Is Marilyn Vos Savant wrong on this probability question?

[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%.