Simple probability problem

[thesurge]

"Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?"

Let's see if this sparks a debate.

 

[hypn0tik]

Yes.

 

[FoBoT]

No.

 

[xSkyDrAx]

Maybe.

 

[SonnyDaze]

Who really gives a sh*t?

 

[bolomite]

here

 

[aux]

Yes, but where is the poll?

edit: too slow...

 

[FP]

Can I buy a vowel?

 

[nycxandy]

Yes... and did you get it from Numb3rs?

 

[thesurge]

Originally posted by: nycxandy
Yes... and did you get it from Numb3rs?

No, I've known of this problem for quite a while.

 

[Pantoot]

I would switch you have a 50% chance, rather than the 33% chance you had the first time.

To expand the question: you have a deck of 52 cards face down, and you are asked to select the ace of spades. After you pick a card I remove all of the cards but one and tell you that one of the two cards is the ace of spades. Would you switch, or would you stick with your original pick?

Edit: I should have read the wikipedia, I guess you really have a 2/3 chance, rather than the 1/3 chance you had the first time.

 

[aux]

Originally posted by: thesurge

Originally posted by: nycxandy
Yes... and did you get it from Numb3rs?

No, I've known of this problem for quite a while.

old problem indeed

 

[MrPickins]

Originally posted by: Pantoot
It would switch you have a 50% chance, rather than the 33% chance you had the first time.

QFT

It doesn't matter what happened before, there are now 2 doors, and 1 car. That means it's 50/50 which will have the car.

 

[FoBoT]

if its 50/50, why does it matter which one you picked? your original pick could be the one, or not. if its 50/50 , i don't see why changing from one of the 2 choices is more chance than staying with the other 50/50

 

[hypn0tik]

Originally posted by: FoBoT
if its 50/50, why does it matter which one you picked? your original pick could be the one, or not. if its 50/50 , i don't see why changing from one of the 2 choices is more chance than staying with the other 50/50

Did you even bother to read the Wiki article that someone linked a few posts above?

 

[kranky]

I never would have believed you should switch!

 

[MrPickins]

After reading the full rules of the game, I'd have to switch my answer, and my door.

 

[FoBoT]

so, it is the same no matter which of the 2 goat doors monty picks?
*edit* oh, i get it, opening the door doesn't change the number of possibilities, it just reveals one condition
:light:

 

[slayer202]

Originally posted by: kranky
I never would have believed you should switch!

I still don't. if you picked a door other than the open one with the goat, that leaves 2 doors left. 1 has a goat, the other has a car. either one you pick gives you a 50% chance at a car. switching does not change jack ******

 

[Cawchy87]

Originally posted by: MrPickins

Originally posted by: Pantoot
It would switch you have a 50% chance, rather than the 33% chance you had the first time.

QFT

It doesn't matter what happened before, there are now 2 doors, and 1 car. That means it's 50/50 which will have the car.

Incorrect. You actually have a 66.6% chance.

You are betting that your first choice (33.3% chance) was wrong. We covered this in my stats class this year, and it was also in an edition of "Discover Magazine" under the "fuzzy math" section.

 

[chambersc]

The probability goes up from 33% to 50%. No, I wouldn't switch.

 

[BigJ]

Originally posted by: slayer202

Originally posted by: kranky
I never would have believed you should switch!

I still don't. if you picked a door other than the open one with the goat, that leaves 2 doors left. 1 has a goat, the other has a car. either one you pick gives you a 50% chance at a car. switching does not change jack ******

At the point when the player is asked whether to switch, there are three possible situations corresponding to the player's initial choice, each with equal probability (1/3):

* The player originally picked the door hiding goat number 1. The game host has shown the other goat.
* The player originally picked the door hiding goat number 2. The game host has shown the other goat.
* The player originally picked the door hiding the car. The game host has shown either of the two goats.

If the player chooses to switch, the car is won in the first two cases. A player choosing to stay with the initial choice wins in only the third case. Since in two out of three equally likely cases switching wins, the odds of winning by switching are 2/3. In other words, players who switch will win the car on average two times out of three.

 

[MrPickins]

Originally posted by: Cawchy87

Originally posted by: MrPickins

Originally posted by: Pantoot
It would switch you have a 50% chance, rather than the 33% chance you had the first time.

QFT

It doesn't matter what happened before, there are now 2 doors, and 1 car. That means it's 50/50 which will have the car.

Incorrect. You actually have a 66.6% chance.

You are betting that your first choice (33.3% chance) was wrong. We covered this in my stats class this year, and it was also in an edition of "Discover Magazine" under the "fuzzy math" section.

Originally posted by: MrPickins
After reading the full rules of the game, I'd have to switch my answer, and my door.

OP, you need to state the problem fully:

" A thoroughly honest game-show host has placed a car behind one of three doors. There is a goat behind each of the other doors. You have no prior knowledge that allows you to distinguish among the doors. "First you point toward a door," he says. "Then I'll open one of the other doors to reveal a goat. After I've shown you the goat, you make your final choice whether to stick with your initial choice of doors, or to switch to the remaining door. You win whatever is behind the door."


(from wikipedia)

 

[Number1]

Originally posted by: kranky
I never would have believed you should switch!

Same here but after reading the wiki solution, it all make sense.

Cool puzzle.

So yes, you increase your chances if you change your choice.

 

[Special K]

Originally posted by: slayer202

Originally posted by: kranky
I never would have believed you should switch!

I still don't. if you picked a door other than the open one with the goat, that leaves 2 doors left. 1 has a goat, the other has a car. either one you pick gives you a 50% chance at a car. switching does not change jack ******

Let's change the scenario a bit. Suppose there are 10,000 doors, and only one of them has the prize. You intially pick one of the doors. The game show host then opens 9998 of the doors, leaving the door you choose, and the other one. The door you chose has a 1/10000 chance of having the prize, because you could have picked from any one of the 10,000 doors when you made the choice. The door the host left has a 1/2 chance, because the host eliminated 9998 of the wrong answers. Clearly you would be better off changing your guess.