I don’t understand how the answer could be anything but 50⁄50. I know the right answer, but if you deleted it from my brain I never would have figured it out. I guess I’m looking for an explanation that isn’t just following through examples from every scenario. Does an explanation like that exist?
Sure. Think of things in terms of distributed probability mass. I think we all find in intuitive that each door starts at 1⁄3 of the probability mass. So you chose a door, and the host opens another to reveal no_prize. Once the door was opened, the probability of that door goes to zero. And that probability mass must go somewhere because probability must sum to 1.
So far, so good. Depending on one’s intuitions about probability, the 1⁄3 mass from the wrong door either (a) splits between the remaining closed doors, or (b) stays together so that some door is now has 2⁄3 mass. So which is it?
Imagine a small modification to the procedure. Before the no_prize door is opened, a curtain is closed that covers the unchosen doors. If this curtain was closed before any door was chosen, 2⁄3 chance the prize is behind the curtain. This is still true if the curtain is only closed after you choose a door.
Once the curtain is closed, the no_prize door is opened. But you still don’t know curtain-door is open. From your POV, the curtain is still 2⁄3 likely to include the prize door.
Now imagine a bystander walks in at this point, knowing nothing about the game. She hears the host ask if you want the door or the curtain. If she thinks the game is fair, she things you have a 50-50 chance. But you know more information that the bystander—all that curtain-covers-two-doors stuff. It is this knowledge that justifies the difference between your probability estimate and the bystander’s estimate.
Perhaps this story will help: Imagine a criminal defendant, represented by a lawyer. The lawyer says, “If we win the motion, your case will be dismissed. Otherwise, you will be convicted.” Defendant: “So, I have a 50-50 chance.” Lawyer: “Absolutely not. The motion is totally lacking in legal and factual merit. There’s a 1% chance that the judge will be so confused that the motion will be granted.”
The lawyer’s knowledge of the decision mechanism allows a more accurate prediction than (1 / # of possible outcomes). In the same way, you have seen the whole process (doors, chose, curtain, no_prize opened but unseen), which allows (and requires) you to make a different allocation of probability mass.
I don’t understand how the answer could be anything but 50⁄50.
I know the right answer, but if you deleted it from my brain I never would have figured it out.
I guess I’m looking for an explanation that isn’t just following through examples from every scenario.
Does an explanation like that exist?
Sure. Think of things in terms of distributed probability mass. I think we all find in intuitive that each door starts at 1⁄3 of the probability mass. So you chose a door, and the host opens another to reveal no_prize. Once the door was opened, the probability of that door goes to zero. And that probability mass must go somewhere because probability must sum to 1.
So far, so good. Depending on one’s intuitions about probability, the 1⁄3 mass from the wrong door either (a) splits between the remaining closed doors, or (b) stays together so that some door is now has 2⁄3 mass. So which is it?
Imagine a small modification to the procedure. Before the no_prize door is opened, a curtain is closed that covers the unchosen doors. If this curtain was closed before any door was chosen, 2⁄3 chance the prize is behind the curtain. This is still true if the curtain is only closed after you choose a door.
Once the curtain is closed, the no_prize door is opened. But you still don’t know curtain-door is open. From your POV, the curtain is still 2⁄3 likely to include the prize door.
Now imagine a bystander walks in at this point, knowing nothing about the game. She hears the host ask if you want the door or the curtain. If she thinks the game is fair, she things you have a 50-50 chance. But you know more information that the bystander—all that curtain-covers-two-doors stuff. It is this knowledge that justifies the difference between your probability estimate and the bystander’s estimate.
Perhaps this story will help:
Imagine a criminal defendant, represented by a lawyer. The lawyer says, “If we win the motion, your case will be dismissed. Otherwise, you will be convicted.”
Defendant: “So, I have a 50-50 chance.”
Lawyer: “Absolutely not. The motion is totally lacking in legal and factual merit. There’s a 1% chance that the judge will be so confused that the motion will be granted.”
The lawyer’s knowledge of the decision mechanism allows a more accurate prediction than (1 / # of possible outcomes). In the same way, you have seen the whole process (doors, chose, curtain, no_prize opened but unseen), which allows (and requires) you to make a different allocation of probability mass.