The Wizard will bank this bet: 1/6 vs 1/11

[redietz]

The question was intentionally worded that way, just as the GRE logic portion was intentionally written so as to provide a verbal maze and just as Alan's law exam used invented language to provide obliqueness.

The question was written to divide the reading audience into those tricked and those not tricked.

I despise when the best and brightest spend their time with these kinds of intellectual masturbations. I mean, really? This is the best use of your time, energy, skills and logic? To write convoluted opaque crap in efforts to demonstrate cleverness?

The WoV crew is a math club, not some group of gambling experts. If they want to be spend their days being clever by writing purposefully torturous prose, let them. As far as actual gambling goes, that entire forum appears woefully overrated.

[Count Room]

The question was written to divide the reading audience into those tricked and those not tricked.

I despise when the best and brightest spend their time with these kinds of intellectual masturbations. I mean, really? This is the best use of your time, energy, skills and logic? To write convoluted opaque crap in efforts to demonstrate cleverness?

I readily admit I, myself, was tricked by the question into believing the answer was 1/6 for a few minutes after Alan first presented it here. It was only after a more careful look did I become a 1/11 convert. There's no shame for me in admitting I was wrong when I first looked at the question. I actually liked the question! It indirectly provides a valuable life lesson about how the beautiful subtleties of mathematics can fool the instinctive mind in a practical application.

[redietz]

I'm not so impressed. Our minds are designed to prioritize personal experience and anecdotal evidence over the abstractness of probability because, well, we don't really have a "probability sense." So bad judgements are made every day by all of us all of our lives. The last thing we need is for those who can do so to manipulate language to cause further confusion.

[arcimede$]

OMG Arc you can't use the same die twice to solve the problem.

I used the red die once and the blue die once. That is obvious from my description. And that is obviously the meaning of "at least one die". If they meant just one die then that is what they would have said. The fact you intentionally lie about this tells me you are not being honest in this entire discussion. Bad Alan.

[Alan Mendelson]

Arc, why are you accusing me of lying? You said I "intentionally lie" -- and what did I lie about?
Where does the word "lie" come from?

[arcimede$]

Alan, you lied when you said "OMG Arc you can't use the same die twice to solve the problem." Since I never used the same die twice, your statement is an outright LIE.

And no, this statement is NOT TRUE ... ""At least one" means that one or both of the dice in the problem can be a 2." ... What it means is what I told you above and you keep denying because it destroys your silly logic. It means EITHER die can be a two (or both). Now, if you would quit lying and redefining what other people are saying maybe we could have an intelligent discussion. However, you continue to lie and misquote others. Quit it, Alan.

[Alan Mendelson]

I just tried to post and somehow the posts got screwed up with what I think were Arc's posts. So I deleted what I posted and if it deleted what Arc wrote, I am sorry.

[Alan Mendelson]

And no, this statement is NOT TRUE ... ""At least one" means that one or both of the dice in the problem can be a 2." ... What it means is what I told you above and you keep denying because it destroys your silly logic. It means EITHER die can be a two (or both).

Arc I am going to disagree with you about the meaning of "at least." When you have two dice and tell me that at least one is a 2 it means that one or both. It doesn't mean "either." Now if you want to say that it's either, we have a problem. Because "either" means only one and it can't mean both. "At least" can include "both" dice.

Consider this:

Idioms
6.
at least,
at the lowest estimate or figure:
The repairs will cost at least $100.
at any rate; in any case:
You didn't get a good grade, but at least you passed the course.

And the word "lie" is not appropriate. I have not lied nor have I msquoted. You quit it.

[Rob.Singer]

I invite everyone to read thru redietz' posts. The original problem was a play on words meant to be taken in two different ways depending on if your life is driven by conflict or if you're at peace in general. Math people tend to get a special chill up their leg when they smell an argument. And no one is lying here, that's what's obvious. Arci hasn't much tolerance these days for views opposite his, and for him to be calling someone a liar....now THAT'S special.

As I said earlier, the way wizard presented his challange, it's a 1in11 proposition. I'd still be inclined to take on his 1in9 offer, simply because it's similar to playing -EV video poker, and everyone knows, anything can and will happen in a short term application of long term theory. But the way Alan interpreted the question--which in my opinion is the more grounded of the two, he is absolutely correct about the question strongly suggesting that the 2 shown "on at least one of the dice" was there to stay. In short, the geek who proposed the original question wasn't the super deep thinker he thought he was. He simply got outsmarted by his own question, the wizard then went all in with another of his bad choices, and the math club had no option but to follow and persist. When you think about it, and how they will never admit their error in judgement, it's quite entertaining.

[redietz]

An intentionally confusing framing device is being used to point out math or logic errors by readers, but the authors of the framing device seem to think their ability to be confusing should be applauded. This is like anti-teaching.

Again, Alan, myself, and Rob all have writing backgrounds. We avoid this kind of Don King trickeration at all costs. It gives me the willies when people intentionally do this. If nuns were teaching technical writing and you did this, your knuckles would be smacked.

[arcimede$]

Arc I am going to disagree with you about the meaning of "at least." When you have two dice and tell me that at least one is a 2 it means that one or both. It doesn't mean "either." Now if you want to say that it's either, we have a problem. Because "either" means only one and it can't mean both. "At least" can include "both" dice.

There you go again. I said " It means EITHER die can be a two (or both)". And you reply with "you want to say that it's either ... it can't mean both" ..,. Did you fail to see my words "or both"? Looks like you are lying once again. You just quoted me wrong. It happens continually.

Look I already said this above. The red die could be a 2 or the blue die could be a 2 or they could both be 2s. That is what "at least" means.

Consider this:

Idioms
6.
at least,
at the lowest estimate or figure:
The repairs will cost at least $100.
at any rate; in any case:
You didn't get a good grade, but at least you passed the course.

And the word "lie" is not appropriate. I have not lied nor have I msquoted. You quit it.

You've done both, repeatedly. Now, I cannot tell if this is intentional since I can't read your mind. But, your typed words are lies. That is not in doubt.

[Alan Mendelson]

Hold it Arc, I admit I am totally mixed up and confused.

Let me start at the beginning with the term "at least" and your red and blue dice.

At least means the red die can be a 2 or both the red and blue dice can be a 2.
At least also means the blue die can be a 2 or both the blue and red dice can be a 2.
BUT at least does not mean you can consider the blue die to be a two and then switch it to the red die to be a 2.

Do we agree?

[arcimede$]

Hold it Arc, I admit I am totally mixed up and confused.

No kidding.

Let me start at the beginning with the term "at least" and your red and blue dice.

Oh, so now you are actually going to read what I wrote. How interesting.

At least means the red die can be a 2 or both the red and blue dice can be a 2.
At least also means the blue die can be a 2 or both the blue and red dice can be a 2.
BUT at least does not mean you can consider the blue die to be a two and then switch it to the red die to be a 2.

Do we agree?

You can't switch it on any given throw but the sum of all the DIFFERENT throws can have either the red or the blue die being a 2.

You throw the dice ... it has a red 2 and a blue 3. That is one possibility.
You throw the dice again ... it has a blue 2 and a red 3. That is another possibility. (Do you agree?)

No one "switched" anything. They are two distinct possibilities.

Now, you keep on doing this and you find out you can have 11 distinct possibilities. A red 2 with a blue 1,3,4,5,6 .... a blue 2 with a red 1,3,4,5,6 ... and a red 2 and a blue 2.

All of these are based on different throws of the dice when "at least" one of the dice has a 2.

On this we can agree or you can continue to ignore reality.

[Alan Mendelson]

Yes, Arc, I understand what you are saying. But please put it in the context of the original question because that is what I am willing to discuss. And I will continue to discuss it.

Now there was some discussion that we didn't know which of the two dice contained a "2" and I explained it didn't matter. Choosing either die as the 2 gives you six other options. Choosing both dice as 2 also gives you six other options. But when you say Dice A is a 2 and then say Dice B is a 2 to justify the 1/11 answer -- that is plain wrong.

[arcimede$]

Yes, Arc, I understand what you are saying. But please put it in the context of the original question because that is what I am willing to discuss. And I will continue to discuss it.

Now there was some discussion that we didn't know which of the two dice contained a "2" and I explained it didn't matter. Choosing either die as the 2 gives you six other options. Choosing both dice as 2 also gives you six other options. But when you say Dice A is a 2 and then say Dice B is a 2 to justify the 1/11 answer -- that is plain wrong.

No one said " Dice A is a 2 and then say Dice B is a 2" on the same throw. We are talking about odds, probabilities ... do you understand what that means? As I stated above,

One possible throw has a red 2 and a blue 3.
Another possible throw has a blue 2 and red 3.

Both cases meet the definition in the original question. At least one 2 is present.

These are two separate possible throws. As such, they make up the part of the list that needs to be considered when discussing probabilities. Once you list all the possibilities you can then compute the odds. Since there are 11 possible throws as I listed above, with only one of them being 2-2, you end up with the odds being 10:1 (or 1 in 11). You aren't "choosing" any die to be a 2. You are considering the total number of possible throws where either die (or both) can be a 2.

This gets back to what redietz has been saying all along. It's a trick question. You have been misinterpreting/redefining the question to fit you initial erroneous answer. You were wrong. Admit it and move on.

EDIT:Alan, the question you are answering would need to be stated differently. To get 1-6 the question would need to ask when the die that was viewed is a 2 what are the odds the other die is a 2.

[RS__]

Well said blount2000. And indeed, for every roll you've seen with at least a 2 in it, you probably saw a 2-2 1/11 of the time (or close to 1/11).

Interesting that Alan agrees that it satisfies the original question. But then he follows up saying the answer is still 1/6. I'm not sure if that's really funny or really sad....

[Alan Mendelson]

RS___ I am not saying that when you see a 2 playing craps that you will see another 2 1/11 times. I said when you see a 2 playing craps, the chance of seeing 2/2 is a 1/36 event.

Here's what I said before:

Your statements 1 and 2 are correct, blount2000. Whether or not you saw 2-2 when at least one of the two dice is a 2 one-sixth of the time doesn't matter. Frankly, when you roll two dice the chance of getting 2-2 is 1/36.

The original question and the question to be answered is simply this: if you have at least one 2 showing, what are the odds for the second die to be a 2 as well. And that is the same as rolling one die with a 2 showing 1/6 times.

If you are not seeing 2-2 with one 2 showing one-sixth of the time you're just like me. Like I said, that would be a 1/36 chance.

Rolling two dice at the craps table was not the question either. And I guess it will never be settled. Thanks for stopping by.

[Alan Mendelson]

No one said " Dice A is a 2 and then say Dice B is a 2" on the same throw. We are talking about odds, probabilities ... do you understand what that means? As I stated above,

One possible throw has a red 2 and a blue 3.
Another possible throw has a blue 2 and red 3.

Both cases meet the definition in the original question. At least one 2 is present.

These are two separate possible throws.

Thanks Arc you made that clear.

[regnis]

One die is a 2. For both to be a 2, the other has to be a 2. What are the odds of that----1 of 6. The one that was a 2 can't suddenly become a 1,3,4,5, or 6 to create more combinations. I once got beaten pretty badly for having dice like that!!!!

Once a 2, always a 2. Arci's dice are chameleons.

[Alan Mendelson]

One die is a 2. For both to be a 2, the other has to be a 2. What are the odds of that----1 of 6. The one that was a 2 can't suddenly become a 1,3,4,5, or 6 to create more combinations. I once got beaten pretty badly for having dice like that!!!!

Once a 2, always a 2. Arci's dice are chameleons.

That's interesting. But I think your comments are more fitting for the videos posted by the WOV folks. They both had the dice changing.