Statistical Mindfuck

[embirrim]

Nothing can be wrong with those tests.
Make as many tests as you want, and you'll find out that first choice was right in 33% of cases and wrong in 66%.

All you need to understand then is that changing your choice means inverting the result.
Whether your first choice is wrong or right, there are only two cases regarding the two other doors :
- two wrongs (first choice right)
- one right and one wrong (first choice wrong)
There is always a wrong door left to open, that mean you are in this situation once it's opened :
- first choice right : remaining door is wrong
- first choice wrong : remaining door is right
If you change your choice and choose the remaining door, you invert the result of your first choice.
If you had 66% chances of being wrong with your first choice, then you have the same 66% choice of being right by choosing the last door.

My point exactly. This is like those proofs that .9999=1 and pi=4

[walker_boh_65]

My point exactly. This is like those proofs that .9999=1 and pi=4

well, .999. . . does = 1, but pi in now way equals 4. That is more like the 1=2 proofs.

[Eriottosan]

My point exactly. This is like those proofs that .9999=1 and pi=4

well, .999. . . does = 1, but pi in now way equals 4. That is more like the 1=2 proofs.

It's not like the 1=2 proofs: this has no gaps in its theory, it is correct.

The 1=2 proof is cleverly done so that you don't realise that you divide by zero ... which obviously you can never do ... And so doesn't actually work. :)

EDIT: Of course, when talking about proofs, you have to remember the two that 'prove' "women are evil" and "the more you know, the less you earn".

[Catox]

My point exactly. This is like those proofs that .9999=1 and pi=4

Nope, it's like proof that one chance out of 3 equals 33% chance.
You can't take a logic out just because you never experimented long enough to get numbers that will be perfectly perfect.

now if you want some talking experiment, you can have it.
what I did : take a random number from 1 to 3. Assume 1 is "right" and 2/3 are "wrong". That's "first choice". If first choice is right, then one of the two wrong doors is open. If first choice is wrong, then the other wrong door is opened. It's "Revealed door". There is only one other door : it's "second choice"

Run it 50000 times, count the numbers of first choices and second choices that are right (=1), and calculate percentages. Do it again, and here we are :

50000 runs = 16533 first choices right (33,07%) and 33467 second choices right (66,93%)
50000 runs = 16680 first choices right (33,36%) and 33320 second choices right (66,64%)
50000 runs = 16620 first choices right (33,24%) and 33380 second choices right (66,76%)
50000 runs = 16801 first choices right (33,60%) and 33199 second choices right (66,40%)
50000 runs = 16664 first choices right (33,33%) and 33336 second choices right (66,67%)
50000 runs = 16559 first choices right (33,12%) and 33441 second choices right (66,88%)
50000 runs = 16835 first choices right (33,67%) and 33165 second choices right (66,33%)
50000 runs = 16705 first choices right (33,41%) and 33295 second choices right (66,59%)
lets add them all :
400000 runs = 133397 first choices right (33,35%) and 266603 second choices right (66,65%)

Do you need more runs ?

[Eriottosan]

<snip>

Yup. That's my favourite thing about this: in practice, it actually works, even though "common sense" makes our heads scream "NO!" :).

[Whisp]

For me it helped to extend this experiment a little to get my mind to stop screaming:
Assume you don't have 3 doors, but 100 of them.
Now you choose one and 98 of the remaining 99 doors are opened and you can choose to switch.
You have a chance of 1% to get the right door in the beginning. In every other case switching means getting the right door. :)

[Catox]

<snip>

Yup. That's my favourite thing about this: in practice, it actually works, even though "common sense" makes our heads scream "NO!" :).

honestly, I don't remember how my common sense told me how to deal with this problem before, but now that I understand it, it clearly screams a big "YES" in my head. Like "I was wrong, I can only be right if I change" ^__^

[Mason11987]

The reason people misunderstand the monty hall problem is because they think the first guess and the second guess are unrelated, as they are asked at separate times.

The thing is the host intentionally picks a door which is empty and NOT your door. Since that's how it's picked it no longer makes the second guess independent of the first, they are necessarily a single guess, a single guess which initially you had a 1/3 chance of getting right, and later still have that 1/3 chance of having guessed right.

The underlined point is absolutely critical, but it's not a situation that people commonly encounter, but since it looks like ones we do regularly encounter we try to use simplistic math to figure it out, but it doesn't apply in this case.

[Ribky]

Ugh... math.

This is why I stuck with linguistics.

[BinoAl]

Ugh... math.

This is why I stuck with linguistics.

I love math, and I find language not near concise enough :)
I've always been good with language, whether it's my own or learning another, but I just don't find it interesting. Math, however... :)

[Ribky]

I love math, and I find language not near concise enough :)
I've always been good with language, whether it's my own or learning another, but I just don't find it interesting. Math, however... :)

Math always made my head hurt, especially when I do figure out the right answer. Language isn't as concise, that's why I love it, you get a bunch of words in any given language and are generally able to play with them a bit.

[Folrig]

And yet, that's not true.
When you chose the first door, it had a 33% chance of being the right one.
When someone opens an empty door, there's only two left, so each has a 50% chance.
When you're asked to switch or not, if you do, you'll have a 50% chance of chosing the right one. If you keep your choice, it's still 50%.

Sorry pal, but you are wrong on this one. Let's put out a list of all you possible options if the prize is behind door number 3. (maybe this can help people see it better :) )
If you pick door 1, (then door 2 is opened) you switch to door three, you WIN..
If you pick door 2, (then door 1 is opened) you switch to door 3, you WIN.
If you pick door 3, (then door 2 is opened) you switch to door 1, you LOSE.
so that's a 2/3 chance you win if you switch doors.
If you pick door 1, (then door 2 is opened) you stay door 1, you LOSE.
If you pick door 2, (then door 1 is opened) you stay door 2, you LOSE.
If you pick door 3 (then door 2 is opened) you stay door 3, you WIN.
so that's a 1/3 chance of wining if you stay.

:)

Wow! That is the best explination of this problem I have ever seen, landing the answer firmly into the relm of logic for me! Thank you sir!

[Eriottosan]

Ugh... math.

This is why I stuck with linguistics.

I love math, and I find language not near concise enough :)
I've always been good with language, whether it's my own or learning another, but I just don't find it interesting. Math, however... :)

BinoAl, I'm the same, but that's why I love Japanese - have you ever tries your hand at it? It's rather consise compared to other languages as you don't have to so things like repeat an established topic &c.
That being said, I *do* like using mathematical abbreviations in everyday writing. I'm talking things like "iff".

[Twitch]

For those of you who need an actual test or a little visual proof:

Spoiler

Show

http://math.ucsd.edu/~crypto/Monty/monty.html

We had to do that assignment for my probability class over spring semester. Had to take it 100 times on two seperate equations, one test switching, the other test staying, and put the results together.

Logically it makes you go "BWUAH?" but mathematically, it makes perfect sense!

[SterlingRed]

To me, math is the same thing as logic. If it is proved by math, it is logical. I don't get how you all are able to think of it as both mathematical and illogical at the same time. Then again, any day of the week if you ask me to add simple numbers, I will get it wrong. But if you ask me to calculate the time of death of a body in a room given two known temperatures, or the growth rate of say rabbits given multiple environmental conditions and their relationship with prey, I will be right every time. After getting a minor in math, I lost the ability to do the easy stuff...

[Catox]

Given Twitch's last comment, it seems to me that there would be 3 level of understanding/accepting this problem :
- proved by experience : "all right it works, I'm forced to admit it given the number of results"
- proved by maths : a mathematical demonstration, through probability formulas and calculations leads to the conclusion that the theory is indeed right, even if it doesn't fill like it is.
- proved by 'logic' : a 'deeper' comprehension of the mechanism at work leads you to (finally) be able to how right the theory clearly is.

I must say I'm quite surprised by the number of people who seem to understand the 'maths' level and still don't grasp the 'logic' one.

Really, how "unlogic" is it that when you invert the result of a choice for which you were probably wrong, you obtain a new result that will make you probably right ?

--
ninjad by SterlingRed > Thank you I feel less lonely ^__^; (and I'm not even mathematician :p)

[Zhil]

What I don't get is how people refuse to accept the truth, even after it has been rigorously explained. Monty Hall is not intuitive, but writing down the logic tables results in a very easy solution that shows that switching doors doubles your chances.

Think of it this way:

Scenario 1: You pick the car first try, switching always leaves you with nothing
Scenario 2: You pick a goat first try, monty reveals the other goat, switching always wins the car

So, if you only get to pick once, that's 33%
Picking twice and choosing to switch throws you into Scenario 2, which happens 66% of the time

Easy, switching doubles your chances.

When Marilyn vos Savant first gave the answer to this riddle, she received tons of letters from mathematicians and statisticians, flat out calling her wrong. Something about this problem makes people say: "NO! The math is incorrect!"

[BinoAl]

Another way to look at it that I heard is this: Say a guy is running a shell game. He has 3 shells, puts a ball under one, and switches them around so you don't know which ball is where. So, you pick one of the shells. Now, he says that if you double your bet, he will reveal a shell with nothing under it, to make your chances 50/50. It's an obvious mistake, and being unable to switch is no different than not switching

[Zhil]

Another way to look at it that I heard is this: Say a guy is running a shell game. He has 3 shells, puts a ball under one, and switches them around so you don't know which ball is where. So, you pick one of the shells. Now, he says that if you double your bet, he will reveal a shell with nothing under it, to make your chances 50/50. It's an obvious mistake, and being unable to switch is no different than not switching

Now you're confusing the issue by posting an unrelated riddle.

Apart from having 33% at the start, that problem doesn't even resemble Monty Hall

[Eriottosan]

Another way to look at it that I heard is this: Say a guy is running a shell game. He has 3 shells, puts a ball under one, and switches them around so you don't know which ball is where. So, you pick one of the shells. Now, he says that if you double your bet, he will reveal a shell with nothing under it, to make your chances 50/50. It's an obvious mistake, and being unable to switch is no different than not switching

Now you're confusing the issue by posting an unrelated riddle.

Apart from having 33% at the start, that problem doesn't even resemble Monty Hall

Actually I think he's right (excluding the obvious exception that these games are often cheated anyway). In this case, you choose one of 3, and have a 33% chance of being right. When one empty shell is revealed you are happy to double your bet as you believe it is 50:50, but as you have no chance of switching, it is 33:66 to the dealer's favour.

[BinoAl]

Now you're confusing the issue by posting an unrelated riddle.

Apart from having 33% at the start, that problem doesn't even resemble Monty Hall

No, the only difference between the 2 is that in the shell game, you don't have the choice to switch, and in the monty hall riddle, you do

[Catox]

which in the end result on who is given an advantage.
That's an interresting one, thanks BinoAl

[Twitch]

Personally I find it both logically and mathematically easy to explain to myself (which leads to interesting discussions with my wife, as I can't seem to comprehend how she doesn't comprehend heh) But I think part of the problem stems from peoples perception of the choices themselves.

It seems (to me at least) that people are viewing the 'switch' as simply starting over with only two doors. The shell game example correlates this. If he mixes it up, you never pick one, then he shows you one that doesn't have the ball, your chances are now 50/50 that you can pick the right one. EDIT: Upon re-reading this example, I realize that you did pick one first.

The difference being that in the monty hall problem, you've already chosen a door, which factors into the probability that switching will net you a car. Whereas most people seem to be thinking "Well he's shown me a door, so now if I have a 50/50 chance of being on the right door the first time" which is, simply put, wrong.

And i'm with you on the simple math thing.... I had lots of red ink on my last statistic test because I said 2 X 16 is 24 >_>

[SgtChuckle]

Spoiler

Show

I stay with door #1. I made my choice, why change it now? Besides, once it has been revealed that door #3 is empty, it's a 50/50 chance as to which door has the big money behind it, so changing will do absolutely nothing to increase my chances.

[Elevatator]

Spoiler

Show

I stay with door #1. I made my choice, why change it now? Besides, once it has been revealed that door #3 is empty, it's a 50/50 chance as to which door has the big money behind it, so changing will do absolutely nothing to increase my chances.

You just gave me an immense headache remembering me to someone I tried to explain it, and that one was not willing/able to understand that. He even argumented the same way.