The crowd gasps! Someone from the audience shouts, "That's not correct, switching doesn't matter since each door has an equal probability of hiding the car!" Only one of us is right, but which one is it?