Ignorance and Illumination: The Boy or Girl Paradox

Today's I&I segment comes via my younger brother, who directed me to this thread on the Something Awful Forums, which after some time devolves into arguing about how probability works with respect to the particular problem. My aim here is to point out the problem, and elucidate the proper solution. So without further ado, I present to you the Boy or Girl Paradox:

You are walking around town when you come upon a mother and her son. She makes chit chat with you, and mentions that she went to the park on a picnic with her two children the other day.

Assume for a moment that the odds a girl or boy is born are equal. Now, what is the probability the other child is a girl?

There are two answers that crop up continually throughout the thread: Either the probability of the unobserved child being a girl is 1/2, or the probability of same is 2/3. Now, there are all kinds of ways you can organize the information when you work this through, but the two ways it's done most in the thread is age, and whether or not the child is observed. The 2/3 answer comes up as a result of a mistake in the interpretation; either there's a missed piece of information, a misinterpretation of what the mother is actually saying, or it's mistaken as a Monty Hall Problem (which I won't be getting into here; that's a whole different can of worms.).

The major misinterpretation of this problem is as follows: Some people read this and think the question is asking what the probability of a child being a girl will be given that at least one of the two is a boy. In that case, the probability IS 2/3; There are four possible configurations, BB, BG, GB, GG, and based on what you're told, only one of them, the GG case, is invalid. The problem with this? This isn't what the question is asking.

One of the issues with ordering the children by age is that you're not given any information about the ages, which can lead to this false interpretation, as outlined by HHHH:

Here's how you get 2/3:

There are four configurations of sexes, each with equal probability:

BB, BG, GB, GG

Obviously the last one is impossible since at least one is a boy:

BB, BG, GB

So this means that there are only three possibilities. Since a girl shows up in two of the three possibilities, and we've already established they're equal, the probability of the other child being a girl is 2/3!

Yes, I know it's wrong, but the more challenging question is what is the fallacy with this logic?

The fallacy here is that although you know one of the children is a boy, you also don't know which child she brought, either child A or child B. Because of that, there are two possible cases for the BB arrangement. So, labeling this by age, and first-born with a subscript 1, second-born with a subscript 2, we get the following cases for the boy being brought to the park:

Brought	Left
B1	B2
B2	B1
B1	G2
B2	G1

The crowd that are insisting that the probability is 2/3 are assuming in the BB case that the woman is bringing the first son, and forgetting that it's possible that the second son could be with her, and the first at home. They're arranging it by age, but forgetting that there are two possible arrangements for the two-boy case, because there are two distinct boys in that case. As such, the probability of the other child being a girl is in fact 1/2.

Now, let's look at the problem another way, also equally valid: Observed or unobserved. The table here goes as follows, first child is observed, second is the unobserved:

BB, BG, GB, GG.

Again, four possibilities here, and we can scrub the last two, since the observed child is not a girl, thus leaving BB and BG as the two remaining possibilities, and the probability of the unobserved child being a girl is once again 1/2.

In truth, this problem is trivial; the question gives us the answer straight up. The probability of a child being either a boy or a girl is equal. Since the gender of the observed child has no bearing on that of the second, children being independent trials, we can say straight off that the probability of the second child being a girl is 1/2.

This problem was even attacked by one poster using Bayes' Theorem, and not only did the answer turn out to be 1/2, but when he used the probabilities of 2/3 for a girl and 1/3 for a boy, Bayes' theorem yielded a probability sum of 4/3, which is impossible; by definition, the sum of all probabilities in any given scenario must be 1.

Of course, given that the only two possibilities for a chromosome pair (trisomies notwithstanding) are XX (female) and XY (male), biology on its own bears out that the probability must be 1/2. Why do you think there's a roughly even split of men and women in the world? SA user semicolon said it best:

It's 50% because biology doesn't give two shits about your tables and theorycrafting.

No, it most certainly does not.

A Brief Public Service Announcement...

From blogger Hunter of "Climbing Out of the Darkness...":

If you want to successfully protest, make sure hits home with the silent majority, and is easy for everyone to participate in, without having to go out and march in the streets. Try these two protests.

Bank fees - pick one day for everyone to stop using their credit and debit cards. If you need money, actually go into the bank to get it. Using an ATM machine is a big no no. Can you imagine how our banking system would react? It's not illegal to not use your debit card, nor is it illegal to walk into a branch of your bank and ask for the cash from your account. Every Canadian could participate without having to take time off work.

Now, aside from his comment that the "TEA parties worked", of which there is no evidence, he seems to suffer under bank fees. So I thought I'd do my good deed for today and provide solutions.

• TD Canada Trust - Value Plus Chequing Acct. - Keep a balance of $2000, and you get 25 transactions/month with no charges at all.


• BMO - Performance Plan - Keep a balance of $2500, and you get unlimited transactions/month with no charges at all.


• Scotiabank Basic Banking Plan - Keep a balance of $2500, get 30 transactions/month with no charges at all.

Now, neither Royal Bank nor CIBC offers no-strings attached fee waiving for a monthly balance, so I'm not including them in the list, but there's one other service I thought I'd mention:

• President's Choice Financial No-Fee Chequing Account. This is my own banking provider of choice. And they truly mean "no fee". Unlimited transactions/debit card uses per month, and cheques are free.

So there you are, Hunter. Adopt any one of the above and you can never pay a bank fee again. How's that for a protest, hmmm?

(H/T to CC for Hunter's idiocy.)

Afterthought: Hunter also uses the redundant phrase "ATM Machine". Doubtless, when using the "ATM Machine", Hunter also uses a "PIN Number". It's not an uncommon mistake to make, linguistically, but it's still painful.

Asshat of the Day - April 11th, 2009

Today's Asshat of the Day? Paul Edwards.

Now, were it not for a case of mistaken identity in a Pharyngula comment thread, I'd never know of the man. Mr. Edwards is a pastor and radio host for WLQV, an evangelical Christian radio broadcaster operating out of southern Michigan. Given their transmission wattage, they cover nearly four states and southern Ontario with their broadcast.

Now, Mr. Edwards was fortunate enough to have the esteemed Christopher Hitchens on his show (hence the confusion in the comment thread from a similar interview), and, for the most part, was respectful. Granted, he repeatedly said that he wasn't out to win a debate, but to "show the Gospel to Mr. Hitchens", but pretty much all evangelists are like that, so I suppose that cannot be held against him.

However...

Now, to digress for a moment, I was under the impression that Pastors were, among their various theological studies, trained in the sort of interpersonal techniques that in which psychologists and counselors are trained. If so, Mr. Edwards clearly failed that part of seminary, because he totally fails to see the points at which he says things that are so abhorrent to Mr. Hitchens as to prompt the later comment, "I felt as though I wanted to hang up, just then." In fact, he goes on, after the interview is over, to ask his audience to "show [him] the point at which Christopher Hitchens got emotional".

I can tell you the things that you said that disgusted him, Mr. Edwards (though I'm not dead certain about the order):

• Mr. Hitchens had already made it clear earlier in the program, that the Bible's acceptance (and mild support) of slavery was absolutely revolting in his eyes, and that he felt that there are few greater evils than one person claiming to own another. And then you went and said that you are a "willing slave to Jesus Christ", and moreover, that you "Do what God wants, without question". Slavish thinking, slavish action - both of these, Christopher Hitchens made it clear he despises and wishes to see eradicated, and you held them up as virtues. If you wanted to deliver the Gospel to your guest, that was not the way to go about it. Specifically, he made reference to Abraham and Isaac, with regards to the slavish obedience to the Lord, with the question being this: What if God hadn't stayed Abraham's hand? Abraham would have killed his own son, just because God told him to do so.


• You also clearly missed the point Mr. Hitchens was making when he said you owed the American Armed Forces an apology for your comparison. You attempted to spin it, in the after-talk, to sound like he was accusing you of suggesting that the US Army had committed war crimes, but this was not the case. In Iraq, the invasion was partially for oil, partially as a victory when Osama Bin Laden couldn't be found, but ostensibly to free the citizens from a tyrannical dictator. The only people they were aiming to kill (and even then, only after a fair trial) were Saddam and his ruling council. If enemy combatants faced them, certainly, there could be deaths. But, tragic though it may be, battlefield deaths are a generally-accepted part of war. What were "God's orders" to the Israelites regarding the Alamekites? Oh, right:

  Now go and smite Amalek, and utterly destroy all that they have, and spare them not; but slay both man and woman, infant and suckling, ox and sheep, camel and ass. (1 Samuel 15:3)

  
  Now, assuming that we're treating the Bible as a fairy story (and what better way to treat it?) wherein every single Alamekite was not only evil, but born evil, perhaps this is acceptable. But if you want to claim that the bible represents parts of history, this is religion-motivated genocide. An effective means of ending an opponents ability to pose a threat, true, but you can't claim it to be good, nor can you draw a parallel between it and modern warfare conducted by any nation that follows the Geneva Conventions. According to the Story Book, God told the Israelites to kill every man, woman, and child in Alamek, and not one Israelite stopped and said, "Hang on, they can't all be evil." Mind you, later, when Moses instructs them not to kill everyone, it gets even worse.
  


• Finally, I think the third major point wherein you got Mr. Hitchens angry was in stating that you were not his enemy, and alluding to Matthew 5:43. On the face of it, this is just fine. Hate is, after all, a destructive emotion - very little good has come of it. But doing good for those who hate you is a recipe for an abusive relationship! Feeling empathy for your enemies is fine, and good - but loving them as you would love your neighbour is both impractical and dangerous!
  
  In many ways, this passage (and, later, Mark 9) are representative of one of the truly frightening messages espoused by believers of many stripes - that what happens in this life doesn't matter. That suffering under your enemy's hands while you love him, or that mutilating yourself rather than admit fault are good options, because you'll be rewarded once this life is over. Those, however, who do not believe, view this as insanely self-destructive - and worse, it can impact them as well, even though they don't think they're getting any sort of benefit after death other than decomposition. It's what makes religious fanatics so unstoppable - death is merely a transition to reward. It seems to me that is the reason why suicide, at least in the Christian doctrine, is a sin. They needed a way to stop people from taking the shortcut to the good life. This is what horrified Christopher Hitchens.

So there you are. Now you know. And, to top it off, you're the Asshat of the Day.

Ignorance and Illumination: The Fine Tuning Argument

So, (he said conversationally), this video, courtesy of Pharyngula, got me thinking about a popular apologist argument of late - one that even showed up in the most recent Discover magazine (this being the International Year of Astronomy) - that of Fine Tuning.

For the unfamiliar, I will elucidate. Fine Tuning is a universal variant of the Argument from Design. Rather than looking inward (the "I sure didn't evolve from a monkey!" crowd), the groups that posit fine tuning look outward. Look at the universe, they say. If there had been a trifle less matter at the beginning, the matter/antimatter imbalance wouldn't have worked out in our favour. Look at the density of matter and the Cosmological Constant. If those were even a trifle different from how they are, the universe would never have lasted long enough for life to form.

On a more local scale, look at the solar system and our place in it. If there weren't gas giants to scoop up most of the incoming interstellar debris, Earth could've been pulverized by an asteroid long ago. If Earth weren't the precise distance from the sun that it is, it would've been too hot or too cold. If Earth's moon weren't in the orbit it is, there would never have been the tidal pools that life needed to get out of the oceans, or, conversely, the tides would've been catastrophically intense - in either case, no tidal pools. If Earth's gravity weren't exactly what it is, the earth might have lost its atmosphere. And so on, and so forth. I'm sure, thinking about it, you can come up with hundreds of stellar and universal constants without which life as we know it could never have come to be.

Now, some apologists find this particularly enticing, and it may seem a more moderate view than most, simply because it presupposes that a great deal of astronomy is, in fact, accurate, and that geological "Deep" time is a fact, rather than, as the YECs would have us believe, a hypothesis. Don't be fooled - this is just Paley's tired old argument, dressed up in fancy new clothes. How can [object] come to have been, when it is so [intricate/beautiful/sensitive to change]? Surely there must have been an intelligence behind its design!

Now, the Watchmaker concept has been thoroughly debunked, but this latest variant on Paley's postulate hasn't been around long enough to get its own book as yet. However, it has a number of arguments against it - one of which is a bit of a knockout.

First, there is a counterargument against those who might argue that a benign creator designed the universe for life (or for some creators, especially for Human life) - they did a really, really poor job of it.

Look at every other planet in the solar system. Look at any of the hundred nearest stars. Look, for that matter, at the vacuum of space. None of these are capable of supporting life. Look, for that matter, at our own planet. While unicellular life is found just about everywhere, the polar extremes sure as heck don't support much of anything - and if your opponent is arguing in favour of a human-centric deity, more than three-quarters of the planet is not very conducive to human life.

In this argument, much depends on the postulated creator being omnipotent. If your opponent in such a debate is willing to grant that the postulated creator was neither omnipotent nor omniscient, or, alternately, that they were either non-benevolent or actively malevolent, then the problem of incompetence (that is, for someone tuning the universe to favour life, they did an awful job) doesn't come up. Now, the question arises as to exactly how much they could have to do with things like the Cosmological Constant if they don't have omnipotence on their side... but I digress.

Regardless, both the Tuning argument and its major opposing arguments spring from the same source - the Anthropic Principle. (The universe we are in is capable of sustaining intelligent life at least at one location in space/time. Us, if you were wondering.)

To the theist, the answer to "Why is this the case?" is very simple. God made the universe with the intent that we might live in it.

However, there are a number of more plausible explanations - and a few that are slightly less plausible but still more feasible in Occam's eyes than "God did it".

The first is comprised of four sub-clauses, but the final point of each is this: if a universe exists/existed where the conditions were not conducive to the emergence of intelligent life, no one would be around to ask the question. That we are guarantees that the universe in which we ask it has these properties. Ah, say the theists, but how?

Well, the simplest answer is coincidence. Yes, a stunning display of coincidences, but if they hadn't occurred, we'd never know about it, since life (including the questioning folks) would never have arisen. So, say the proponents of coincidence, we're here, so it did happen. Q.E.D.

A second group, with a few more followers than the first, suggest that part of the Theory of Everything that we've not yet encountered includes strictures on the laws of physics such that if a universe exists, various constants must be the way they are, or that all the constants are dependent on some other variable (possibly discrete), which forces them into their current values. Therefore, if a universe exists, and isn't radically different from ours, it would have to have laws, topology, and constants similar to our own. This argument is given limited credence, as it is, as yet, only a hypothesis, with little evidence to support it.

A third group uses the multiverse theory to suggest that there are, in fact, googols of other universes which DO have different constants, or different distributions of matter, and so forth. If everything that can happen must happen, it is then a certainty, rather than a possibility, that in at least one universe, life must arise. In fact, since there are many ways life could have arisen without intelligent life, or without any one creature, an infinite number of universes must have life, by this argument. Recent developments in quantum physics - particularly quantum computing - lend credence to this argument. It also suggests interesting things about human thought, as decisions might very well simply be bifurcations between new universes in the multiverse, with what you perceive to be "yourself" progressing down a particular branch of the tree.

The fourth and final group suggests that while there is only one universe at any given time, it has repeatedly appeared and collapsed, been and then not-been, on a sort of quasi-timescale (as linear time cannot exist outside the bounds of the physical universe). After however many quadrillion iterations, the chance of the universe having the features necessary to engender life becomes more than reasonable. In a way, this is like the multiverse theory, in that as the number of iterations approaches infinity, the chance of a universe coming into being that can support/engender life approaches one.

There is one other, more recent explanation for the principle, but this one is both a bit more out-of-left-field and a bit more difficult to wrap one's head around. (At least my own head. Those with more capacious heads might find it simple. In any event, read on.)

This last group postulates that life, and intelligence, is in fact necessary to allow the universe to exist. As has been demonstrated in quantum physics, the act of observation changes the observed. Known as the Participatory Anthropic Principle, a simplification of this point of view would be that the universe itself can be interpreted to act like a Schrodinger waveform, and unless or until life arises to act as the observer, it takes on all possible properties. This, in a way, is similar to the multiverse theory, in that if the universe takes on all possible properties, it is a given that life will arise.

In summary - while both "how" and "why" are nebulous at the moment, (and in fact, the "why" may never be more satisfactorily answered than "because"), the argument for Fine Tuning by an intelligence is far less plausible than any of those offered by the Anthropic Principle. It's just Paley's Watch meeting the God of the Gaps.

Logos!

To spruce things up around here a bit, I've created a couple of logos to correspond to our regular features.

First, Ignorance and Illumination:



Featuring Pharyngula (and Monty Python)'s inimitable Mr. Gumby to represent ignorance, and a brilliant CFL (just to really piss off the AGW-deniers) for the purposes of illumination. I think it gets the point across.

Next, Asshat of the Day:



Fairly self-explanatory, I should think.

And that's it! Smaller versions of these logos will adorn their corresponding posts. Cheers!