Is Marilyn Vos Savant wrong on this probability question?

Studiot

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

Agreed, good point.

chiro

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

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

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

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.

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

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/201...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

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

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

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.