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Is Marilyn Vos Savant wrong on this probability question? 
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#91
Feb1712, 08:29 AM

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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?? But, ok, are you prepared to do the computer simulation I proposed?? I'll even code it for you. 


#92
Feb1712, 12:55 PM

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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. 


#93
Feb1712, 10:36 PM

P: 4,572

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 nondomain specific (general statistical understanding) as well as domainspecific 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. 


#94
Feb2412, 03:21 PM

P: 263

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. Your failure includes misunderstanding 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 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! 


#95
Feb2412, 04:40 PM

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Are you now making your case by using linguistics?? This is not good...



#96
Feb2412, 04:42 PM

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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 lowIQ 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 highIQ grammar? 2) It is antecedent, not antecedant, mr highIQ grammar. If you want to talk down to others you may want to spellcheck before replying. 3)Learn the _actual names/handles_ of others : I, with my lowIQ 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 email when I asked ), nor in her site. Moreover, none of the Guiness bookofrecord 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 backtracked in a very nongracious 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 . 


#97
Feb2412, 05:04 PM

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#98
Feb2412, 07:07 PM

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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. 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. 


#99
Feb2412, 09:06 PM

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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 1in75 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. 


#100
Feb2412, 09:50 PM

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Your premise is not clear. If I operated according to the procedure
But if I operated according to the procedure
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
If I operated according to the procedure
*: The particular method doesn't matter, so long as it satisfies the given constraint 


#101
Mar912, 12:32 AM

P: 263

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 supercomputer 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 coprocessors 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. :) 


#102
Mar912, 12:50 AM

P: 263

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? 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 NONACADEMIC 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. 


#103
Mar912, 04:48 PM

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



#104
Mar912, 04:57 PM

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#105
Mar912, 09:19 PM

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Firstly, my code written in Scheme:
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 5050. Running it more than 1000000 times will only reinforce this, but I don't got the time to do so. 


#106
Mar912, 10:08 PM

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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? 


#107
Mar912, 11:26 PM

P: 4,572

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. 


#108
Mar1012, 02:42 PM

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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