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.

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.

Ignorance and Illumination: Going Galt

Apparently Obama's announcement of a 4.9% tax increase for those earning more than $250000 (and 3% increase for those making more than $200000) has people making claims that he is a socialist or, alternately, a fascist. You'd think that people would have difficulty combining the two, given that their respective positions on the political spectrum, but some people manage the feat.

Regardless, this has some people talking about "going Galt". For those of you unfamiliar with Ayn Rand's Atlas Shrugged, a quick synopsis:

Atlas Shrugged centres around Dagny Taggart, Hank Reardon, and the titular John Galt. The novel is set in a sort of idealistic dystopia (as contradictory as that sounds) populated by two (maybe three) types of people. Producers, like Reardon and Taggart. Looters/moochers - those who either demand things from the producers or appeal to their sense of pity. And the third category are those who have the right ideals, but aren't themselves producers. At best, they are assistant producers - Eddie Willers, Dagny's assistant, for example - a man who does his job, but isn't a visionary.

At any rate, the moochers/looters wind up in control of the government, systematically punishing the producers because they hate them, and yet depend on them. Psychologically speaking, it tells us a lot about Ms. Rand's mind - she emigrated from Soviet Russia, and therefore had a certain amount of vitriol for "from each according to his abilities, to each according to his needs". The titular character, fed up with the looters/moochers' activities, establishes an enclave in the Colorado Rockies, and entices other producers to stop "empowering" the parasites, and join him there. With this handful of industry giants removed from the scene, society collapses.

Cheery, eh? Well, this is the hero that people (predominately right-wingers) supposedly wish to emulate. However, the manner in which they do it would be enough to make Ms. Rand cringe...

The meme really got its start from a post by Dr. Helen Smith. Now, the post itself is rubbish - "rewarding those who overspend, underwork, or are just plain unproductive" indeed! - but the real proof of the pudding (sewage pudding with arsenic frosting, in case you were wondering) were the comments this post engendered. It was a veritable Baskin Robin's of wignuttery - every flavour showed up:

• The very first comment suggested refusing to live up to a contract (anathema for Rand's characters), abuse of a system designed to help ailing family members, and communes. If anything is likely to make Rand turn in her grave, it's people proposing communes in her name!



• Here in Tenneessee we not only have the Caney Fork and the Little River in which to fly fish but we also make our own cars. Lots of them.
  ...
  It does my heart some good as a southerner to know that this time it is the South that is on the productive and responsible side of the issue.

  This gentleman seems to forget what industry was asking, even then, for a fairly colossal bailout, because it wasn't economical...



• Minimalism. Work all you want. Make all the money you want but spend as little as possible. Put your cash in a safety deposit box so it's yours but the bank and the rest of the financial system can't count is as an asset and use it to make loans, etc.

  This gentleman has a bit more of a Galtish feel to him, but doesn't seem to get that this only helps the US by increasing the value of the currency, increasing the worth of the money the US does collect in taxes. That said, I am all for this technique. I wish the government did it.



• Anyone who stands against replacing public education with a private educational system or home schooling (and immediate reduction in school based taxes at all levels) should be voted out.

  And here we sample a different flavour of wingnuts - the home-schooling nuts. The idea of "home skooling iz bettar!" is so ridiculous it deserves a post all its own - but to summarize: there's something that manifested in the twentieth century private sector more than any period before it - specialization. While I'm fairly certain I could repair a cart and keep a horse fed and healthy in addition to my other skills, I wouldn't have the first idea of how to repair a hybrid automobile. While my math and computer science skills are fairly well-developed, I know virtually nothing of electrical engineering. I'm sorry, but unless you are in the top 0.5% of the population, there's no way you can teach your child every subject to the same depth they would receive in a classroom. I'll stop here before I really get into it... this will definitely be a later post, however. the stupid, it burns!



• In business I can lower my salary and either leave it in the business or pay it out in dividends (no FICA taxes). Buy gold instead of stocks and CDs. Drop all subscriptions to MSM. Buy antiques, used guns, etc. instead of new products.

  Once again, we have the stupid people who don't seem to get the concept of a marginal tax rate. If you want to reduce your income out of spite, to deny the government taxation on your earnings... well, that's your prerogative, and good for you for sticking to your guns, I suppose. But diminishing returns?



• The final straw was the last debate, when the One proclaimed that health care is a "right". Screw him and his minions. Let them try to provide this "right" to the masses, when those who produce it by their labor and intellect refuse to do so.

  And herein we see the benefit inherent in a socialized healthcare system - no idiots like this who commoditize health care. No pay, you die, is that it?



• 
  I went to dinner recently with a client. He works his butt off. She has never worked and never will work, because she is far, far better than her husband or me. I got the drift.
  ...
  And why on earth are these house-pigs praised while the working men are slammed in society?

  Strangely, all the right-wing debates usually feature at least one comment of this nature. Odd, isn't it?



• Hold your breath, or you may be breaking the law. Any scientist knows that CO2 is not pollution, nor can it be classified as a pollutant. One more reason to vote against Obama

  This stuff too.

And on, and on, and on.

Let's make things simple, folks. Here's how you "Go Galt". First, you have to truly be a producer. One poster suggested that lawyers were producers. I think Ms. Rand would probably disagree with you. Doctors also seem to get short shrift in her novel. In fact, the only people who get respect are the industrialists - the ones who manufacture things or facilitate their manufacture, and get joy out of so doing. So if you work just to pay the mortgage - you never were a Galt, so don't pretend otherwise.

Second, you have to be willing to withdraw your services from society altogether. This may mean working as a fry cook, it may mean living on a self-sufficient farm - but whatever your choice, you cannot contribute to society in a productive manner. You cannot, however, become a moocher or looter yourself; that would run entirely contrary to Objectivism. You cannot join a commune, either. Ideally, you would join a community with its own currency, or which trades by barter.

But the fact of the matter is, even those of the would-be Galters who aren't moochers or looters themselves still aren't Galts. At best, they are Willers. They get their jobs done, but were they to vanish, there would always be others to fill their shoes.

Finally - when Ms. Rand wrote her book, the marginal tax rate on the highest tax bracket was 91%. 39%? Dagny Taggart would laugh at you and tell you to grow a pair.

Ignorance and Illumination: The Airplane and the Treadmill

Samson Effect here with a new segment called Ignorance and Illumination here on Not Fit to Print, where we find some situation with a major misconception or ambiguity, and make efforts to dispel the fog surrounding it.

For the inaugural event here on I&I, I'll be discussing the ever-confusing Airplane-on-a-Treadmill problem. There are a few typical descriptions of it, but the key points are this:

1. You have a typical airplane on an infinitely long treadmill.

2. The treadmill can accelerate to any speed.

3. Structural integrity will not be considered as a restriction (i.e. No structural failures of the airplane, tires, or treadmill.)

4. The axles on the airplane's wheels are, for this case, frictionless.

5. The airplane will start its engine and attempt a takeoff roll as the treadmill begins moving in the opposite direction, matching speeds with the wheels of the plane.

6. The airplane will either take off as normal, or will be stopped by the treadmill.

Now, the major source of confusion with this problem seems to be in item number 5; many people have differing interpretations of just what this statement about 'the speed of the wheels' means, as follows:

1. The speed of the wheels as measured by a speedometer connected to the wheel. This is dependent on the rotation rate of the wheel.

2. The speed of the wheels as measured by a stationary observer off the treadmill as they move through space, i.e. The translation speed of the wheels.

3. The ground speed of the airplane, which is identical in principle to case b.

So we'll deal with the misconceptions that abound in case a. Many people think that since the treadmill speeds up, it's going to draw the airplane back as it tries to accelerate; this is not the case. The issue here is that a false analogy to a car is being drawn. A car engine delivers its power to the wheels, which rotates them; that rotation then translates into a static friction force between the tires and the ground, which is what propels the car forward. A car on a treadmill would be stopped by the treadmill because it operates with respect to the ground. An airplane, and the wheels thereof, however, behave differently.

What IS the difference, I hear you cry? Well, the difference is an airplane isn't driven by rotating the wheels; it's driven by pushing the entire aircraft through the air via direct, and the wheels simply rotate freely on the axle to allow the plane to move more easily along the ground. A car's driving force goes through the ground it rests on, while an airplane's driving force goes through the air. The airplane operates in a different reference frame than a car, speficially, with respect to the air, not the ground.

Now, time to analyze the forces involved in the plane powering up. Brakes are off, throttle wide open, which means at our initial situation of a stationary plane and a stationary treadmill, we have two forces acting – thrust, and static friction. Thrust acts through the structure of the plane, accelerating it forward, and the friction acts on the tires. That friction, however, translates very little into the structure of the plane, and mostly only serves to rotate the tires along the ground. Additionally, that friction exists whether the treadmill is moving or not; it exists just the same on a paved, static runway, and airplanes take off from those all the time.

Now, what's the biggest problem with case a? This situation fundamentally defines the speed of the airplane as zero for all time; that's the only case where this situation would be true, and the only case where it can be true is if the engine is not running, or if you're picky, at low idle. Since it makes that definition, it's not physically significant, and once the airplane begins moving through space in reality, then you get a situation where the treadmill is constantly accelerating to match the speedometer connected to the wheel, which accelerates the wheel (but fails to decelerate the plane), which accelerates the treadmill. So with this feedback loop happening, you'll eventually get a situation that is eventually going to become physically untenable, but still failing to stop the acceleration of the plane through space.

So, that being said, case a is physically trivial. There's no important physics happening there because we've demanded in the question that the airplane remains stationary, and it has essentially reduced to “Demand that the airplane remains stationary. Does the airplane take off?” It creates a tautology by framing the question poorly or incorrectly.

So now that we've shown why that definition not only runs afoul of sense, but makes no difference to the physics of the problem as well, let's talk about case b, where the treadmill runs at the speed of the wheels through space. So, for a plane traveling at any given speed, v, along the ground, the treadmill will be running at the same speed, v, in the opposite direction. Now, taking into account what I mentioned well above about the friction, the wheels, will be rolling – and freely spinning on the axle, remember – at a relative speed of 2v, so for a plane traveling at 50 knots, the treadmill rolls back at 50 knots, and the wheels rotate at the equivalent of 100 knots. All the treadmill does is speed up the wheels, slow down the plane.

So, after all that, no matter how you rig it up, as long as your axle is frictionless and your wheels don't explode, then you can get moving forward, get lift, keep accelerating, and take off, no matter how you rig it up.

Still don't believe me? Well, here is some experimental verification, with a real plane on a moving tarp acting as a conveyor belt. You don't even need the ideal scenario!

So hopefully after reading this, you'll be able to find the signal for all the noise on this subject. Fare thee well, and hopefully we've learned something interesting!