# Is Marilyn Vos Savant wrong on this probability question?

by CantorSet
Tags: marilyn, probability, savant
Mentor
P: 18,019
 Quote by andrewr 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.
P: 1,716
 Quote by Hurkyl 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.
P: 4,572
 Quote by lavinia 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.
P: 263
 Quote by pwsnafu 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!
 Mentor P: 18,019 Are you now making your case by using linguistics?? This is not good...
 Sci Advisor P: 1,169 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 .
P: 1,169
 Quote by micromass 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.
P: 820
 Quote by andrewr 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.

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 "[i]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.
Emeritus
PF Gold
P: 16,099
 Quote by lavinia 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.
Emeritus
PF Gold
P: 16,099
 Quote by andrewr "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
P: 263
 Quote by micromass 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.
:)
Attached Files
 MarilynCasinoPack.zip (9.9 KB, 5 views)
P: 263
 Quote by Hurkyl 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.
 Mentor P: 18,019 Mod note: Let's please keep this thread on-topic. The topic is a probability question. Off-topic posts will be deleted
Mentor
P: 18,019
 Quote by andrewr 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. :)
 Mentor P: 18,019 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.
 Sci Advisor HW Helper P: 9,453 If you think 1,1,1,1,1,1,1 has essentially no chance of occurring as the winning numbers in a lottery, then you have just answered why the lottery is not a good bet. I.e. every other choice is just as unlikely as this one in a fair lottery. It is ironic that Ms. Vos Savant would make this simple mistake since she rode to fame on a probability question that stumped some mathematicians (including me) as follows: Suppose there are three doors and a prize lies behind one of them, and you have one choice. After you indicate your preferred choice the moderator opens another door with nothing behind it, leaving two doors still closed, yours and one other. Then you have the opportunity of keeping to your original choice or changing it. What should you do, and why?
P: 4,572
 Quote by mathwonk If you think 1,1,1,1,1,1,1 has essentially no chance of occurring as the winning numbers in a lottery, then you have just answered why the lottery is not a good bet. I.e. every other choice is just as unlikely as this one in a fair lottery. It is ironic that Ms. Vos Savant would make this simple mistake since she rode to fame on a probability question that stumped some mathematicians (including me) as follows: Suppose there are three doors and a prize lies behind one of them, and you have one choice. After you indicate your preferred choice the moderator opens another door with nothing behind it, leaving two doors still closed, yours and one other. Then you have the opportunity of keeping to your original choice or changing it. What should you do, and why?
I've said this before, but I think it's important to bring this up.

The differences IMO that Ms. Vos Savant is talking about is the comparison of an underlying process vs the estimation of process parameters using likelihood techniques based on existing data.

Hurkyl is right in saying that if the underlying process is random, then every combination will be as unlikely (or likely) as every other possibility. No argument there.

But an important thing that statisticians have to do is 'guess' the probabilistic properties of a stochastic process using data. For a process that is binomial we use things like MLE estimation and using this we get the estimator to be t/n +- std where t is the number of 'true' or 'heads' and n is the number of trials.

My guess is that Marilyn is talking about likelihood estimation in the very last statement as opposed to true underlying probabilistic properties that Hurkyl is referring to.

Again if the dice are really and truly from a purely random process then Hurkyl is right, but if we have to measure some kind of 'confidence' by taking existing data where we do not know the real underlying process and have to make a 'judgement' about the probabilistic properties of the process where we don't actually know them, then if a likelihood procedure was done on a space with 6 possibilities per trial with 20 trials and we get all 1's, then given this data we have to say that we are not 'confident' that this data comes from a process that is purely random.

It's important to see the distinction: the likelihood results do not say that it doesn't come from a particular process, but rather gives evidence for it either coming or not coming from a particular kind of process.

Statisticians have to do this kind of thing all the time: they get data and they have to try and extract the important properties of the underlying process itself. We don't often get the luxury of knowing the process in any great detail so what we do is we say 'this model looks good, lets try and estimate its parameters using the data'.

People have to remember that the probabilistic properties of the true underlying stochastic process that is known and the exercise of trying to measure distribution parameters for a process that is not known are two very different things.

One specifies properties for a process that is known and the other tries to 'figure out' using sound statistical theory 'what the specifics of the process should be given the data since we don't actually know the underlying process'.

Again, two very different things.
 P: 129 Both probabilities are equally likely. On a side note, if I roll a fair dice 999999999999 times and get 1 each time, and I roll it again, the probability of rolling a 1 is still 1/6. (Empirically, we might dispute that the dice was fair, however! ;)) Here is a nice quote from Feynman: "You know, the most amazing thing happened to me tonight. I was coming here, on the way to the lecture, and I came in through the parking lot. And you won't believe what happened. I saw a car with the license plate ARW 357. Can you imagine? Of all the millions of license plates in the state, what was the chance that I would see that particular one tonight? Amazing!"

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