Probability Question.

Michael Collins

Colonel Sanders wrote:

"We know that monty will open one door and this door will be a door containing a goat. therefore by opening the door he tells us nothing that we already don't know."

But we don't know which door has the goat behind it until he opens it!

Colonel Sanders

no but we know at least one of them has a goat and he will revel a goat.

another way of thinking is i you originally pick a door the probability of picking the correct door is 1/3, the probability of an incorrect door is 2/3, no one argues with this.

think of it this way:

Scenario 1: We change doors.

If we originlly picked the car we lose if we change. Therefore if we change we lose if we originally picked the car (p=1/3) and we win if we dont originally pick the car (p=2/3) as monty will reveal a goat (he knows which door the car's behind) and the only door remaining will contain the car. Therefore if we change we win the car if we havent already picked the car origianally (p=2/3). Therefore picking a door and then changing will win 2 out of every 3 times.

Scenario 2: We don't change:

Therefore we will only win the car if we originally picked the car and the probability of this is 1/3.

Buffer

Your latest explanation is correct, Colonel. The only point on which I disagreed with you in your earlier post is your claims that opening the door he doesn't give us any new information.

Buffer.

Michael Collins

Buffer wrote:

Your latest explanation is correct, Colonel. The only point on which I disagreed with you in your earlier post is your claims that opening the door he doesn't give us any new information.

Buffer.

Me too.

Colonel Sanders

I still don't see how opening one of the other 2 doors gives you any new info.

you know one door has a car behind it.

you know he will open one door

you know the door he opens will have to have a goat behind it (as he knows which door the car is behind and can't show us the car).

I don't understand how doing what you know he'll do gives you and posterior info!!!!!

2Scoops

Colonel Sanders wrote:

I don't understand how doing what you know he'll do gives you and posterior info!!!!!

He will eliminate one of the two doors you didn't pick that has a goat behind it. We knew he would do that... but we didn't know specifically which door it would be (door #1, door #2 etc.)

Now that one of the two doors has been eliminated, the total probability of the prize being behind either of these doors (2/3) becomes the probability of being behind the one remaining unpicked door.

cast_iron

Colonel Sanders wrote:

I still don't see how opening one of the other 2 doors gives you any new info.

Well, you're arguing over a definition.

It doesn't technically "give you information" in the literal sense. But it increases the odds of you winning, and that act alone has given you information (you know you are more likely to win if you change). You didn't have that knowledge of increased odds beforehand.

Colonel Sanders

true enough I suppose, thanks.

Buffer

cast_iron wrote:

It doesn't technically "give you information" in the literal sense. But it increases the odds of you winning, and that act alone has given you information (you know you are more likely to win if you change). You didn't have that knowledge of increased odds beforehand.

I would say that technically, it does give you information, literally, since you now know something that you didn't previously know (if he opens Door 2. you now know for certain that that's the wrong door) and this changes your knowledge of the state of the game.

For anybody who's still wrestling with this, it might be helpful to think about why this game is different from one where the host begins by opening a 'dud' door, before you choose a door yourself. If you can figure out for yourself why it makes a difference that you've chosen a door first, then it might be clearer that the host's actions depend on yours and his actions give you new information.

Buffer.