Making Money from Tuesday's Child

Here's a way to make money from the Tuesday's Child problem, assuming you can convince people that John Derbyshire is right and the probability in question is 13/27 (or 0.48, nearly even odds).

First, we need to convince seven of your neighbors of Derb's analysis, to wit, that the probability in question is 13/27. Then we need to find a large number of fathers with two children, at least one of whom is a boy. It doesn't matter on what days they were born. Have them bring the birth certificates. I hope you will agree with me that the probability that any father in this group has two boys is 1/3 (or 0.333, not 0.48).

Then we segregate the fathers into seven groups according to what day of the week the boy was born on. If the father has two boys born on different days of the week, have him flip a coin and join son #1's group if heads, son #2's group if tails.

Have the group with a boy born on Tuesday line up, and then repeatedly knock on neighbor #1's door and say:

"I have two children. At least one is a boy. He was born on Tuesday. Two dollars will get you three if I have two boys."

Hopefully neighbor #1 will take the bet, having been convinced that he's getting good odds. He's getting a 3:2 payout on a bet that he thinks is approximately 1:1.

Have the group with a boy born on Wednesday line up, and then repeatedly knock on neighbor #2's door and say:

"I have two children. At least one is a boy. He was born on Wednesday. Two dollars will get you three if I have two boys."

Do this with the groups for the remaining days of the week and remaining neighbors. As far as we are concerned, we are paying out at 3:2 a bet that is actually 2:1! (A fair payout would be four dollars for every two dollars bet. We are paying out as though the odds were 2/5 or 0.40. Our advantage is 7%, better than the house advantage in a typical casino game. ) 

As far as any single neighbor is concerned, he's simply seen a repetition of the Tuesday Child problem. Everybody who knocks on his door says the same day of the week. Once you've got all their money, they might get suspicious, but you've got the birth certificates to back it up... and the (bogus) mathematical analysis. Some people are just lucky, you will tell them.

The Tuesday's Child Game

In this post I discussed the so-called Tuesday's Child problem in probability theory. The theory (with which I disagree) is that the probability of that the speaker has two boys is 13/27. I think the probability is actually 1/3. I've made a crude Java applet, the Tuesday's Child Game, to demonstrate the point.

Since the theory claims the odds are approximately even that the speaker has two boys, if I give you anything better than even odds, you are probabilistically ahead of the game. The applet pays off at 3:2 so,
according to the theory, you should win a lot of money.

If you are still interested in this problem, and can get $100 before going broke, let me know.

Tuesday's Child

Here is a post at the corner concerning a probability question. It leads to a number of other posts and to Derbyshire's analysis here. If you follow the chain of posts back, it leads to other sites and considerable debate over the interpretation of this problem.

The analysis is tricky only if the problem is interpreted in something other than its straightforward, plain meaning. The statement of the problem is:

"I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?"

Now the problems all come from that "born on a Tuesday" clause in the middle sentence. Take that out, and everyone agrees on the answer:

"I have two children. One is a boy. What is the probability I have two boys?"

This is the classic coin-flip enumeration problem. Having children is like flipping a coin, with heads = boys and tails = girls. The possible outcomes for two consecutive coin flips are:

Heads - Heads
Heads - Tails
Tails - Heads
Tails - Tails

or, in the boy girl terms:

Boy - Boy
Boy - Girl
Girl - Boy
Girl - Girl

Since we know that at least one of the children is a boy, the last case is ruled out and the probability that the speaker has two boys is one in three.

Returning to the original problem, the analysts all seem to think the phrase "born on a Tuesday" is very significant, but they can't agree on its significance. I don't think it adds anything to the problem at all. In the straightforward, obvious interpretation, it is only a statement after the fact of birth concerning the day of birth. It's like saying the boy was 8 lbs at birth, or was born with blue eyes. It doesn't say anything about the prior possibilities of weight or birthdays; it is only a statement about what in fact occurred. It doesn't say that one or both boys couldn't have been born on a Wednesday. If that had happened, the consequence would be that the problem would say:

"I have two children. One is a boy born on a Wednesday. What is the probability I have two boys?"

The answer to this question is the same as the answer to the Tuesday question and to the simpler question that does not refer to a day at all: 1/3.

Derbyshire calculates the probability as 13/27. He can only get there by interpreting the "Tuesday clause" as affecting the prior probabilities of birth. In other words, the case of two boys born on Wednesday need not be included in our enumeration of cases because it wasn't possible for both boys to be born on Wednesday, since we know one was born on Tuesday! I hope everyone can see the post facto fallacy in this reasoning. Anyone who would buy this line of reasoning is playing the role of the father in the following comic scenario:

One day you get a letter from the town correcting your son's birth certificate. He was born two seconds after midnight so he was actually born on a Wednesday rather than a Tuesday. With a heavy heart, you sit your son down and tell him the unfortunate news: "I'm sorry to tell you this son, but I'm not your real father. My son could only have been born on a Tuesday, and I've just learned you were born on a Wednesday."

By the way, this problem is not comparable to the Monty Hall problem. The Monty Hall problem is a genuinely counter-intuitive probability result. The Tuesday's Child problem is more like a riddle or joke that depends on deceptive or ambiguous language.

Is Free Will a Contingent Possibility

With respect to this post at the Secular Right, Kierkegaard cannot be bettered:

Freedom is never possible. It is either actual or it is not at all.

Put another way... anyone who wonders if he is free (or even could possibly wonder if he is free) is already free.

Toy Story and Religion

I just saw the wonderful Toy Story 3 with my wife and daughter, and it put me in mind of an argument I read years ago at the Internet Infidels. I haven't been able to find the article (it was in the "Agora", which they no longer seem to have), but the gist of it was straightforward. Toy Story, the argument goes, is a parable of atheism. It is the story of Buzz Lightyear, a man living in a false world of imaginary Space Rangers and Evil Emperors, finally brought back to reality when his illusions are punctured. Buzz hangs on to his illusions as long as he can but, finally summoning the courage to find out the truth one way or the other, puts them to empirical test. One of his "special powers" is supposed to be an ability to fly, so he jumps off a second floor bannister in an attempt to prove it. Naturally, he falls to the floor, and is broken both physically and spiritually. But the story has a happy ending as Buzz is not only physically repaired, but learns to accept the non-dramatic and mundane truth that he is but a child's toy. Would that the Buzz Lightyears attending Mass every weekend could follow his example.

The argument is a good example of how atheist arguments can be perfectly sound but miss the target. The Christian can accept the argument in its entirety, and even applaud with the atheist Buzz's breakthrough to a true understanding of his nature. For it is not in his dreamworld as a Space Ranger battling Emperor Zurg that Buzz has found religion (or, at least, religion in the sense of a metaphysical religion like Christianity), but rather when he recognizes the true cause and source of his being; and that cause is a Creator who made him in light of a final cause: To be of service to a child in providing him joy in the form of a toy. And it is only when Buzz comes to terms with his destiny (a destiny created for him) that he can be truly happy.

Buzz Lightyear is no product of an atheist universe. If Toy Story were an atheist parable, then Buzz and the other toys would be the accidental result of a brute physical process. In those terms, their destiny as a child's plaything would have as much purchase as any other destiny; which is to say, none. Indeed, it would have no more purchase than Buzz's Space Ranger worldview. We can reimagine Toy Story in atheist terms in the following way: Finally tiring of Woody's attempts to "enlighten" him out of his Space Ranger fantasy, Buzz pulls Woody aside and lets him in on something. Of course, Buzz says, I know there is not an Emperor Zurg in the sense you think I think there is, and that I can't defy gravity. So what? Your insistence that I am "meant" to be a child's plaything is as much a fantasy as my Space Ranger worldview. The difference between us is that I know whatever purpose I give my life is purely of my own fantastic creation, while you are under the illusion that you "know" the "true meaning" of every toy's existence. You are, in a word, naive.

Why isn't the atheist version of Toy Story produced? It certainly isn't because Hollywood is afraid of offending religious believers. It's just because few people would want to see it. The story is boring. It's a story that can be told only once, and it was told long ago. It's the story of the discovery that, in the end, there isn't really anything worth discovering; a discovery that, if it puts an end to anything, it puts an end to storytelling.

Philosophy, Results, Kierkegaard and Socrates

In his recent article in the Fortnightly Review, Philosophy as a personal journey., Anthony O'Hear reflects on the meaning for philosophy that it has been unable to produce "results" with respect to its most fundamental questions:

But, and here is a second worry, given that, notoriously, most of the big disputes in philosophy remain unresolved – and have been unresolved since the time of the ancient Greeks who first raised them in systematic form – what can we actually learn from philosophy?

It is in light of the failure of philosophy to produce results that O'Hear develops his interpretation of philosophy as a personal journey. This is his last paragraph:

Of course, some of the people who write and practice philosophy in these ways will see their tightly focused work as contributing to a larger vision, but it seems to me that the overall direction is false to the true nature of the subject. And although we can all agree that our endeavours are directed to the truth, and guided by reasons and arguments that bear on the truth of what each of us believes, we each have to face the fact that we will not achieve complete rational convergence on premisses, because it is not there to be achieved. Nor will we come to a set of truths which will be so evident that they will command the assent of all who embark on the journey and pursue it in a rational and reasonable manner, aiming as best they can to seek the truth. It is just this picture which our earlier considerations on the nature and history of philosophical disagreement seem to undermine. In the beginning and at the end, philosophy is a personal journey, crucial to the examined life Socrates thought so integral to human flourishing.

I am afraid that, as edifying as I find O'Hear's article, I cannot agree with this paragraph. In fact, I think the paragraph clearly contradicts itself. On the one hand, O'Hear tells us that we cannot arrive at a set of truths that will command the assent of the rational and the reasonable. On the other hand, he proposes just this truth as one that every reasonable man should accept as the basis of philosophy. In other words, the proposal that "philosophy cannot arrive at a set of truths that will command the assent of the reasonable" is itself a purported reasonable truth that the proposal denies.

O'Hear's proposal is of the type that formed the original basis of Enlightenment philosophy, and was exposed by Kierkegaard (and, through him, Socrates) as failing to respect the true nature of subjectivity. Enlightenment philosophers concluded, like O'Hear, that the long history of philosophy proved the futility of the classical philosophical approach. Rather than continuing the fruitless dialog, they imagined various ways to found philosophy anew, from the rationalism of Descartes to the empiricism of Hume. But what all the Enlightenment philosophers failed to recognize is that if knowledge as it had been traditionally conceived was not possible, then their knowledge of the futility of philosophy was also not possible. Remember, it was their alleged conclusion to the futility of philosophy that justified their breaking with tradition and creating a new foundation to philosophy. Their knowledge of the futility of philosophy was therefore both logically and temporally prior to the "new" knowledge they arrived at through their new methods. To take a specific case, Descartes in the beginning of the Discourse on Method discusses his reasons for abandoning classical philosophy and inventing the Method; reasons that, naturally, refer to the futility of philosophy. But as soon as the Method of universal doubt is proposed, then Descartes' doubt of classical philosophy should also be subject to doubt. But it never is; like O'Hear, the futility of philosophy is the one non-futile result that Descartes knows in the old-fashioned way.

Socrates has not been improved on in his understanding of ignorance. If I am ignorant - and surely I am if philosophy is futile - then I am ignorant. Whether anyone else is ignorant, or whether everyone throughout history was necessarily ignorant (as Englightenment-inspired philosophers suppose), must be one of the things of which I am ignorant. This is the authentic Socratic way in which subjectivity enters philosophy and true philosophy is born; and it is the way philosophy is born in all philosophers following the Socratic tradition. When Aristotle says that philosophy begins in wonder, he doesn't mean only or even primarily that the historical origin of philosophy happened when certain men of leisure began to wonder. He means that philosophy is only alive to the extent that it is born in wonder in each individual soul. It is one thing to speculate about the nature or reality of final causes as an abstract problem that has no necessary relation to my life; quite another to recognize that the question of final causes, if truly asked, must primarily involve the question of the final cause of my own being. If I have a final cause, which means a purpose that informs my existence whether I recognize it or not, then this cause judges every moment of my existence - including the moments when I speculate about final causes.

This, I believe, is the primary lesson of Plato's Crito. Socrates is in prison and is told by Criton that all the arrangements have been made for Socrates to escape prison and repair to another city, where he would be welcomed and could continue to philosophize. The guards are sympathetic and the populace generally recognizes the injustice of his conviction. But Socrates will have none of it. He recognizes the truth of his subjectivity with respect to the laws of Athens. It doesn't matter whether, in an abstract sense, the jury decided his case correctly. Justice for Socrates means that he must respect the decision of the jury whatever it is. Were he to avail himself of the opportunity to escape, and continue to "philosophize" in another city, his philosophy would be reduced to a language game, and himself to a comic figure, spending his time in apparently serious conversations about justice, when he has made it clear that whatever he thinks about justice, it doesn't include justice for Socrates.

In what sense, then, can philosophy have "results"? Not in the sense that it can produce answers that must be recognized by any rational person, if by "rational" we mean the existentially indifferent "objective" reason characteristic of modernity. But it can produce answers that are true for everyone and for all time, if we acknowledge that those answers will be recognized only by those who have first absorbed the subjective truth necessary to philosophize; or, in Kierkegaard's words, if they have become subjective thinkers.

Dirk Pitt is Fictional, and so are Grand Conspiracies

If there is one thing that the oil disaster in Gulf has demonstrated, it's that there is no real life character corresponding to Clive Cussler's Dirk Pitt. If you don't know, Dirk Pitt is Cussler's recurring hero, a sort of combination of James Bond and Jacques Cousteau. Pitt works for the fictional NUMA (National Underwater and Marine Agency) of the Federal Government, and regularly finds himself involved whenever any underwater derring-do is called for. Well, the Gulf disaster is just the sort of crisis Pitt and his trusty sidekick Al Giordino would solve in thrilling fashion, just barely escaping with their lives and Pitt (never Al) landing the requisite hot babe. Unfortunately for us and the Gulf coast, this disaster has shown there is no one like Pitt on the vast Government reservation.

It also shows the silliness of conspiracy theories like the 9/11 "truth" movement - the idea that G.W. Bush somehow orchestrated the 9/11 attacks without leaving a trace of evidence. If the government can't stop an oil leak in a couple of months, how could it possibly pull off a massive conspiracy like 9/11? They just aren't that good.