Is Marilyn Vos Savant wrong on this probability question?

[chiro]

[Double post]

 

[Studiot]

I was talking about the "believability" of getting something like 20 1's in a row.

But that stems from common experience which can lead the unwary to inappropriate conclusions.

Continuing my example,

Although P(4,3) = P(1,1) it also = P(3,4)

So there are two ways of throwing a three and a four, but only one way of throwing two ones.
So if we don't differentiate between (4,3) and (3,4) then obviously you are twice a likely to throw a three and a four as two ones.

Taking this further there are 30 ways to throw two different numbers as against 6 for throwing two the same.

So throwing two different numbers in any order is five times as likely as throwing two the same.

 

[Phrak]

"In theory, the results are equally likely. Both specify the number that must appear each time the die is rolled. (For example, the 10th number in the first series must be a 1. The 10th number in the second series must be a 3.) Each number—1 through 6—has the same chance of landing faceup."

"But let’s say you tossed a die out of my view and then said that the results were one of the above. Which series is more likely to be the one you threw? Because the roll has already occurred, the answer is (b). It’s far more likely that the roll produced a mixed bunch of numbers than a series of 1’s."

Bait and switch. Paraghaph 2 has nothing to do with paragraph 1.

 

[Studiot]

But let’s say you tossed a die out of my view and then said that the results were one of the above.

Agreed, good point.

 

[chiro]

But that stems from common experience which can lead the unwary to inappropriate conclusions.

Continuing my example,

Although P(4,3) = P(1,1) it also = P(3,4)

So there are two ways of throwing a three and a four, but only one way of throwing two ones.
So if we don't differentiate between (4,3) and (3,4) then obviously you are twice a likely to throw a three and a four as two ones.

Taking this further there are 30 ways to throw two different numbers as against 6 for throwing two the same.

So throwing two different numbers in any order is five times as likely as throwing two the same.

You can talk about order all you want, I didn't (and still don't) care about the order issue. I understand the order issue and its relation to the combinatoric representation to statistical distributions, but I'm not advocating that order will influence underlying theoretical probability.

If you look in my earlier posts, I advocated the idea of entropy, and the reason I did that was based on the idea that it provides some measure of measuring how "believable" a process is to being pure random (In a purely random process, entropy is always maximized).

If a process is truly random, then conditional orders of entropy are also more or less maximized as well.

Based on the use of entropy as an estimator of randomness, you can use the sample to determine the likelihood estimate of entropy and hence draw a conclusion of whether you "think" or "believe" that sample came from a purely random process like a coin toss or a dice roll.

Entropy measures take care of things like order, especially when you consider first or higher order conditional probabilities. These measures can quantify these accurately and do not need any hand-waving arguments.

Again, with various forms of entropy, you don't need to use any intuition with regard to order and risk making a bad judgement: the different conditional levels will quantify whether the process is really random.

Stop thinking about order, and focus on how you can accurately gauge the likelihood of whether the sample comes from a pure random process (the believability) and how different measures of entropy can ascertain a quantitative level of "likelihood".

 

[Studiot]

Statistical Mechanics, which provides a statistical view of entropy, operates on the same basic principles of probability as casting dice.

The idea of 'likelihood' is another statistical process or technique established for when we do not have the exact probabilities.
A substantial amount of statistical theory is available to replace exact probabilities with a best estimate of liklihood using the known parameters of the situation, probability distributions and so forth.

In this case where exact probabilities are available they are not appropriate.

go well

 

[chiro]

In this case where exact probabilities are available they are not appropriate.

Why is that?

This whole post is about judging how relevant both the real theoretical probability and the likelihood is in terms of "probability of an event" and "likelihood that it comes from a random process". This is the basis for the thread!

 

[Hurkyl]

It was this very idea of believability that I thought that Marilyn Vos Savant was talking about in the last quote, and that is why I defended her take on it.

Then you're wrong too. What you or she personally believes doesn't change probabilities, making one more likely than the other.The only possible loophole is if she's guessing ways in which the die roller is a flawed random number generator -- but her phrasing very, very much doesn't sound like she's talking about that.

 

[DIABEETUS]

Did anyone besides Fredrik even read my post on page 3??

 

[Studiot]

Did anyone besides Fredrik even read my post on page 3??

Yes but it seemed rather confused.

I think we have all taken implied in the first statement that the rolls of the dice are unbiased.

By definition that means the die roller does nothing to influence the outcome.

Marilyn's first statement confirms this.

Marilyn's second statement concerning the behaviour of the die roller gives a conditional situation which corresponds to the second part of my analysis. She is correct in stating that under these new conditions the latter outcome has a higher probability. However she is incorrect in her reason for this, which has nothing to do with the timing of the roll, as she claims.

It is simply a matter of comparing apples with pears. The first and second outcomes refer to different situations.

Say you plan to roll a die 20 times. Which of these results is more likely: (a) 11111111111111111111, or (b) 66234441536125563152?

and Marilyn (high IQ record holder) answers:


In theory, the results are equally likely. Both specify the number that must appear each time the die is rolled. (For example, the 10th number in the first series must be a 1. The 10th number in the second series must be a 3.) Each number—1 through 6—has the same chance of landing faceup.

But let’s say you tossed a die out of my view and then said that the results were one of the above. Which series is more likely to be the one you threw? Because the roll has already occurred, the answer is (b). It’s far more likely that the roll produced a mixed bunch of numbers than a series of 1’s.

 

[DIABEETUS]

1) I know, but technically it does not explicitly say that, it is just presumed (along with some other assumptions, of course). Which is fine, I’ll go along with that; to assume otherwise would be ridiculous and missing the point of the puzzle. The reason I mentioned that is because many people tend to hang on her phrasing word-per-word and interpret it in the strictest sense.

2) You meant: timing has NOTHING to do the probability of rolling a mixed bunch of numbers... not: timing has NOTHING to do with the probability that you actually rolled that specific, mixed sequence, right??

3) Ehhh, yes and no. Her reason was that the roll already occurred AND that it is far more likely that the roll produced a mixed bunch of numbers than a series of 1's, which both statements are true. The only thing that she really left out are detailed explanations that she probably considered to be obvious and shouldn't require mentioning. That first part is just useless information because it’s a tautology. But that’s why I really don’t think she mentioned that to be the explanation for the second part as a stand-alone question. It makes more sense that she mentioned that in reference to the first part of the puzzle; to compare and explain why the probabilities from the first and second parts are different.

So, I would definitely agree that it is a poor explanation because it is over simplified and vague.

 

[chiro]

Then you're wrong too. What you or she personally believes doesn't change probabilities, making one more likely than the other.

I think you think that I think that believability changes the probabilities. I didn't say that!

I don't know how many times I have to tell you this, but I said that the probability of getting any sequence under this random process assumption is the same! I'll say it again: It is the same! One more time: It is the same!

Believability is related to likelihood! Likelihood doesn't change the underlying process: it's used to try and estimate characteristics of the process!

I don't understand how you don't get this!

If someone rolled the dice and got a million ones one after the other, even if the dice rolling was a pure random process, do you really think it is more "likely" given "likelihood" that the dice comes from a pure random process or not?

Likelihood doesn't change the underlying theoretical probabilities at all! It's used to make inferences based on the sample you are given. You can still make incorrect inferences based on your likelihood methods and in context with this problem, a inference saying that 20 or even 100 ones in a row don't come from a random process could well be wrong!

I think you need to study what likelihood is, and how it is used in statistical inference.

 

[DIABEETUS]

http://www.parade.com/askmarilyn/2011/08/Likelihood-of-Die-Tosses-15.html

 

[Bacle]

I don't understand your point, DIABEETUS, in posting this. Should we just accept her
answer?

 

[DIABEETUS]

haha! You should never "just accept" an answer because someone says so. I posted that for a couple of reasons:

1. At the beginning of the thread, someone posted there was apparently NO follow up discussion to this particular puzzle (especially from her).

2. Hopefully this will shed at least SOME light to some of y'all's questions about her answer.

 

[pwsnafu]

Marilyn writes

In the die toss, the mixed bunch of numbers was much more likely to have occurred than the string of ones because the event happened in the past.

which is of course nonsense. The mixed bunch of numbers are more likely because there are less constraints. If I asked "which is more likely: string of ones or mixed bag?" the answer is the latter even if we are talking about future dice rolls.

 

[DIABEETUS]

which is of course nonsense. The mixed bunch of numbers are more likely because there are less constraints. If I asked "which is more likely: string of ones or mixed bag?" the answer is the latter even if we are talking about future dice rolls.

yeah, but I don't think she was referring to "mixed bunch numbers" series collectively in general, but rather the specific one mentioned, or any specific one for that matter (at least in THAT quote).

 

[Fredrik]

http://www.parade.com/askmarilyn/2011/08/Likelihood-of-Die-Tosses-15.html

I think this makes it more clear than before that what she had in mind all along is that one of the sequences was obtained by rolling the die repeatedly, and the other is just a lie, thought up by a human. But she still hasn't mentioned the correct explanation for why the "random looking" sequence is more likely to be the one that was actually obtained: Because a bad random number generator like a human is more likely to come up with a constant sequence than the die. It certainly isn't because it has already happened.

 

[Bacle]

I think she is mixing up outcomes and events ( of course, an outcome is an event, but not viceversa) , and the string with all 1's is an outcome, and a collection of different values is an event, i.e., there are more events in a throw with a variety of outcomes, then there are with all outcomes equal to each other.

Idot-Savant then tells us that the events happen in the past. I wonder how one
would deal with events that happenned in the future. My point is that, after her having created an artificial controversy by ill-posing the problem with the three doors ( Jim Morrison and two others ) , it would have made sense for her to be more careful with her use of language a second time around, or to at least give a more detailed explanation. Fat chance, it seems.

 

[pwsnafu]

I think this makes it more clear than before that what she had in mind all along is that one of the sequences was obtained by rolling the die repeatedly, and the other is just a lie, thought up by a human.

Which of course, wasn't what OP was asking. The original question was

Say you plan to roll a die 20 times. Which of these results is more likely: (a) 11111111111111111111, or (b) 66234441536125563152?

nowhere is it stated that one of the events is human generated. Heck, it specifically says "plans", which completely rules out Marilyn's reasoning!

Oh and she has the gall to write

You can't say a solver is incorrect because you didn't tell the truth!

*facepalm*

 

[Bacle]

You should have seen the tread marks she left when she tried so hard
to back-pedal from her claim that Wiles' proof of Fermat's last theorem was wrong,
bringing up Euclidean and non-Euclidean geometry. Not a pretty spectacle.

 

[Fredrik]

Which of course, wasn't what OP was asking. The original question was

Right. She answered that question correctly, and then went on to describe a different scenario that she came up with herself, and offered a really strange motivation for her solution to that problem. That's what's the discussion is about, not the original question.

 

[pwsnafu]

Right. She answered that question correctly, and then went on to describe a different scenario that she came up with herself, and offered a really strange motivation for her solution to that problem. That's what's the discussion is about, not the original question.

So...we were just wasting our time?

 

[Fredrik]

So...we were just wasting our time?

Not sure if you're asking if it was a waste of time to discuss the original question, or if it was a waste of time to discuss the problem she came up with herself. Either way, if you have learned something, or helped someone else understand something, I wouldn't say that you have wasted your time.

 

[andrewr]

Assuming your quote is fair (I can't see the article), Marilyn made an unfortunately common psychological mistake. The question she answered is very different from the question that was asked.

I am preparing to ask a question concerning fair dice, and found this thread attempting avoid wasting people's times on trivialities and mistakes I might make.
I searched for "fair" dice.

As physics forums is dominant (by funding and perseverance in the internet arena), comments from its famous posters are sure to reach Marilyn's review, eventually -- I wonder if she already has seen this...

I don't grasp how Marilyn necessarily made a psychological mistake as you mention it here.
She did answer a question which might not have been asked, but it also might have been asked. You yourself indicate I haven't seen the original article question -- so I judge your response here ONLY on the quote given in the first post of the thread; and I find that very curious.

Please review that for context, as it has been a while.

This statement is quite true. But can see why this has no bearing on the following question?

Would you show how you derive your own "following" question from the original quote of Marilyn?
Are you answering Marilyn's question, the original poster's question, or another of your own?

(I think chiro is making the same mistake -- answering the question of "all 1's versus a mix of all numbers" instead of answering the question "all 1's vs that other specific sequence of numbers")

Hmmm, why?
Marilyn stated a hypothetical Q, which is interpretable: (paraphrase):
If you prepare to roll a dice 20 times, and THEN (consequently) provide a sequence of all 1's vs a series of mixed numbers; which is more likely to be the true answer about what was rolled?

She could be asking about the psychology and also the statistic about which *sample* from a single run of the test would be more likely to be a lie/outlier? She does say that Both are equally likely as a specific answer according to THEORY, but she NOTES that the signature of mixed digits is seen far more often than the signature of a single repeated digit.
(She is aware of the Hemholtz principle.)

By a fair "Runs" analysis, I am absolutely certain the odds of getting answer (a) would lead a Casino to reject (a) as a loaded dice, but allow (b) as a "fair" dice. (This is one of the questions dealt with regularly when measuring a "Fair" dice.)

The problem is that Marilyn is judging the outcome based on a single example; For this, one would need to analyze based on Chi**2 analysis or an EXACT TEST of the variance of the 20 INDEPENDENT rolls. As a statistics run analysis --- "111111111111" 20x times would certainly be rejected as a loaded dice; whereas the other value would not.

May I ask, what school did you study probability and statistics at, and what text?
I'm curious if I learned from an equal source...

If you still maintain a case after my gentle cross examination -- I will bump Marilyn herself, as she does accept my e-mail, and ask her for her own take on this issue.

I do believe it is only right that everyone accused from an entrenched position should be allowed to face their accuser.

That's also why I try to avoid accusing until backed into a corner; I like to practice the virtue of truth in disclosure among disagreeing parties;eg: as a way to come to consensus and NOT compromise.

Cheers.

 

[Bacle2]

"don't grasp how Marilyn necessarily made a psychological mistake as you mention it here.
She did answer a question which might not have been asked, but it also might have been asked"

You would have thought that after the confusion she caused by stating the Monty Hall problem ambiguously, that she would make an effort to avoid ambiguity. Fat chance.

"If you still maintain a case after my gentle cross examination -- I will bump Marilyn herself, as she does accept my e-mail, and ask her for her own take on this issue."

If you do, ask her to support her claim that she appears as "highest IQ" on any book, and to clarify the meaning/context of that statement. She never answered my e-mails.
I looked for many years on Guiness and other record books and never saw her listed.

For someone who takes strong positions on ethical issues, and has often strongly chastised certain behaviors, you would think she would be more careful with her own actions.

"I do believe it is only right that everyone accused from an entrenched position should be allowed to face their accuser.
"

Maybe if she actually answered my/others' questions at all, I would back down from my statements. I e-mailed her a few times and she never bothered to reply, nor to post an answer in her site.

 

[pwsnafu]

She did answer a question which might not have been asked, but it also might have been asked.

Here's the question again:

Say you plan to roll a die 20 times. Which of these results is more likely: (a) 11111111111111111111, or (b) 66234441536125563152?

Marilyn was not asked about what would happen after the rolls were made. The original question was unambiguous.

Edit: Just a note: "As a statistics run analysis --- "111111111111" 20x times would certainly be rejected as a loaded dice; whereas the other value would not."
If you obtained "66234441536125563152" exactly 20 times in a row, you would be worried as well.
Observe the fact that 20 trials is far to small to do Pearson's chi (you would want at least 2 more orders of magnitude).

 

[micromass]

If you prepare to roll a dice 20 times, and THEN (consequently) provide a sequence of all 1's vs a series of mixed numbers; which is more likely to be the true answer about what was rolled?

Both are equally likely, according to theory. Human psychology dictates us however to say that 1111111111111 is from the faulty dice. But human psychology can be wrong.

but she NOTES that the signature of mixed digits is seen far more often than the signature of a single repeated digit.

Now you're falling in the trap. You're comparing "single digit" versus "all mixed digits". Of course mixed digits are more likely, because there are more possible outcomes.
However, you should test 'single digits' versus 'specific other outcome'. Then both are equally likely.

Since you mention statistic, you should know that it's impossible to prove anything with statistics. It is merely possible to give a likelihood statement or to make the chance on a type I error small. It is impossible to show, using statistics, that a dice is faulty.

 

[alan2]

I think she's just silly. Consider this question: In a roll of 20 dice, which is more likely, 66234441536125563152 or something else. The answer is clearly something else. But that's not the question. Which is more likely, 20 consecutive ones or 20 consecutive twos? Any two specific results are equally likely.

 

[chiro]

It might be helpful to consider estimation as opposed to true underlying process probabilities.

The estimation in this context refers to estimating the probabilities from the data and the actual process probabilities are the actual probabilities that represent the complete process.

 

[andrewr]

I think she's just silly. Consider this question: In a roll of 20 dice, which is more likely, 66234441536125563152 or something else. The answer is clearly something else. But that's not the question. Which is more likely, 20 consecutive ones or 20 consecutive twos? Any two specific results are equally likely.

Of Course Marilyn is precious; what has that to do with the question she answered?
A woman has a right to be silly -- AND right.

Again, the question:

(Paraphrase)
If YOU roll the dice 20x; and YOU report these two numbers xxx, yyy, which is MORE likely to be true? 1111111111111111 or a random sequence of mixed digits?

Please quote the question MARILYN was answering and show your paraphrase is identical in meaning to her words; eg: don't change the wording to make her wrong. I am giving you and her the benefit of the doubt.

I call anything else, out-lie-r; to be blunt -- a LIE.

Now, I ask you separately from Hurkle (let him speak for himself, too) How do you DERIVE your question as equivalent to hers?

Hurkle says he did not have any information except what was stated the first post, perhaps you are different ?

 

[Loren Booda]

Hi everyone,

Someone posed the question:



and Marilyn (high IQ record holder) answers:



What do you guys think? You can find the original in the link below.

http://www.parade.com/askmarilyn/2011/07/Sundays-Column-07-31-11.html

I was the one who posed the question, last summer, I believe.

 

[chiro]

I'm sorry guys for getting slightly off track but based on the posts (including the most recent ones) here is my 'conjecture' of what she was getting at:

When she was talking about getting all 1's and saying that it was not a 'fair die' I think what she was talking about was the process of likelihood and using likelihood and estimation to show how 'unlikely' it would be that the dice were random given the initial data that she received.

For the other part, well I interpret that to mean basic probability in the context of the actual underlying process that if given a real distribution that encapsulates the entire process probabilistically, does reflect the true probabilities of the entire process and not a selective subset.

A big chunk of statistics is based on the idea that you are given a 'snapshot' of data and from that, try to extrapolate probabilistic properties of the underlying process.

To me Marilyn Vos Savant is emphasizing an important caveat of this process that relates likelihood and estimation procedures back to the real underlying probabilistic properties of the underlying process in a way that highlights a statistical procedure in a psychological context.

Mathematically Marilyn Vos Savant could easily be wrong with her conclusion (as is pointed out by many members), but all of this is a standard well known part of statistical theory that scientists and others that use statistical techniques have to acknowledge in the form of Type I and Type II errors.

Again this is my interpretation and would welcome any feedback or further debate.

 

[pwsnafu]

Please quote the question MARILYN was answering and show your paraphrase is identical in meaning to her words; eg: don't change the wording to make her wrong. I am giving you and her the benefit of the doubt.

For the second time now: the original question is available at http://www.parade.com/askmarilyn/2011/07/Sundays-Column-07-31-11.html

Say you plan to roll a die 20 times. Which of these results is more likely: (a) 11111111111111111111, or (b) 66234441536125563152?
—Loren Booda, Arlington, Va.

You write

If YOU roll the dice 20x; and YOU report these two numbers xxx, yyy, which is MORE likely to be true? 1111111111111111 or a random sequence of mixed digits?

The original question was not a random sequence of mixed digits. 66234441536125563152 is a specific sequence of 20 six sided dice rolls.

 

[andrewr]

I was the one who posed the question, last summer, I believe.

Hi Loren!

Do you mean, you are the one who posed the question to Marilyn?


In the original question to Marilyn, it does not say whether or not the number to the right (the non 1111111111) throw was an a-priorori or a-postiori determined number.

It merely says, "which is more likely".
When the numbers became part of the test (before or after the roll) was not clearly specified in the opening post; They could have been arranged in many ways.

Marilyn has discussed this difficulty in the past:

For example, in a three shell game with an item hidden under one of the cups -- if a person points to a cup as their "choice", and then the shell master (helpfully) removes one of the non-chosen cups which is empty of the prize; The probability is not changed for whichever cup the item was (and still is) hidden under.
(It doesn't magically *move* after the choice...)
Therefore: The a-priori probability of a fair shell player is 1/3.

But the a-postiori probability after having a specific cup is removed means that it IS still random between two cups -- But it is no longer NECESSARILY of EQUAL probability; Eg: it is NOT 50/50%. (Nor is the dice "11111" vs. ANY Random sequence)

eg: I don't believe a person who is allowed to choose again NECESSARILY has a 50/50 chance of being right since there are two cups, and the actual cup is not known for certain; and I can write a Python program to DEMONSTRATE the assertion statistically.

This problem IS Marilyn's hallmark of fame against academic minded people in the past, BTW.

Bringing this back to the dice throw:

In the problem specified at the beginning of the thread, the question to Marilyn does not clarify whether the sequence given is an a-priori value or an a-postiori value. Hence, I think Marilyn's claim hinges on the ambiguity of the English of the question poser.

She DOES indicate that theoretically, the specific events are equally probable. (Right answer for the question interpreted as a FUTURE prediction among choices)

She then moves on to the question of "you" giving a Lie and a True answer to her, and asking her to a-postiori, determine which answer is more likely to be true about you throwing a dice 20x. Therefore, she is dealing with logic which you give her a "FALSE" answer and a true one -- eg: it isn't just randomness.

"But let’s say you tossed a die out of my view and then said that the results were one of the above."

Either you have lied to her twice by reporting two false numbers that your fair dice did not roll, or you are telling her the truth about one of the numbers.

It is this question that determines whether you are a liar or a truth teller.
In the end, you are either a liar totally -- at which point, she succumbed to a lie while giving you the benefit of the doubt about a "REAL" dice roll -- or else, you have told the truth -- and she knows statistically that the number on the right is more likely to be true of what YOU actually did with a dice 20x.

If you did not actually roll a dice 20x and report a real sequence of dice rolls to Marilyn, she can't be wrong -- for her premise is that you actually rolled the dice for the question, and reported that number AFTER the roll; (a postiori).

I can write a python program to test "11111111" vs. a random sequence -- and we know that it isn't psychology, but experience of gamblers which say "111111111" 20x (OR ANY FIXED SEQUENCE OF DIGITS) is the hallmark of either a liar, or an unfair dice.

You can't guess a set of digits in "advance", and have it happen to that many places of precision, without it statistically finding a crook, or rigged dice. I Could, for example, take the random sequence on the right -- and be safe in assuming that never in my lifetime I would see that exact sequence repeated in a gambling casino IN THAT ORDER. (I don't gamble that much, but ask people who do... THEY would remember if they saw a streak of 1's 20x long; That part is psychology. )

I am not advocating throwing someone into prison for rolling a "11111111" 20x; (They could). but I am advocating escorting them out of the casino and revoking their right to come back -- EVER.
Likewise, if they rolled the "random" number given to Marilyn in the OP, having now had this discussion -- I would be equally likely to suspect that person of having specially rigged dice.

Marilyn, however, did not say "11111111111111" vs ONLY "66234441536125563152"; she said
It’s far more likely that the roll produced a mixed bunch of numbers than a series of 1’s.

What say you?

 

[andrewr]

For the second time now: the original question is available at http://www.parade.com/askmarilyn/2011/07/Sundays-Column-07-31-11.html

For the THIRD time, HURKLE claimed he only had what was written in the FIRST post;
not what was said at parade.

I am not asking about the actual question, but about the one HURKLE saw.
The rest of you might be talking about a different subject; if HURKLE was off topic (The Opening POST?), let me know before punishing me.

eg: Let HURKLE answer for himself, please? :!)

 

[pwsnafu]

She then moves on to the question of "you" giving a Lie and a True answer to her, and asking her to a-postiori, determine which answer is more likely to be true about you throwing a dice 20x. Therefore, she is dealing with logic which you give her a "FALSE" answer and a true one -- eg: it isn't just randomness.
<snip>

Please explain how any of that is relevant to Loren's question.

For the THIRD time, HURKLE claimed he only had what was written in the FIRST post;
not what was said at parade.

What is on post #1 is what is on Parade. It is a word for word copy and paste. That's the point! AFAIK Marilyn has not responded to the criticism. She has not made a post in the comments.
You wrote "Please quote the question MARILYN was answering" . Sheesh.

 

[andrewr]

Please explain how any of that is relevant to Loren's question.

Simple, you go get a dice. Roll it 20x, and fairly (use a can to shake it rigorously before dumping). Record the 20x results. Then ask me whether or not you rolled a sequence of repeating digits "11111" "22222" "33333" ... "6666" (20x), as opposed to what the dice rolled.

We can do 10 posts with this game EXACTLY as Marilyn *allows* by her ambiguous answer.
I am allowing "2222","3333", etc, as choices for you because another poster in the thread understood the symbol "11111" to be an example of repeating digits. To use Marilyn strictly, I would have to force you to chose only "111111" vs, whatever you actually roll on the dice.

eg:
Let's actually test the GAME as Marilyn suggested, and see who is right statistically (eg: in a sample of 10 games.)

I will guess, every time that you rolled whichever sequence has the maximum variance.

A fair dice has a mean of 3.5; So, all ties can be broken; and in the case of duplicate numbers (left==right), I can't be wrong for you will have rolled the same value I pick.

If you don't report to us the/an actual fair dice roll sequence, you are violating the premise of Marilyn's answer to *one* possible interpretation of the question to her. (detect True roll vs. Lie.)

Also, re-read my post to Loren. It wasn't to you; and it asks for clarification regarding the question -- not in terms of what was written, so much as what was in Loren's mind, and in what way is she (or not) involved in asking Marilyn the question.

 

[pwsnafu]

Let's actually play the GAME as Marilyn suggested, and see who is right statistically.

Why?

Loren wrote: "Say you plan to roll a die 20 times." Clearly there has been no rolling done.

I fail to see how Marilyn's "game" is relevant to the question Loren posed.

 

[alan2]

@andrewr

Her original question, which appeared in the magazine was: "If you roll a die 20 times, which is more likely, 11111111111111111111 or 66234441536125563152"? Clearly neither. The question that she answered was 100% equivalent to the question that I posed: "If you roll a die twenty times, which is more likely, 66234441536125563152 or any other random sequence"? Please read the original question as posed in the magazine.

She has frequently given incorrect answers to probability questions. A drug testing question recently ran and she answered a different question in the same manner that she did this one and later apologized for it. Her response was that she misinterpreted the question. That doesn't make her correct. The question, as I recall was: "If a company randomly tests 25% of their employees each quarter for testing, what is the probability that any individual will be chosen over the course of the year"? The answer is about 68%. She replied, in a national magazine, that the answer was 25%.

 

[micromass]

Simple, you go get a dice. Roll it 20x, and fairly (use a can to shake it rigorously before dumping). Record the 20x results. Then ask me whether or not you rolled a sequence of repeating digits "11111" "22222" "33333" ... "6666" (20x), as opposed to what the dice rolled.

No, that's not what's going on here. The deal is: go get a dice and roll it 20x, then see whether you rolled the specific sequence 14325231542341632165. The answer will be no most likely.

Let's continue with the analysis. Let's write a computer program and let's do billions of dice rolls and let's measure whether 14325231542341632165 and 11111111111111111111 is more likely. Are you willing to accept the answer of a computer simulation??

eg:
Let's actually test the GAME as Marilyn suggested, and see who is right statistically (eg: in a sample of 10 games.)

LOL, a sample of 10 games. You know very well that you need to roll it many more times to have something statistically significant.

But, ok, are you prepared to do the computer simulation I proposed?? I'll even code it for you.

 

[lavinia]

What is "confidence"? Is it anything other than "I know the math says one thing, but I don't want to believe it"? (edit: I don't mean to be condescending, but it is really easy to try and rationalize one's intuition when faced with the fact it's simply wrong)
The mistake I mentioned earlier -- here is one way to make that mistake:

I'm going to invent a statistical test: my statistic T is the entropy distribution of observed frequencies. Computing T for 1111... gives a result less likely than computing T for 6623... Therefore, I infer that 6623... is what was rolled​

Hurky I see your points and agree but something is bothering me that maybe you can explain.

If I take independent samples from a distribution with finitely many values then for a large sample wouldn't I expect the frequencies in the sample to be close to the frequencies in the distribution? So forgetting the order of the digits in the not all 1's sequence - wouldn't it be more expected since its frequencies are more like the underlying uniform distribution? And I guess it is being assumed that the distribution is uniform in this case or at least very far from constantly 1.

 

[chiro]

Hurky I see your points and agree but something is bothering me that maybe you can explain.

If I take independent samples from a distribution with finitely many values then for a large sample wouldn't I expect the frequencies in the sample to be close to the frequencies in the distribution? So forgetting the order of the digits in the not all 1's sequence - wouldn't it be more expected since its frequencies are more like the underlying uniform distribution? And I guess it is being assumed that the distribution is uniform in this case or at least very far from constantly 1.

It depends on the specific probabilistic properties of the process.

If the process has very complex conditional probabilistic properties of any order that are known, then this information can be incorporated when you are trying to get likelihood information for a parameter.

This problem is essential in statistics. What we usually do is we assume that our data fits a specific model and then based on the data we find out how likely this really is.

Again with this kind of problem there are many perspectives you can take and a large amount of statistical work deals with the task of trying to get representative samples or design processes where a real representative sample can be obtained that 'represents' the real process in the best way possible (i.e. the distribution of the sample is a good representation of the underlying process distribution).

Statisticians have to do this all the time and consider the kinds of things that the OP has brought up and because of situations like this, we have to use a combination of solid mathematical foundations in statistical theory as well as some kind of 'inner judgement' that includes non-domain specific (general statistical understanding) as well as domain-specific knowledge to know when we should 'repeat the experiment just to be sure' or to 'look at the data and process it further' if we don't have the time or resources to do the experiment again.

 

[andrewr]

Why?

Loren wrote: "Say you plan to roll a die 20 times." Clearly there has been no rolling done.

I fail to see how Marilyn's "game" is relevant to the question Loren posed.

Emphasis mine:

Oh come forth(right) and use an English grammar book.
Loren said "YOU" and she used the infinitive "to".
Therefore, there is a colloquial expression and a variable interpretation of the hypothetical question involved.

Marilyn has the right to use her own opinion(eg: the YOU) about how Marilyn would roll and when/how she would report the results.

Her reply has a conditional answer for a given variation of the original question's meaning.

But let’s say you tossed a die out of my view and then said

The colloquial expression "But ... you" is a hypothetical question, meaning "if you"; and notice, Marilyn casts it in the PAST tense instead of the equivocal infinitive.

Your failure includes mis-understanding the sphere of discourse problem Marilyn was confronted with in the "OP" (I still haven't and won't read the parade article itself before reading Hurkle's response.)

The infinitive does not strictly define "when" an event happens. Connotation is NOT the same as denotation.
http://en.wikipedia.org/wiki/Infinitive

They do not have tense, aspect, moods, and/or voice, or they are limited in the range of tenses, aspects, moods, and/or voices that they can use. (In languages where infinitives do not have moods at all, they are usually treated as being their own non-finite mood.)

I read several languages, and the question Loren asked is a trick question.

As you (pf...) falsify the antecedent of Marilyn's SECOND response (as you clearly do) then her consequent statement SHOULD NEVER HAVE BEEN DISCUSSED AT ALL by you. eg: Marilyn is thus *CORRECT* in her evaluation of your interpretation of Loren's question, (for her answer STOPS before the BUT can be evaluated as TRUE -- no "BUT" about it.)

Anyone who judges Marilyn according to the consequent by saying the antecedant of Marilyn's reply can only be true in one way, is making a psychological and logical error. (by a fallacy...!)

Again, I was asking Hurkle how he judged the antecedent of Marilyn's hypothetical as TRUE;
He might have a legitimate answer -- but YOU do not, so far!

As you persist in attacking Marilyn -- tell me, how do you show her antecedent *is* DEFINITELY True in order to evaluate the consequent as an error?

No court would vindicate a judgment of Marilyn based on the low IQ grammar understood by most people in this thread.

Marilyn scored high in English as well as math; Take it all into account!

 

[micromass]

Are you now making your case by using linguistics?? This is not good...

 

[Bacle2]

Anyone who judges Marilyn according to the consequent by saying the antecedant of Marilyn's reply can only be true in one way, is making a psychological and logical error. (by a fallacy...!)

Listen, I usually make an effort not to carp on others' grammar unless it is egregiously wrong, given my own imperfections. Still, considering you're accusing us here of using "low IQ grammar" ( ever heard of punctuating as low-IQ grammar, before chiding others' grammar?), an unclear term, I will make an exception and will carp on every small innacuracy of yours. I like to do that with those who claim to be smarter than others.

1)"... by a fallacy"? Is that high-IQ grammar?

2) It is antecedent, not antecedant, mr high-IQ grammar. If you want to talk down to others you may want to spell-check before replying.

3)Learn the _actual names/handles_ of others : I, with my low-IQ can tell it is HURKYL.

4)How do you know the errors are of a psychological nature?

5)Do you have a copy of Marylin's IQ test? I have asked her to support her claims of having the highest IQ, and she has not replied, neither personally (I included my e-mail when I asked ), nor in her site. Moreover, none of the Guiness book-of-record editions of the last few years include her --in any category. Still, VS repeatedly takes strong ethical positions, chiding others' behavior. Maybe she would care to live by the standards she wants to enforce in others.

Now, would you please include a copy , or at least tell us her score, and some details of her test?

6)"Marilyn scored high in English as well as math; Take it all into account!"

Beside the above point, _you_ may want to consider that Marylin back-tracked in a very non-gracious way when her claim that the proof of Fermat's last theorem
was challenged.

And I doubt there is any relation between the level of math in an IQ test and advanced mathematics, tho..., maybe there is (sic) "by a fallacy"

Sorry for muh, rekuest, IQ majesty I is no have low IQ .

 

[Bacle2]

Are you now making your case by using linguistics?? This is not good...

Don't forget his use of appeal to authority--a fallacy -- by his mention that he knows several languages.

 

[pwsnafu]

Her reply has a conditional answer for a given variation of the original question's meaning.

Doesn't change the fact that she doesn't explain what her assumptions of the second half was. If you are going to change the intention of the question then be clear in stating the assumptions. If you andrewr had read the first half of this thread you would know that's what the bulk of the discussion boils down to.

The colloquial expression "But ... you" is a hypothetical question, meaning "if you"; and notice, Marilyn casts it in the PAST tense instead of the equivocal infinitive.

Your failure includes mis-understanding the sphere of discourse problem Marilyn was confronted with in the "OP" (I still haven't and won't read the parade article itself before reading Hurkle's response.)

The infinitive does not strictly define "when" an event happens. Connotation is NOT the same as denotation.
http://en.wikipedia.org/wiki/Infinitive

Yes, I understand all that, that's why I am able to make the claim she shouldn't have done so in first place.

I read several languages

As others have noted that's an appeal to authority, but I'll just say: so do I.

and the question Loren asked is a trick question.

Trick question (and I disagree on that) or not, she's still wrong.

Again, I was asking Hurkle how he judged the antecedent of Marilyn's hypothetical as TRUE;

That is why we have PMs on this forum.

He might have a legitimate answer -- but YOU do not, so far!

Apart from the fact that I'm not the only one arguing the irrelevance angle (see Fredrik's post #72), I already have given a criticism of Marilyn's second answer (see the end of post #84).

But because you clearly don't chase up references, to make this explicit (again): Marilyn is right when she claims that "t’s far more likely that the roll produced a mixed bunch of numbers than a series of 1’s." But she is wrong when she claims that 66234441536125563152 is a mixed bag of numbers. It is a very specific sequence. That's why it is equal odds.

No court would vindicate a judgment of Marilyn based on the low IQ grammar understood by most people in this thread.

What court? Courts are for legal issues.
Apart from being a backhanded argumentum ad hominem, the use of "vindicate" is an appeal to emotion. You are stooping low when you have to resort to these tactics.

Marilyn scored high in English as well as math

Clearly you have not.

 

[Hurkyl]

Hurky I see your points and agree but something is bothering me that maybe you can explain.

If I take independent samples from a distribution with finitely many values then for a large sample wouldn't I expect the frequencies in the sample to be close to the frequencies in the distribution?

Yes. The set of sequences whose frequencies are flat^*, for example, contains around 5 \cdot 10^{13} elements. Each element is just as unlikely as 11111111111111111111, but there are so many of them.

Of course, the odds of picking something from this set is still only 1 in 75...

*: Well, they can't be flat because 20 isn't divisible by 6, so I mean the frequencies are 333344


Let me repeat that, for emphasis. When picking the sequence of 20 digits at random, you have a 1-in-75 chance of getting the flat distribution. The reason is entirely because there are many sequences whose frequencies are flat. Each individual sequence with this property is just as unlikely as any other sequence -- do not get the idea that the individual sequences with this property are somehow more likely than any other sequence.

 

[Hurkyl]

"111111111111" 20x times would certainly be rejected as a loaded dice;

Replace 11111111111111111111 with any 20-digit sequence -- chosen before the dice are rolled -- and the same is true.

If you prepare to roll a dice 20 times, and THEN (consequently) provide a sequence of all 1's vs a series of mixed numbers; which is more likely to be the true answer about what was rolled?

(what does "mixed" mean? every number appears at least once?)

Your premise is not clear. If I operated according to the procedure

• Roll 20 dice and write down the sequence
• Come up some other sequence of 20 digits uniformly randomly
• Present both sequences to you

then under the hypothesis that I present to you 11111111111111111111 and 66234441536125563152, the odds are 50% - 50% that the dice really did roll 20 1's in a row.


But if I operated according to the procedure

• Roll 20 dice and write down the sequence
• If the dice roll was not all 1's, write down 11111111111111111111, otherwise write down 66234441536125563152
• Present both sequences to you.

then under the hypothesis that I present to you 11111111111111111111 and 66234441536125563152, the odds are still 50% - 50% that the dice really did roll 20 1's in a row.

Of course, if I presented you with 11111111111111111111 and 66234441536125563125, the odds are strictly 100% that the latter is what was actually rolled.


If I operated according to the procedure

• Roll 20 dice and write down the sequence
• If the dice roll was not all 1's, write down 11111111111111111111, otherwise select another sequence of 20 digits uniformly randomly
• Present both sequences to you.

then under the hypothesis that I present to you 11111111111111111111 and 66234441536125563152, then the odds that the latter is what was actually rolled is 3.6 \cdot 10^{15}


If I operated according to the procedure

• Roll 20 dice and write down the sequence
• Think up^* some other 20-digit sequence that contains every digit at least once
• Present both sequences to you.

then under the hypothesis that I present to you 11111111111111111111 and 66234441536125563152, the odds are strictly 100% that the former is what's rolled.

*: The particular method doesn't matter, so long as it satisfies the given constraint[/size]