Is Marilyn Vos Savant wrong on this probability question?

[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

 

[andrewr]

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??

It was a computer simulation that taught me the three shell problem; And I did accept it although I disagreed with my room-mate before I tried the program.

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.

Thank you, yes I would like to see how you code the program and verify it is at least algorithmically correct. I had some minor trouble in mine; for much of the tests, it is indeed nearly impossible to get an answer in "10" tries and so it is *very* difficult to verify that I coded the success counting section correctly for a 20x dice (so, if it ever does succeed, the program might just crash -- but I'm generally pretty good at debugging...)

For the 3 shell game I described, 10 runs is sufficient to notice a bias in the randomness, if there is one. I got 50/50 on my first try using the digits of pi mod 2 to choose among the two remaining shells. Not exactly random, but a good enough test.

I include the 3 shell casino, just as an example of how I code a probability demonstration, and a little fun. Let's have everyone play... ! and gather cumulative statistics...
I don't know about the 20x dice throw; but it won't hurt for a few thousand people to see if they can manually outguess python's well tested shuffling randomizer. Mercen? whatever twister core -- but pretty good.

If you catch a bug, let me know where and why it a bug in the code. :)
I'll fix it, if it is indeed a bug.

And, again -- Thank you for your offer to code something for me.
I love integrity, Micromass, it *always* impresses me; and it will save me some time.

I know C,C++,Java,Python,Fortran,Cobol,Snobol,assembly -- but here at the Farm (just a small one) we mostly have power processors free to do number crunching. Don't get me wrong, this isn't IBM's Haupage New York super-computer room; but I do have some spare computing... However, I can't use x86 based binaries; I *do* need source code.

If you read my thread on converting a binomial/normal data distribution, you'll note that even at 500,000 data points, that the Python gaussian random number generator has a inexplicable defect near the mean value; it can be seen in all three graphs, although it is a very small bias.

I *do* believe this is a problem with the math co-processors on the Intel platform. I also had to borrow one to run a test of the casino under windows. Intel's fpu has a minor underflow problem in the log function, and when used to produce a univariate random variable by inversion (e**-0.5x**2) by anti/inverse -function-- the problem shows up in the graph.

I tried to work around that in the casino by using shuffling of an unbiased deck in my example program -- and I have commented lines that allow you to see the random numbers generated and verify they are reasonably "fair", or to even replace the random number generator with one of your own. (not that it's really important for a three shell game...)

But for the 20x dice, a bias in the random generator might be suspect, right?

I'm looking forward to your program... I'm sure to learn something about you from it.
:)

 

[andrewr]

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

I already noted that in a previous post.
In fact, if the sequence mentioned in the OP were to come up at a casino -- I WOULD be checking for loaded dice; and I would be justified in doing so... DO you ever think I will?

(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.

That is the premise of "future" roll. I do include it in the casino... It is, as you say -- 50/50; even Marilyn agrees to that.

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.

This is exactly what I was wondering about how you think. I don't care to judge the rightness or wrongness of your response -- I just wanted to know how *you* personally approached the problem.

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

Which constraint is that?
A child playing dice with a friend, say a cup rolling dice game, refuses to show the roll sequence to their mate; but claims, it is '1111111111'; So the father comes over to stop the fight, and looks in the cup which was bumped. He sees a sequence of numbers and says to the other child, "it is either 1111111111' or '5248232123'; Then the father says to the less favored child, they are "both" equally likely. Now, we don't know what happened -- but it isn't about the probability of '5248232123' being rolled in the future. It's about what happened in an actual roll of the dice in a past game -- and cheating is suspected.

What would the other child do? (It's fair, he got all ones and that was perfect to win the game?), or would the child say "Marilyn, suppose you decided to roll dice; and then you told me '111111111' or '5248243123'; which would be more likely to be the true roll?" )
Obviously, one of the rolls is a lie -- for a dice can't be both; and it was already rolled as far as the child is concerned.

Clearly, the first child "COULD" have cheated. The total probability of the problem includes the number of ways a child could cheat according to *any* algorithm that is reasonably possible. (Let's ignore space aliens, although they *ARE* theoretically possible, they are as unlikely as 11111111111111111...).

The issue in my mind is that a child could have asked the question to Marilyn through their parent in a NON-ACADEMIC way; EG: The supposed asker of the question to Marilyn hasn't told us publicly how she came up with the question. I rather wonder if you will appreciate it if she does...

I just wanted to know how you personally thought through to an answer.
I'm not saying you're wrong or anything, I don't know your IQ score in comparison to Marilyn anyway. Why should I believe you aren't equals?

Peace. --Andrew.

 

[micromass]

Mod note: Let's please keep this thread on-topic. The topic is a probability question. Off-topic posts will be deleted

 

[micromass]

It was a computer simulation that taught me the three shell problem; And I did accept it although I disagreed with my room-mate before I tried the program.



Thank you, yes I would like to see how you code the program and verify it is at least algorithmically correct. I had some minor trouble in mine; for much of the tests, it is indeed nearly impossible to get an answer in "10" tries and so it is *very* difficult to verify that I coded the success counting section correctly for a 20x dice (so, if it ever does succeed, the program might just crash -- but I'm generally pretty good at debugging...)

For the 3 shell game I described, 10 runs is sufficient to notice a bias in the randomness, if there is one. I got 50/50 on my first try using the digits of pi mod 2 to choose among the two remaining shells. Not exactly random, but a good enough test.

I include the 3 shell casino, just as an example of how I code a probability demonstration, and a little fun. Let's have everyone play... ! and gather cumulative statistics...
I don't know about the 20x dice throw; but it won't hurt for a few thousand people to see if they can manually outguess python's well tested shuffling randomizer. Mercen? whatever twister core -- but pretty good.

If you catch a bug, let me know where and why it a bug in the code. :)
I'll fix it, if it is indeed a bug.

And, again -- Thank you for your offer to code something for me.
I love integrity, Micromass, it *always* impresses me; and it will save me some time.

I know C,C++,Java,Python,Fortran,Cobol,Snobol,assembly -- but here at the Farm (just a small one) we mostly have power processors free to do number crunching. Don't get me wrong, this isn't IBM's Haupage New York super-computer room; but I do have some spare computing... However, I can't use x86 based binaries; I *do* need source code.

If you read my thread on converting a binomial/normal data distribution, you'll note that even at 500,000 data points, that the Python gaussian random number generator has a inexplicable defect near the mean value; it can be seen in all three graphs, although it is a very small bias.

I *do* believe this is a problem with the math co-processors on the Intel platform. I also had to borrow one to run a test of the casino under windows. Intel's fpu has a minor underflow problem in the log function, and when used to produce a univariate random variable by inversion (e**-0.5x**2) by anti/inverse -function-- the problem shows up in the graph.

I tried to work around that in the casino by using shuffling of an unbiased deck in my example program -- and I have commented lines that allow you to see the random numbers generated and verify they are reasonably "fair", or to even replace the random number generator with one of your own. (not that it's really important for a three shell game...)

But for the 20x dice, a bias in the random generator might be suspect, right?

I'm looking forward to your program... I'm sure to learn something about you from it.
:)

Can you post an outline of your program in pseudocode, please??

 

[micromass]

Firstly, my code written in Scheme:

(define (MakeRandomList)
  {local [(define (MakeRandomList-iter n)
            {local [(define x (+ (random 2) 1))]
              (if (= n 0)
                  (list)
                  (cons x (MakeRandomList-iter (- n 1))))})] 
    (MakeRandomList-iter 10)})

(define (ListEqual List1 List2)
  {local [(define (ListEqual-iter l1 l2)
            (if (empty? l1)
                true
                (and (= (car l1) (car l2)) (ListEqual-iter (cdr l1) (cdr l2)))))]
    (ListEqual-iter List1 List2)})

(define list1 (list 1 1 1 1 1 1 1 1 1 1))
(define list2 (list 1 2 1 2 1 1 1 2 1 2))(define (Test n)
  {local [(define (Test-iter n amount1 amount2)
            {local [(define CurrentList (MakeRandomList))]
              (if (> n 0)
                  (if (ListEqual CurrentList list1)
                      (Test-iter (- n 1) (+ amount1 1) amount2)
                      (if (ListEqual CurrentList list2)
                          (Test-iter (- n 1) amount1 (+ amount2 1))
                          (Test-iter (- n 1) amount1 amount2)))
                  (list amount1 amount2))})]
    (Test-iter n 0 0)})

(Test 1000000)

A disclaimer first: the original post worked with "rolling the dice 20 times". This is unfeasable. Therefore, I changed the problem to "flipping a coin 10 times".

I worked with the two sequences 1111111111 and the supposedly random sequence 1212111212.

Now, what I did was:
Each test, I flip a coin 10 times. If the result is not one of the two sequences above, I discard the test. If the result is one of the two sequences above, I add 1 to the amount of times I saw the sequence.
This I do a million times.

Why is this a good representation of the test?
The original test was that I flip a coin 10 times. Then I get a choice which one of the above sequences was rolled. Of course, to get that very choice, I actually need to get one of the sequences. This is why every experiment where I do NOT get one of the sequences, I discard it.

After I got one of the sequences, I can choose which one of the sequences I get. Adding 1 to the amount of times I saw sequence 1 corresponds to getting it right if you guessed 1. Adding 1 to the amount of times I saw sequence 2 corresponds to getting it right if you guessed 2.
Eventually, the two amounts correspond to the number of times you got it right.

So, after iterating it a million times, I get
Sequence 1: 948
Sequence 2: 995

A subsequent test yielded:
Sequence 1: 1015
Sequence 2: 1001

These two are so close together that it seems plausible that the actual amount you get things right is indeed 50-50. Running it more than 1000000 times will only reinforce this, but I don't got the time to do so.