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Lunaria

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Posted on 07-25-10 08:40:06 PM Link | Quote
Near the end of the last term my math teacher put out one of those "fun" other things to do during math class. (During this time we where studying chance and such aspects) (And mind you, I'm not into high level math) Now, the thing in question is quite easy to understand in general, it's just that I think he's wrong. Mind you, I find that odd as while I'm quite natural talented in math, he have been an university teacher before this. Anyway, on to the issue.

Now it's simple. Imagine you have three cups, under one of them there's a die. Now, you pick one that you want to keep/etc. Then the one who did set this whole thing up removes one of the other cups that did not have the die under it. You now have the option to change what cup you want to keep before looking under it. And this is the part we disagree on.

While I say that as there are two cups left I should have 50% chance of having the right one, so it should really not matter if I change or not.

My teacher, however, argues that as you picked the first cup while having a 1/3 chance of picking the right one, you still have a 1/3 chance that it's the right one, while if you change cup, then you have a 50% chance. Something that I myself find heavily stupid. But regardless I could be wrong so I might as well throw this out here so you guys can put in your thoughts on the matter.

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Posted on 07-25-10 08:48:35 PM Link | Quote
It's the Monty Hall problem, and neither you nor your teacher are correct, actually.

Here's how it works. You start with 3 choices for which cup you pick. Let's call the cups A, B, and C, and let's assume that you pick A. Now, the die can be under any of A, B, or C, all with P = 1/3.

If the die is under A (P = 1/3), and you pick A, the host can remove either B or C.
If the die is under B (P = 1/3), and you pick A, the host must remove cup C.
If the die is under C (P = 1/3), and you pick A, the host must remove cup B.

Notice that which cup he removes actually has nothing to do with anything. If you were wrong, the host must remove a particular cup, but if you were right, the host can remove either, so you can't tell from the removed cup whether you were right or wrong. There was a 1/3 chance that you were right to begin with and a 2/3 chance you were wrong. If you were right (P = 1/3), you should keep cup A. If you were wrong (P = 2/3), the other cup will always have the die under it and you should switch to the remaining non-A cup (either B or C). Therefore, there's a 2/3 chance, not 1/2 or 1/3, that you should switch. Notice that the number 1/2 never factors into the problem at all!

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Lunaria

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Posted on 07-25-10 08:53:33 PM Link | Quote
Oh I see, that explenation is sure as hell alot better then the one he gave me. Also, in retrospect, maybe he only said the chance was higher if you switched, I don't really recall that well on that part. :<

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NSNick
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Posted on 07-25-10 09:59:33 PM Link | Quote




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Another explanation I like for this problem is as follows:

Imagine that instead of 3 cups (or doors), now you have 1,000. You make your choice, and then the host lifts 998 cups/doors. Would you switch then?

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Higsby

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Posted on 07-25-10 10:05:15 PM (last edited by Higsby at 07-25-10 07:05 PM) Link | Quote
That was an awesome explanation Terra. I had always thought the same thing as Lunaria because whenever a friend and I talked about this problem he always sucked at explaining why there was a 2/3 chance that the other cup was correct instead of 1/2 .

And that's definitely an interesting way of thinking about it as well NSNick.

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ReiNi
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Posted on 07-26-10 02:04:18 AM Link | Quote
At least until I can do something better...
Originally posted by NSNick
Another explanation I like for this problem is as follows:

Imagine that instead of 3 cups (or doors), now you have 1,000. You make your choice, and then the host lifts 998 cups/doors. Would you switch then?

About that, I don't understand why many teachers refuse to use exaggeration to explain... Maybe becouse of wanting to get students used to dealing with smaller numbers whenever possible, but to deal with the problem at hand, it's always much simpler to put it in a larger scale.
Nevertheless the answer is usually that [trope=ViewersAreMorons]large numbers are too complex for kids to understand[/trope].

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Posted on 07-26-10 02:46:37 AM Link | Quote
When it comes to probability, you can throw common sense out the window, as humans are typically really bad at calculating that. Even with years of study and a significant amount of natural talent in math, one would still be far from a walking probability calculator.

Anyway, when explaining this problem, I usually go with the traditional explanation first. If they still don't understand / believe me, then I introduce the "1,000 doors" thing.
I find that the traditional explanation can provide a greater understanding of the problem than the other one, and that the other one is generally understood by a greater amount of people, so employing both seems best to me.
Orlandu


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Posted on 07-26-10 01:43:55 PM (last edited by Orlandu at 07-26-10 10:44 AM) Link | Quote
This was explained to a point in the movie "21". (good movie, by the way)

<object width="480" height="385"><embed src="http://www.youtube.com/v/6Rl_p3JlSd0&hl=en_US&fs=1" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="480" height="385"></embed></object>

And a simple image I found that displays it pretty well.



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ReiNi
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Posted on 07-26-10 03:49:44 PM Link | Quote
At least until I can do something better...
Probability is useless anyways. Where the McGuffin IS, never relies on any sort of "luck" nor "randomness", but on the different events that you don't pay attention to nor care about knowing. Knowing the entire enviroment the "randomization" takes place, it would be very possible to know the result in pretty much the same way one can figure a pseudorandom number generator.
If you don't know, you pick one based on psychological factors. You being right or wrong may be a coincidence, influenced by an ability to sense the result somehow and paired with a desire of getting either a succesful or unsuccesful result, or mostly both influenced by the human ability to see the future or and the inherent desire of doom that makes you pick the wrong or "right" one, which will always cause disastrous consequences due to chaos theory.

In short, picking the wrong or right choice deppends on the current universe you are in. Probability is the branches of parallel universes that vary from that very result alone, which may or may not coincide with other universes.

No, I am not serious with any of this.

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Orlandu


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Posted on 07-26-10 03:56:17 PM (last edited by Orlandu at 07-26-10 12:56 PM) Link | Quote
Originally posted by Reimu
*bunch of funny BS*

This wins the thread.

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Xkeeper

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Posted on 07-27-10 09:23:14 PM Link | Quote
Originally posted by The Red Snifit
When it comes to probability, you can throw common sense out the window, as humans are typically really bad at calculating that. Even with years of study and a significant amount of natural talent in math, one would still be far from a walking probability calculator.

A good example of this: You flip a coin that has an even chance of falling heads or tails. You begin flipping it, and every time for 50 times it comes up heads. What is the probability that it will come up heads on the next throw?

Many people will get this wrong. (It's 50%, by the way.)


Interestingly, the Monty Hall problem only applies if the host knows which one has the prize. If he opens a door at random, the probabilities are equal. It is only if he knowingly reveals a loss.

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Taryn

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Posted on 07-27-10 10:42:23 PM Link | Quote
Originally posted by Xkeeper
You flip a coin that has an even chance of falling heads or tails. You begin flipping it, and every time for 50 times it comes up heads. What is the probability that it will come up heads on the next throw?


The sad thing is that I've tutored someone who didn't even know the probability of a coin landing heads at all. They guessed 0% and 100%

University student, BTW.

But yeah, I think that's the "gambler's fallacy" or some such, although I could be wrong. When people expect streaks to either continue or be broken when there's no reason for them to, like betting on the colours in roulette.

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Orlandu


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Posted on 07-28-10 02:02:26 PM (last edited by Orlandu at 07-28-10 11:32 AM) Link | Quote
Originally posted by Xkeeper
Interestingly, the Monty Hall problem only applies if the host knows which one has the prize. If he opens a door at random, the probabilities are equal. It is only if he knowingly reveals a loss.


Thus it being called the Monty Hall problem or Game Show Host problem. Otherwise its just normal probability, like you said.

I definitely would not want to be on a game show where the host didn't know what he was giving away, haha. "Whats behind door number one? I don't have a fucking clue, lets find out!"

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Xkeeper

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Posted on 08-02-10 12:55:26 AM Link | Quote
Originally posted by Terra
The sad thing is that I've tutored someone who didn't even know the probability of a coin landing heads at all. They guessed 0% and 100%

University student, BTW.

But yeah, I think that's the "gambler's fallacy" or some such, although I could be wrong. When people expect streaks to either continue or be broken when there's no reason for them to, like betting on the colours in roulette.

Welcome to the education system.

And yeah, it is, though the larger name for it is the Law of Averages.

Living in Vegas, this is stuff you either learn the easy way (by, you know, learning) or the hard way (going bust).

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Posted on 08-02-10 12:58:59 AM Link | Quote
Originally posted by Orlandu
I definitely would not want to be on a game show where the host didn't know what he was giving away, haha. "Whats behind door number one? I don't have a fucking clue, lets find out!"

I would imagine most don't, simply because it would be too easy to give off clues as to what the correct door is.

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Sunny

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Posted on 08-02-10 01:06:13 AM Link | Quote
POKER FACE
You can't handle it, so you don't get to know
(get what I mean?)

To hell if I can hold a poker face while concentrating
I'm better off keeping my mind occupied with other things instead of trying to manage my own face.


And as far as the probabilities go...an average is an average, but a coin can have enough wear and tear to skew the results somehow. Have to go factory fresh for pure averaged results, and even then you're skewing it with your oily fingers... Don't know what I mean? Well just think about it for a bit.

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Xkeeper

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Posted on 08-02-10 01:10:55 AM Link | Quote
Originally posted by Madman200
And as far as the probabilities go...an average is an average, but a coin can have enough wear and tear to skew the results somehow. Have to go factory fresh for pure averaged results, and even then you're skewing it with your oily fingers... Don't know what I mean? Well just think about it for a bit.

Any skew outside of actually damaging/bending the coin is likely to be negligible.

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