“There is a
principle which is a bar against all information, which is proof against all
arguments and which cannot fail to keep a man in everlasting ignorance—that
principle is contempt prior to investigation.” – Herbert Spencer
Our Question:
A woman and a man
(unrelated) each have two children. At least one of the woman’s children is a
boy, and the man’s older child is a boy. Do the chances that the woman has two
boys equal the chances that the man has two boys?
Assumptions:
Boys and girls are equally likely.
Our problem statement is true.
Marilyn answers no, I answer yes. She is wrong; she is on the wrong side of the paradox - on the naive side of a counter-intuitive question - and refuses to know it. I told her I would send $1000 to her favorite charity if “my argument is wrong.” She said, “Eldon, I take the bet, you’re wrong, send the money to the American Heart Association.” To my knowledge she has not even looked at my argument. She offered the conventional argument, then has remained silent amid proofs of its falsity. Put her argument into logical steps; I’ll show you where the false statements are.
Eldon’s $1000.00 challenge:
Prove my logic wrong and I’ll send the money to the American
Heart Association in your honor. (Whoever you are but disprove the logic,
typos don’t count). When my argument is put into logical steps, there are no
false statements. To claim the challenge, show me a false statement, in my
argument. Email me Eldon Moritz
Email me? Certainly. But first, understand that two coins flip four equally likely ways. To flip two coins three equally likely ways, the null must be selected, prior to the flip. Concede this point, or argue it first. There is a subtle difference between, "Two coins were flipped and Outcome (n) was flipped around," and "Two coins were flipped and Outcome (n) didn't happen." Hopefully, you'll understand that difference, before you email; however, People who understand that difference, usually don't even write.
Next understand that when the writer of a question thinks one question, but writes another, the written question prevails.
Herb Weiner, on the Wiskit lists two interpretations.
1.You meet a woman, and ask how many children she has, and she replies "two." You ask if she has any boys, and she replies "yes." After this brief conversation, you know that the woman has exactly two children, at least one of whom is a boy. When the question is interpreted this way, the probability that both of her children are boys is 1/3, as Marilyn has claimed.
2.You meet a woman and her son. You ask the woman how many children she has, and she replies "two." So now you know that this woman has exactly two children, at least one of whom is a boy. When the question is interpreted this way, the probability that both of her children are boys is 1/2, as Eldon has claimed.
Dr. Terry Moore, a Mathematician in New Zealand, and Sean MacLeod, a graduate of Harvard, agree. According to them, we should vote on which of the two questions is the most reasonable. They think that number 1 is the question intended by the questioner. Eldon Moritz a charter pilot in Abilene Texas, and at most, an amateur mathematician, sees the fallacy in this reasoning. Don't worry about what the writer intended, answer the question, as written. If the writer thinks one question, then writes another, the written question prevails.
Political Question? This is not a political question,
it is not a matter of opinion. This is a simple little logic question, it has
an answer and Marilyn is wrong.
Marilyn vs Marilyn! A textbook example with Marilyn on the other side of the fence.
A Maths Solution! With help from Dr. Terry Moore of New Zealand 12/12/'03.
How to win!! Send me a working model that works. Do not send me Marilyn's silly little, incorrect, argument. (several people have done that already)
Moritz Gray Paper! In this paper Dr Gray put my argument into mathematical terms. This will show you that I'm not operating entirely without adult supervision.
A few daffinitions:
The Book: The Power of Logical Thinking. Written by Marilyn. First Edition: February 1996
The Key:
The Key: The real key is in the order of events. Were 'boys' chosen, then the family? Or, was the family chosen, then the problem statement made? There is no evidence for the former. The problem statement itself is evidence for the latter.
For my friends from Georgia Tech. Obviously, as we learned from our statement, boys were chosen. Were they pre chosen?
A Historical Event: Our question is talking about a historical event. Two families were selected, our problem statement was made, our question was asked. We are talking about two families.
The one-thirders. People who like Marilyn, think the lady’s chances for two boys are 1/3. The one-thirders have a tremendous ability to act dumb.
The boy statement. At least one is a boy.
The girl statement. At least one is a girl.
A priori. Without further inspection.
Whatever. Whatever made the boy statement? Whatever made the boy statement. Whatever picked the families. Whatever picked the boy child. Whatever posed the question. Whatever may be man, woman, or machine.
The Original Collection: A collection of families with two children.
Marilyn’s Collection: It is our Original Collection, minus the GG families. This can also be called a Short Collection/
Another Short Collection or The Other Short Collection:
This is The Original Collection with the BB families removed. Note that one half of the families are members of all three collections. All families are members of at least two.
Marilyn states that there are four ways that the woman can have two children, then assumes that the woman’s family was selected from this short collection. Obviously in error.
The Australians I've been involved in a forum with some young scientists from Australia. They are an interesting group, but seem to have gone to Marilyn's one thirder school. Being convinced that our two families have differrent chances, they pay little attention to what I say.
Our two families’ chances are the same.
Marilyn is incorrect on this one. Marilyn said in the book on page 20, "I don't especially mind being wrong, but I do particularly mind being thought wrong when I'm right."
Eldon says unlike Marilyn, I do especially mind being wrong and I particularly mind being thought wrong when I'm right. When I'm wrong I want to correct it, I want to be right. When I'm right I don't want to be thought wrong. Marilyn told the world I'm wrong when I'm right, now she won't answer. I want to correct that, even though in the grand scheme of things, it probably doesn't matter very much;)
Moritz’
Apology
People have written, disagreeing with me, refusing to understand my logic. I am unable to tell who can’t and who won’t. I have had the attitude that people smart enough to be interested in this question would be able to understand and therefore just won’t understand. In this I may have been wrong. I've had correspondence with Collegiate Faculty. He, so far, has not understood my argument. He refuses to follow the examples. This question may be harder than I thought.
To those who have written and honestly can’t understand, I apologize. There is no shame in being dumb.
To
those who refuse to understand; shame, shame. There is no glory in ignorance.
The Experts can be wrong!!
Marilyn is wrong on this one but so far has refused to discuss it. She defers to Martin Gardner as her expert but on this one Martin agrees with me.
Martin Gardner said that he has spent too much time on this question and doesn’t wish to discuss it. I respected his wishes but he has written several times on said question and his writings agree with me. Martin said that when you select a family from the short group, then you get one third. I agree with this, whether it be the man’s or the woman’s family. In his book, Aha Gotcha! , he wrote that to get 1/3 when flipping two coins the flipper had to agree, in advance to always call heads when possible and to reflip on two tails. This agrees with my logic. He wrote in the 1959 Scientific American that when a parent tells us that “at least one is”, then the probability for two is one-half.
Ian
Stewart, a Ph. D. with impeccable credentials, writing in the same Scientific
American but thirty years later used the same, Mrs. Smith said, then went
to the Bayes’ Theorem and got an answer opposite to Martin Gardner’s. One
of them is wrong and to you Scientific American I will bet for Martin
and against Ian on this one. In their defense, they may have
corrected their errors in the intervening years and I missed the article.
Herb
Weiner got stuck in the ambiguous mode. He is wrong. He has many
ambiguities on his web page. This question is not one of them. Herb may have
surpassed the limit of his intelligence, probably not. He could probably
understand this argument if someone in whom he has confidence explained it to
him. He seems awed by Marilyn’s intelligence(he wrote her a patronizing letter)
and reluctant, in this case to understand that she is wrong. He keeps saying
things like, “what if we change the question?” Herb, change the question
and you get a different question, possibly a different answer. He said I don't owe the money because the question is ambiguous. Thanks Herb, I don't owe the money, but because my argument is correct.
Dennis Rakestraw understands the question to an extent but wants to hold out room to ask the lady the question, “Do you have at least one boy?” Dennis, that changes the question.
Fred Moolten, a Microbiologist from Boston U. showed up in my email complaining about my statement on the wiskit. That people who accept the one third argument have no more intellectual curiosity. He made a couple of points, then said, when all we know is that at least one is a boy, then the probability is one third. I said, “Fred, that is the question we are trying to explore.” Since then he has refused to answer my email. He has refused to discuss our question. He has demonstrated no intellectual curiosity to me. He stopped where our question starts. He, a microbiologist, is not too dumb to understand. He has chosen to remain ignorant.
Harry Eaton seems to have impressive credentials. His stuff on the wiskit about seeing the one child is correct. Harry may understand this question. I alienated him too early and he won’t discuss it with me. My fault.
Zelda Klotz sent me an email. Said that she is a mathematical person, that she does not understand my argument but that it is wrong. “Zelda, before you condemn my argument, try to understand it.” I refuse to believe she is too dumb to understand my logic. She seems to have a chip on her shoulder about being female and has not ever considered that my logic is correct.
Dr. H. Eldon Sutton, a Zoologist from The University of Texas, sent me an email and tried to teach me Marilyn’s argument. So far he has refused to look at mine. “H. Eldon, I understand Marilyn’s argument, it is wrong, I have proof. My argument is correct. Do you not even care?” I don’t know what his story is. I like his name. I was in Austin March 20, 2000 and talked to him on the phone. He would not listen. It was our first conversation; he hung up in my ear. I do not understand that kind of scientist.
I will give H. Eldon the benefit of the doubt. I don’t know him. He showed up unsolicited in my email. He is probably very respected in his field. He just butted into my question, then refused to hear my argument. Then again, he may be senile. I just don’t know. If he wants to shut me up all he has to do is show me that my argument is wrong. He is an old man and I’m no spring chicken myself. We should be able to discuss this question sensibly. If it is too late and he is out of it, I apologize to his family. If he’s just too dumb to understand the logic, I apologize to him.
Theodore Hwa of Stanford U. sent an email. He wanted to do a priori. His argument was correct except he couldn’t handle the part about the problem statement is true. When he realized this he should have gone to the correct answer. He did not. He seemed to go into the ambiguous camp, and then went radio silent.
There
was a mathematical person from Georgia named Jud McCranie who sent me an
email and said that I am wrong. He refused to consider my argument. He sent a
couple of foolish arguments that he said he did not want me to publish. I don’t
blame him, but offered him the opportunity to change any argument he was
ashamed of. He has refused further discussion.
I have had an enormous and delightful correspondence with Dr. Terry Moore, a Ph.D. from Massey University in New Zealand. Dr. Moore started out saying that I was wrong. Now he says that my argument is correct but that it is only my interpretation of the question. He holds out that there is another “interpretation” of the question whereas the answer is one third. God bless Terry Moore. He just won’t follow the argument far enough. Dr. Moore is a superb math historian, he is probably an expert teacher; He should be able to understand my argument. For some reason, he won't. Dr. Moore, show me another, valid interpretation and a working model which works. Show me a valid interpretation which doesn’t change the question.
For those of you who came from the wiskit, probability of boys and wish to return. Hello, thanks for coming, good bye and God Bless.
I appreciate Herb Weiner, writer of the The Wiskit. I disagree with much of his logic.
Herb specializes in finding Marilyn wrong. I don't care about Marilyn being wrong, she just stumbled in on the incorrect side of my question. My interest is in getting this question correct. Marilyn is smart enough to understand the logic. I have no evidence that she has looked at any of my arguments.
Herb sees that my answer is correct but then he sees an ambiguity. What he sees as an ambiguity is actually an incorrect assumption. He wrongly assumes that the gender decision was made. As per his homepage he bungee jumped off The Royal Gorge Bridge. Methinks he went A Bridge Too Far.
Return to the wiskit, Marilyn is Wrong page.
Nick understands this question, maybe better than I do. He says that Marilyn has committed the "Fallacy of Equivocation", or "Sophismata Equivocatio" (Aristotle). He says that, "Most, if not all basic logic textbooks explain it, and at least one of them uses this very problem to illustrate it, clearly and indisputably enough for any Freshman to "get it".
I quoted Nick in a way which he didn’t wish
to be quoted. I’ll print any retraction which he sends. I think it’s a moot
point as not very many people read this page.
Nick has written:
For your info, Marilyn’s trail of “authority”
has led to yet another blind alley. She told me to get the proof of her answer
from Gardner. He declined to do so. When I told him that that means he doesn’t
have one, he told me to get it from a Stanford professor named Persi Diaconis.
Diaconis has declined to give it to me. The blind alley has gone far enough to
prove that her answer is based on no proof and false attributions. You may
quote me on that.
Eldon Moritz sticks by his argument. It is correct.