I still don't get it, changing your choice.

[1MileCrash]

I still don't get it, "changing your choice."

21 is on, you know the old situation where you're given three choices, you make your choice, then one of the incorrect options is eliminated, and the question is whether or not it benefits you to change your choice.

The answer is yes, apparently. But why?

I asked a friend in support of the answer "what if I change it, and change it back?" and he says that this also gives me the benefit. What? That changes nothing regarding the probability other than the words I spoke to change my choice twice.

As far as I can tell, once the third, incorrect option is eliminated, it becomes 50/50 probability regardless of whether or not I change my choice.

Can someone explain?

 

[micromass]

21 is on, you know the old situation where you're given three choices, you make your choice, then one of the incorrect options is eliminated, and the question is whether or not it benefits you to change your choice.

The answer is yes, apparently. But why?

I asked a friend in support of the answer "what if I change it, and change it back?" and he says that this also gives me the benefit. What? That changes nothing regarding the probability other than the words I spoke to change my choice twice.

Your friend is incorrect, changing your choice, then changing it back does not give you the benifit.

As far as I can tell, once the third, incorrect option is eliminated, it becomes 50/50 probability regardless of whether or not I change my choice.

The point is that the "quizmaster" will give you information in regarding your choice. Here's to see why:

Say you choose DOOR A, then there are three possibilities:
1) DOOR A contains the price, DOOR B and DOOR C not.
2) DOOR B contains the price, DOOR A and DOOR C not.
3) DOOR C contains the price, DOOR A and DOOR B not.

Obviously, all these three possibilities occur with probability 1/3. Now, the quizmaster opens a door:

1) DOOR B is opened. (doesn't really matter here, the quizmaster can also choose C)
2) DOOR C is opened. (since B contains the price, and you've chosen A)
3) DOOR B is opened. (since C contains the price, and you've chosen A)

Again, 1, 2 and 3 occur with probability 1/3. Now, let's say you stay with your choice:

1) You win!
2) You lose!
3) You lose!

So the probability of winning is only 1/3, while the probability of losing is 2/3! So staying with your choice is not good here!

 

[Jocko Homo]

21 is on, you know the old situation where you're given three choices, you make your choice, then one of the incorrect options is eliminated, and the question is whether or not it benefits you to change your choice.

The answer is yes, apparently. But why?

I asked a friend in support of the answer "what if I change it, and change it back?" and he says that this also gives me the benefit. What? That changes nothing regarding the probability other than the words I spoke to change my choice twice.

As far as I can tell, once the third, incorrect option is eliminated, it becomes 50/50 probability regardless of whether or not I change my choice.

Can someone explain?

Your friend is on crack...


Here's an analogy that can help you conceptually. Instead of just three doors, imagine that there are a million doors. A lot of doors, I know...

The car is behind one of the doors but you don't know which one. What are the chances you will choose the right one? Almost zero but you choose one anyway...

Now, because I'm such a nice guy, I will open 999,998 doors so now there are only two doors left: the door you picked and some other door.

Would you change your guess then?

 

[1MileCrash]

So essentially the difference between the two possible choices left is that the "quiz master" eliminated one because it is certainly not correct, and other is eliminated just because you chose it, therefore preventing him from eliminating it as a possibly surely incorrect answer himself?

 

[micromass]

So essentially the difference between the two possible choices left is that the "quiz master" eliminated one because it is certainly not correct, and other is eliminated just because you chose it, therefore preventing him from eliminating it as a possibly surely incorrect answer himself?

Yes, that makes sense! Do you understand it with that reasoning?

 

[1MileCrash]

Yes, that makes sense! Do you understand it with that reasoning?

Yes, I do. Here's how I'm seeing it go down, logically.

Choice A is either incorrect or correct. 1/3
Choice B is either incorrect or correct. 1/3
Choice C is either incorrect or correct. 1/3

C is actually correct. I choose B.

A is subject for elimination, and is eliminated. - 0%

B would be subject for elimination, but it is not because I selected it.

So, in my mind that means the probability of B being correct is essentially the product of the probability that it was not eliminated because I choose it and that it was not eliminated because it is the correct answer.

(50%) (2/3) = 1/3 of being correct.



C is not subject for elimination, as B, however C's probability of being correct hinges only on it's probability that it was not eliminated because it is the correct answer.

= 2/3 of being correct.

Am I being clear on that?

 

[micromass]

Sounds like you get it

 

[hillzagold]

The short, uninteresting answer is that you can pick the wrong door, and the quizmaster will pick the right door for you. You have a 2/3 chance of picking the wrong door, in which case you should switch.

 

[LCKurtz]

Another way to say it: The only way you can lose by switching doors is if you had the right door in the first place. The probability of that is 1/3. So the probability of winning if you switch is 2/3.

 

[yenchin]

Personally, I think draw a tree diagram helps.