Is Marilyn Vos Savant wrong on this probability question?

[SW VandeCarr]

Hurkyl, I admire your persistence in try to explain this... if I gave you a random sequence of numbers and asked you to roll that SPECIFIC sequence, I would find your getting it to be exactly as unlikely as your getting all ones."

Exactly. Why did it take 30 posts to explain this? The probability is $10^{-20}$ for either sequence assuming independence and every digit is equally probable.

 

[phinds]

Exactly. Why did it take 30 posts to explain this?.

Don't know, but that's why I admire Hurkyl's persistence.

 

[OpenGates]

Phinds got it right. This is apples and oranges. Probability is the apples and believability is the oranges.

The probability that you roll all ones is the same as the probability that you roll any other sequence.

But the believability that you roll all ones is lower than the believability that you roll a less aesthetic sequence.

 

[DaveC426913]

Phinds got it right. This is apples and oranges. Probability is the apples and believability is the oranges.

The probability that you roll all ones is the same as the probability that you roll any other sequence.

But the believability that you roll all ones is lower than the believability that you roll a less aesthetic sequence.

I had a tough time convincing my brother that the lottery numbers 1,2,3,4,5,6,7 had exactly the same odds as any other selection of 7 specific numbers.

But the believability that you roll all ones is lower than the believability that you roll a less aesthetic sequence.

This is simply explained: in the bucket called "rolls of all 1's" there is exactly one result. In the bucket labelled "a less aesthetic sequence" there are many, many, many results. That's many, many, many possibilities that can match it.

 

[SW VandeCarr]

This is simply explained: in the bucket called "rolls of all 1's" there is exactly one result. In the bucket labelled "a less aesthetic sequence" there are many, many, many results. That's many, many, many possibilities that can match it.

I just wanted to correct any misconceptions that might arise from the above. There are not many possible matching results. We are talking about the probability of one unique sequence regardless of what it appears to be. For example, pi has just one correct sequence. There is no apparent pattern to the digit sequence, but the sequence is unique.

 

[LCKurtz]

I had a tough time convincing my brother that the lottery numbers 1,2,3,4,5,6,7 had exactly the same odds as any other selection of 7 specific numbers.

Certainly true. You can't increase or decrease the probability of selecting the winning lottery numbers no matter how they are chosen. But one thing you might be able to control is the probability of sharing the prize given that you win it. This is more a psychological problem than a mathematical one. For example, if it is true that birthday date numbers are chosen by players more often than random chance would dictate, you can perhaps increase your probability of not sharing any pot, should you win it, by avoiding the numbers 1 .. 31 in your selection.

I don't know if it is true, but several years ago I heard about a situation where some TV program had mentioned some numbers in an episode that didn't have anything to do with a lottery. But apparently several people decided to try those numbers for luck. The numbers came up and all those people got to share a much diminished prize. True or not, it illustrates my point.

 

[Hurkyl]

We are talking about the probability of one unique sequence regardless of what it appears to be.

The point is that while we are talking about one unique sequence, many people don't think of one unique sequence and mentally replace the specific sequence with the notion of a 'random' jumble of numbers.

 

[Studiot]

I am with Hurkyl on this one.

Chiro the way to check this is to simplify.

The argument is the same if there are only two throws and thirty six possible outcomes.

I roll the dice, without seeing them, and a friend reads them 2 minutes later.
The result therefore remains hidden for two minutes.

What you are asserting is that

before the roll of the dice
P(4,3)= P(1,1)
After the hidden roll of the dice
P(4,3) > P(1,1)


I assert that
Before the roll of the dice
during the roll of the dice
one minute after the roll of the dice
10 years after the roll of the dice

that

P(4,3) = P(1,1) = 1/36


To suggest that P(4,3) > P(1,1) would admit the idea of considering comparison this for all 36 outcomes and summing to greater than unity.

If you really want to get somplicated you can consider the difference between results (1,1) and (1,1) or (4,3) and (3,4)

go well

 

[DIABEETUS]

No, she's right, depending upon the dice thrower's behavior (believe it or not)...

Assuming that the dice thrower only accounts throws that result in one of those two sequences, then yes, the likelyhoods of either having occurred are the same.

However, assuming that the dice thrower allows for ANY result, then ANY mixed sequence could pop up, and the dice thrower would then declare THAT particular mixed sequence as the one to compare with the sequence of twenty 1's. This would obviously happen WAYYYY more often than the rolling a series of all 1's.

 

[Fredrik]

No, she's right, depending upon the dice thrower's behavior (believe it or not)...

Assuming that the dice thrower only accounts throws that result in one of those two sequences, then yes, the likelyhoods of either having occurred are the same.

However, assuming that the dice thrower allows for ANY result, then ANY mixed sequence could pop up, and the dice thrower would then declare THAT particular mixed sequence as the one to compare with the sequence of twenty 1's. This would obviously happen WAYYYY more often than the rolling a series of all 1's.

I agree that it depends on the dice thrower's behavior. The fact that she doesn't mention that, and the fact that she mentions something else as the reason ("because the roll has already occurred"), makes me inclined to describe what she's saying as wrong.

Her claim: 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.

She's right if one of the numbers I give her is the actual sequence, and the other is a number I just made up. In that case, 66234441536125563152 is more likely to be the sequence I rolled, and 11111111111111111111 is more likely to be the number I made up. The reason is not that "the roll has already occurred". That's a garbage explanation. The reason is that the die is a better random number generator than I am.

If I had used a random number generator that's better than the die, then it would have been more likely that 11111111111111111111 is the sequence I rolled. The crappier the random generator, the more likely it is to produce a constant sequence.

 

[DIABEETUS]

I agree that it depends on the dice thrower's behavior. The fact that she doesn't mention that, and the fact that she mentions something else as the reason ("because the roll has already occurred"), makes me inclined to describe what she's saying as wrong.

...

She's right if one of the numbers I give her is the actual sequence, and the other is a number I just made up. In that case, 66234441536125563152 is more likely to be the sequence I rolled, and 11111111111111111111 is more likely to be the number I made up. The reason is not that "the roll has already occurred". That's a garbage explanation. The reason is that the die is a better random number generator than I am.

If I had used a random number generator that's better than the die, then it would have been more likely that 11111111111111111111 is the sequence I rolled. The crappier the random generator, the more likely it is to produce a constant sequence.

Because, as you said, she never mentioned ANYTHING about assumptions about how the dice are rolled and behavior and so forth, then technically she's wrong only on the account that the inquiry is INCONCLUSIVE... there's no way of determining the probabilities without making one of those requirements (assumptions) first. However, on a more practical level, I think its obvious that she implied to exclude such ridiculous assumptions such as: the dice thrower only accounts for rolls that produce those two particular sequences, and that the dice roller could be lying, and stuff like that. But I also do agree that her explanation is VERY, VERY vague and over-simplistic.

 

[Bacle]

Just to state that Marylin Idiot-Savant has the bad habit of posting problems without clearly explaining the "boundary conditions". It seems too, that her IQ credentials are suspect; I myself have checked in record books, but I have found no official records of her claims. It is disingenous for Idiot-Savant to claim she gives no importance to the IQ matter, and yet repeatedly posts while claimimg to have " The world's highest IQ", and it is arguably this claim that explains why most would care to read her column.

 

[Fredrik]

It seems too, that her IQ credentials are suspect; I myself have checked in record books, but I have found no official records of her claims.

Her Wikipedia page explains this. Link.

Guinness retired the category of "Highest IQ" in 1990, after concluding that IQ tests are not reliable enough to designate a single world record holder.
...
"Miss Savant was given an old version of the Stanford-Binet (Terman & Merrill 1937), which did, indeed, use the antiquated formula of MA/CA × 100. But in the test manual's norms, the Binet does not permit IQs to rise above 170 at any age, child or adult. And the authors of the old Binet stated: 'Beyond fifteen the mental ages are entirely artificial and are to be thought of as simply numerical scores.' (Terman & Merrill 1937)... the psychologist who came up with an IQ of 228 committed an extrapolation of a misconception, thereby violating almost every rule imaginable concerning the meaning of IQs."

 

[Bacle]

Fredrik:
Thanks for the link. What I think is dishonest about Idiot-Savant is her claiming to attribute no importance to IQ, yet including the claim "world's highest IQ" on most of her columns on 'Parade' magazine (at least last time I checked). If she made no such claim, it is likely, I believe, that there would be fewer people asking her questions.

 

[DaveC426913]

Fredrik:
Thanks for the link. What I think is dishonest about Idiot-Savant is her claiming to attribute no importance to IQ, yet including the claim "world's highest IQ" on most of her columns on 'Parade' magazine (at least last time I checked). If she made no such claim, it is likely, I believe, that there would be fewer people asking her questions.

What makes you assume she is the one making that claim?

 

[Bacle]

DaveC (sorry, my quoting function is disabled for some reason):

I don't know if she's the one that makes the claim, but I assume she

knows the claim is being made, and she must know that readership

increases as a result of that claim; people want to know what (allegedly)

amazingly-brilliant has to say, not what just Marilynn Q Anonymous has to say. So, she

either uses that claim to have a larger readership (and likely be paid more), or

allows that claim to be used in her name, and yet she claims that she assigns little

importance to IQ (let alone the fact that she does not mention the controversy behind

the truth of the claim ) . But it is precisely the fact that she (allegedly) has the world's

highest IQ that attracts readers, and she must know this. I think she should either drop

the claim of having highest IQ-- if she wants to say that she attributes no major

importance --or accept that it is this claim that attracts a good portion of her readers.

And there is too the issue that she often gives strong opinions on ethical issues. I would

just like here to be consistent in her position.

 

[DaveC426913]

I think she should either drop the claim of having highest IQ...

I would just like here to be consistent in her position.

I ask again. What makes you think she's making this claim at all?

How can she "drop" something that someone else is saying? She's have to make an effort to make a public announcement that 'those claims people are making about me are false'.

 

[Bacle]

DaveC: I am assuming she has some control of how she is depicted, i.e., I am assuming

she has the ability to tell the paper that she does not want to be depicted as having

the world's highest IQ (it would also be reasonable for her to try to explain away the

controversy surrounding her claim). And , if she does not have that ability, she has the

choice of not writing for Parade at all (she is married to a wealthy heart-doctor, so

I doubt she needs the money). She does ofter pass ethical judgement on

letters sent to

her, so I usually like those passing judgement , who declare others as being unethical, or

describe others' actions as being unethical, to act ethically herself.

 

[chiro]

I am with Hurkyl on this one.

Chiro the way to check this is to simplify.

The argument is the same if there are only two throws and thirty six possible outcomes.

I roll the dice, without seeing them, and a friend reads them 2 minutes later.
The result therefore remains hidden for two minutes.

What you are asserting is that

before the roll of the dice
P(4,3)= P(1,1)
After the hidden roll of the dice
P(4,3) > P(1,1)


I assert that
Before the roll of the dice
during the roll of the dice
one minute after the roll of the dice
10 years after the roll of the dice

that

P(4,3) = P(1,1) = 1/36


To suggest that P(4,3) > P(1,1) would admit the idea of considering comparison this for all 36 outcomes and summing to greater than unity.

If you really want to get somplicated you can consider the difference between results (1,1) and (1,1) or (4,3) and (3,4)

go well

I already said in a prior post in this thread that I agreed with Hurkyl's statement that every chance has the same probability of occurring.

When I was talking about confidence, I was talking about the "believability" of getting something like 20 1's in a row. While it is true that this has the same probability as any other sequence, the "believability" of this sequence coming from a pure random process is not as believable as getting a more "random-looking" sequence.

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.

Again, just to clarify I agreed with Hurkyl that the probability of every possibility is the same.

 

[Bacle]

Sorry for my detour.

Ii wonder if we can formalize the argument by using maximum likelihoods given

the fixed parameters of a multinomial with probability =1/6, using "backward probability"

. instead of the forward type, i.e., Bayes vs. "Standard" .

 

[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.