I never understood this. Well, I understand the theory and the mathematics behind how changing your choice increases the chances of winning, but it just doesn't sit right with my sense of probability and stuff.. It's hard to explain, but I really dislike the way this works haha.
Wait... I'm confused, I don't see how the 3rd door factors in probability-wize because the probability that said door is a goat is 100%. That being said, according to the logic that goes inside my mess of a brain decides that 1 of the doors is a goat, and one of the doors is a car. And therefore, odds are 50/50.
Well... that's just my brain... I am TERRIBLE at math.
Door 1|Door 2|Door 3|result if switching|result if staying
Car Goat Goat Goat Car
Goat Car Goat Car Goat
Goat Goat Car Car Goat
But if there's only 1 goat, and 1 car, isn't that 50%?
Okay. If the player's pick has a 1/3 chance while the other two doors have 1/3 chance each, for a combined 2/3 chance. And with the usual assumptions, player's pick remains a 1/3 chance, while the other two doors a combined 2/3 chance, 2/3 for the still unopened one and 0 for the one the host opened.
Okay. If the player's pick has a 1/3 chance while the other two doors have 1/3 chance each, for a combined 2/3 chance. And with the usual assumptions, player's pick remains a 1/3 chance, while the other two doors a combined 2/3 chance, 2/3 for the still unopened one and 0 for the one the host opened.
It makes mathematical sense, but no logical sense...
FINALLY I CAME TO CHECK ON THIS POST!
You guys have had quite the argument.
The correct answer, in fact, is that switching doors will give you a higher probability of getting a car, because as the door that opens first will always be a goat (otherwise there would be no point in asking whether or not to switch), then it still must be taken into account in the equation.
Let's start from the beginning of the scenario.
Since there are three doors, you automatically have a 1 in 3 chance of choosing the door with the car behind it.
Door 1 Door 2 Door 3
1 Car Goat Goat
2 Goat Car Goat
3 Goat Goat Car
Say you choose door 2 in scenario 1.
Therefore, you have not chosen the car.
Door 3 is opened for you (being the only door left with a goat behind it) and reveals a goat.
You are asked whether or not to switch. You are absolutely aware that one of the doors has a car behind it.
The reason it's not a 1 in 2 chance is because there are still two goats. The door opened will ALWAYS be a goat door. Therefore, you are not sure which goat is behind door number 3.
There are two goats! This is so important, and people think that the goat behind the opened door ceases to matter!
But since the goat is there, we must take it into account in our equations.
Since there are two goats, and 1 car, there is a 1 in 3 chance you chose a car.
That means, there is a 2 in 3 chance the car is behind the other two doors.
OPENING THE FIRST DOOR WITH A GOAT BEHIND IT DOES NOT CHANGE THAT.
The first choice was made, locking in the 2 in 3 chance.
Changing doors doubles your chances of winning.
If you don't believe me, grab a friend, and try this out as many times as you want. Use a deck of cards, where two are the same and one is different. Write down how many times staying made them win, and how many times changing made them win.
It may not make sense at first, but once you get down to the mathematics and probabilities, the numbers add up.
Let's say there are three doors.
One door has a car behind it. Behind the other 2 doors are goats.
You, obviously, want to get a car.
You don't know which door is which.
You are asked to choose a door. You choose door #1. The door remains unopened.
Then, I open door #3. A goat is behind it.
I ask you, "Would you like to keep door #1, or change to door #2?"
Which choice is more likely to make you win a car?
If you opened a door at complete random without knowing what was behind it the chance of me winning is still 50/50
If you know what is behind the doors and intentionally open one that has a goat the chance with switching is 2/3
The increased chance of the player winning the car comes from them first picking the first door, making this a controlled enviornment and breaking the results.
Door 1 Door 2 Door 3 result if switching result if staying
Car Goat Goat Goat Car
Goat Car Goat Car Goat
Goat Goat Car Car Goat
See, this is contingent upon the player selecting the first door originally. What if in scenario 2 I picked Door 2? That would then switch it to this.
Door 1 Door 2 Door 3 result if switching result if staying
Car Goat Goat Goat Car
Goat Car Goat Goat Car
Goat Goat Car Car Goat
I never understood this. Well, I understand the theory and the mathematics behind how changing your choice increases the chances of winning, but it just doesn't sit right with my sense of probability and stuff.. It's hard to explain, but I really dislike the way this works haha.
Edit: If you want to test it for yourself:
http://www.nytimes.com/2008/04/08/science/08monty.html
[simg]http://i54.tinypic.com/4zzw1z.png[/simg]
Door 1|Door 2|Door 3|result if switching|result if staying
Car Goat Goat Goat Car
Goat Car Goat Car Goat
Goat Goat Car Car Goat
2/3 chance of car.
How can the door have a car and 2 goats behind it?
They're different scenarios.
I don't understand...
Okay. If the player's pick has a 1/3 chance while the other two doors have 1/3 chance each, for a combined 2/3 chance. And with the usual assumptions, player's pick remains a 1/3 chance, while the other two doors a combined 2/3 chance, 2/3 for the still unopened one and 0 for the one the host opened.
It makes mathematical sense, but no logical sense...
You really should look at the diagram in the wikipedia page
...Okay, maybe I DID read "The curious incident of the dog in the night-time"
[12:41] Coffeeeeeee!
---
[16:29] "And lo, the tacos were delicious"
You guys have had quite the argument.
The correct answer, in fact, is that switching doors will give you a higher probability of getting a car, because as the door that opens first will always be a goat (otherwise there would be no point in asking whether or not to switch), then it still must be taken into account in the equation.
Let's start from the beginning of the scenario.
Since there are three doors, you automatically have a 1 in 3 chance of choosing the door with the car behind it.
Say you choose door 2 in scenario 1.
Therefore, you have not chosen the car.
Door 3 is opened for you (being the only door left with a goat behind it) and reveals a goat.
You are asked whether or not to switch. You are absolutely aware that one of the doors has a car behind it.
The reason it's not a 1 in 2 chance is because there are still two goats. The door opened will ALWAYS be a goat door. Therefore, you are not sure which goat is behind door number 3.
There are two goats! This is so important, and people think that the goat behind the opened door ceases to matter!
But since the goat is there, we must take it into account in our equations.
Since there are two goats, and 1 car, there is a 1 in 3 chance you chose a car.
That means, there is a 2 in 3 chance the car is behind the other two doors.
OPENING THE FIRST DOOR WITH A GOAT BEHIND IT DOES NOT CHANGE THAT.
The first choice was made, locking in the 2 in 3 chance.
Changing doors doubles your chances of winning.
If you don't believe me, grab a friend, and try this out as many times as you want. Use a deck of cards, where two are the same and one is different. Write down how many times staying made them win, and how many times changing made them win.
It may not make sense at first, but once you get down to the mathematics and probabilities, the numbers add up.
If you opened a door at complete random without knowing what was behind it the chance of me winning is still 50/50
If you know what is behind the doors and intentionally open one that has a goat the chance with switching is 2/3
Good idea.
Im confused.
Door 1 Door 2 Door 3 result if switching result if staying
Car Goat Goat Goat Car
Goat Car Goat Car Goat
Goat Goat Car Car Goat
See, this is contingent upon the player selecting the first door originally. What if in scenario 2 I picked Door 2? That would then switch it to this.
Door 1 Door 2 Door 3 result if switching result if staying
Car Goat Goat Goat Car
Goat Car Goat Goat Car
Goat Goat Car Car Goat