In Defense of Critical Thinking

Steve wrote a couple of articles on Daniel Kahneman’s book, Thinking Fast, Thinking Slow:

Most of the experiments recounted in Thinking, Fast and Slow aren’t as prima facie silly, but Kahneman, a literal-minded soul, often seems to miss the fundamental point of why his subjects fall for his little hoaxes so often:

“Steve is very shy and withdrawn, invariably helpful but with little interest in people or in the world of reality. A meek and tidy soul, he has a need for order and structure and a passion for detail.

“Is Steve more likely to be a librarian or a farmer?”

Wrong! Kahneman scoffs at your intuition: “Did it occur to you that there are more than 20 male farmers for each male librarian in the United States?”

Well, no, actually, I didn’t know.  Come to think of it, nor do I know the measured frequency of the traits listed among either farmers or librarians.  Give me that information, and I can calculate the conditional probabilities.  As it is, though, the only correct answer to the question as written is:  insufficient information.

Steve continues:

Did it occur to Kahneman that anybody who isn’t trying to deceive us would have added, “So, Steve’s just not cut out for his life of slopping hogs” or the like? As con men, conjurors, and comedians demonstrated long before Kahneman, most people trust in the speaker’s good faith. They play along and try to guess what is being implied. So it’s easy to pull the rug out from under us.

Yes, but there is still such a thing a critical thinking.  For instance, here is another Kahneman problem from Michael Lewis’ Vanity Fair review that Steve quotes:

“Linda is thirty-one years old, single, outspoken, and very bright,” they wrote. “She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in antinuclear demonstrations.”

Which alternative is more probable?

(1) Linda is a bank teller.
(2) Linda is a bank teller and is active in the feminist movement.

The vast majority—roughly 85 percent—of the people they asked opted for No. 2, even though No. 2 is logically impossible. (If No. 2 is true, so is No. 1.)  The human mind is so wedded to stereotypes and so distracted by vivid descriptions that it will seize upon them, even when they defy logic, rather than upon truly relevant facts. Kahneman and Tversky called this logical error the “conjunction fallacy.”

Well said, and I would add that this is exactly the kind of problem that tests use to measure critical thinking skills.  Unfortunately, Kahneman mixes bogus problems like the first in with true logic challenges like the second in an effort to show that stereotypes are worthless.

Reflections of an Underachiever

My father came to visit me last week.

I’ve always admired my father.  He is what we in this corner of the blogosphere would call a natural alpha*, although significantly, I didn’t really appreciate this fact until I was well into adulthood.  But I did know that he succeeded wildly at a high-risk profession and as an entrepreneur (and at everything he tried, in fact, except, unfortunately, investing).  I knew that he owned fine houses and expensive foreign cars.  And I knew that he was in charge.

In short, he was everything I expected to be when I grew up.

What I didn’t know was how different we were.

I began succeeding at school much earlier than my father had, as well I should have:  my father climbed up from working-class mill family origins, whereas I always enjoyed high parental expectations.  I excelled at academic subjects of which he had no understanding; ergo, I was smarter.  By high school I could run farther and faster than he could, and by college I could swim faster too; ergo, I was more physically attractive.  It was simple math.

But although my father is only slightly taller than I am, he is much larger.  Imagine, for a moment, Don Draper and Pete Campbell standing next to each other.  I’m Pete Campbell.  A taller, more athletic Pete Campbell perhaps, but still:  Pete Campbell.  It took me until quite recently to understand how that might matter.

Looking back on it, there were plenty of signs.  Inside my father’s stories of his youthful hijinks was the message that he had a gang of friends that did stuff together.  Inside his stories of high-stakes gambling at pool and cards on the rough side of town was the message that he faced dangerous situations with confidence.  Inside his stories of conflict at work was the message that he was the master of his situation.  His was the life of someone who expected to succeed.  It was not the life of someone who’s daily objective was to get through the day without getting beat on, or made an object of ridicule.

Much of this began to sink in around the time I got married.  My father got married at age 25 to a woman not yet 20.  I loved my mother, but growing up it never occurred to me to compare her to anybody else.  I knew she was pretty, but I never stopped to think about what that meant.  Looking back on it, I guess I would say that I unconsciously saw her as the baseline of attractiveness.  I wasn’t, at that point, sure if women were worth the aggravation of being married to one, but I assumed that if I ever did get married that it would be to a woman who resembled my mother in certain salient respects.

It wasn’t until I was engaged myself that this was brought home to me.   I remember attending a party with mother and watching her across the back lawn of the house when it struck me that my mother wasn’t merely pretty.  She was, in fact, strikingly beautiful in an uncommon way.  She does not – or pretends to not – see herself in this light, so I had few cues that this was the case.  She does see herself as disciplined about diet and exercise, perhaps, but even here she fails to appreciate how, in practice, this physicality is not often willed into being among women.

And I realized then that my own success with women would never equal my father’s; despite having four extra years to work with, I was marrying a woman 15 months older than I was.  Don’t misunderstand me:  I did not want to marry my mother.  Nor even did I want to marry someone like my mother.  Mrs. Φ excels in the domestic arts in a way that my mother, an accomplished musician, never bothered, and she has proved her loyalty through circumstances with which I would definitely not test my mother.  But I could nonetheless grasp the difference between the position my mother occupied in the status hierarchy – her SMV, as it were – and the position that my bride-to-be occupied, and what this said about my father’s position relative to my own.

I’m not sure my father has reconciled himself to the fact that his son is, from his perspective, an underachiever.  He doesn’t know that my own adventure as the pointy-haired boss could be charitably characterized as mediocre.  He’s figured out that I’m not especially upwardly mobile with my lifelong employer, and that my houses don’t seem to be getting any bigger, but he doesn’t grasp that being a Company Man is about the extent of my realistic ambition.  He’s proud of the fact that I’ve worked my way to near completion on a PhD, but it’s also clear from his comments that he thinks, why hasn’t this kid struck out on his own?  Thought like a man who expects to succeed, rather than someone who tries to get through the day without getting beat on, or made an object of ridicule.

 

* Parenthetically, I should clarify here that my father has never indicated he is much motivated by sex.  On the contrary, his stories show him to be very much a straight arrow in this respect, straighter than me even, though he is not especially religious.  (This trait I inherited from my mother.)

Tuesday Roundup

The Washington Times this morning was an exceptionally fruitful source of material:

Justice concludes black voters need Democratic Party

KINSTON, N.C. | Voters in this small city decided overwhelmingly last year to do away with the party affiliation of candidates in local elections, but the Obama administration recently overruled the electorate and decided that equal rights for black voters cannot be achieved without the Democratic Party.

The Justice Department's ruling, which affects races for City Council and mayor, went so far as to say partisan elections are needed so that black voters can elect their "candidates of choice" - identified by the department as those who are Democrats and almost exclusively black.

The department ruled that white voters in Kinston will vote for blacks only if they are Democrats and that therefore the city cannot get rid of party affiliations for local elections because that would violate black voters' right to elect the candidates they want.

So can we count this as official legal recognition of black voting tribalism? I'm sympathetic to the complaint: party identification often saves even me the hard work of learning about candidates in advance. But the voters of Kinston ought to have the power to make this decision for themselves, and the DOJ is stretching its mandate under the Voting Rights Act to the breaking point.

Climate bill aids authors' states

A little noticed Environmental Protection Agency analysis shows that the pending climate-change bill in Congress would particularly benefit the states represented by its primary authors.

The analysis, obtained by The Washington Times, shows that the states that would benefit most from the climate legislation that passed the House in June include California and Massachusetts. The bill was co-authored by Rep. Henry A. Waxman, California Democrat, and Rep. Edward J. Markey, Massachusetts Democrat.

As I predicted, "global warming" is just another frontier in government rent-seeking.

U.S. scientist Charged with Spying; Accused of dealing with Israel

An acclaimed former NASA scientist was arrested Monday on charges that he attempted to sell sensitive defense secrets to a person he thought was an Israeli intelligence officer.

Stewart David Nozette, 52, worked in top government jobs from 1989 to 2000 and had access to military satellite programs and nuclear weapons programs, according to court papers that were unsealed Monday after the scientist was arrested at his Chevy Chase home.

I remember having some sympathy for Jonathan Pollard when he received a life sentence for spying for an "allied" country. But then I found out that the Israelis traded some of Pollard's product to the Soviets, which may have cost the lives of several CIA east-bloc operatives. So I'm inclined to throw the book at any foreign spy.

Does Religion Make You Smart?

Via Overcoming Bias, a report on some interesting correlations between undergraduate fields of study and religiosity:

Majoring in biological sciences, engineering, or vocational areas all increase religiosity about the same relative to not going to college. Majoring in education encourages religion even more, while majoring in physical science has about the same effect as no college. Majoring in humanities reduces religiosity relative to no college, and majoring in social science reduces it the most.

With the exception of education, which is widely recognized as the major the collects the dregs of the college-graduate population, it would appear that the majors that increase religiosity are high g-loaded quantitative disciplines chosen by the top scorers on standardized tests: engineering and physics. Meanwhile humanities and social science, whose students are found on the left side of the bell curve, are the least religious.

Somebody with more time and energy should look at these correlations more closely, but on the surface, it appears to refute Half Sigma's GSS-inspired assertion that religion makes you stoopid.

Employment Testing, USAF Edition

This is interesting:

MEMORANDUM FOR TECHNICAL OFFICER SURVEY PARTICIPANT, 4 June 2009

FROM: AF/A1, 1040 Air Force Pentagon, Washington DC 20330-1040

SUBJECT: AFOQT Revision – Technical Officer Survey Participation

We are initiating a major revision of the Air Force Officer Qualifying Test (AFOQT). The AFOQT, in its many forms, has had an effective track record of predicting officer training and pilot training success since it was implemented in 1953. Its screening has saved tens of millions of training dollars by reducing attrition and contributed to an effective officer accession process. Since initial development, the AFOQT has been revised 17 times. The current version, Form S, was implemented in 2005. With our rapidly changing mission and Air Force culture, we want to ensure the next revision assesses all of the current critical officer skills.

To better identify theses technical skills, we are conducting a survey . . . .

The survey goes on to ask participants drawn from technical and aviation military career fields to rate the importance to success in their careers of 54 different mental, physical, and social facilities:

• Cognitive Abilities: Oral Expression, Listening Comprehension, Reading Comprehension, Written Expression, Mathematical Computation, Mathematical Reasoning, Inductive Reasoning, Deductive Reasoning, Memorization, Perspective Prioritization, Task Management (Multi-tasking), Pattern Recognition, Planning Resourcefulness, Foresight, Technology Literacy, Spatial Orientation, Visualization, Adaptability, Situational Awareness, Information Processing/Sensor Management, Electro-Mechanical Science, Earth/Weather Science, Critical Thinking, Perceptual Vigilance, Aviation Knowledge

• Psychomotor Abilities: Static Strength, Physical Fitness/Stamina, Finger Dexterity, Arm-Hand Steadiness, Multi-limb Coordination, Choice Reaction Time, Rate Control, Hand/Eye Coordination, Color Vision, Depth Perception, Auditory Acuity, Visual Acuity

• Interpersonal Abilities: Persuading/Influencing, Mediation, Cooperating, Assuming Responsibility, Responsiveness, Decisiveness, Resilience, Teaching/Mentoring, Work Effectively in Isolation Settings, Work Effectively in Stressful Situations, Empathy, Self-Assessing, Self-Discipline, Integrity, Selflessness

I once had an opportunity to ask a military engineer how much opportunity she had to actually apply her engineering education. The dirty little secret, she said, was that most military engineers did what she did: technical management. Specifically, most of them spent most of their time engaged in some facet of the military procurement process, a byzantine administrative function in which the Armed Services spend a great deal of effort educating officers to perform effectively.

But on the other hand, she went on, their experience in bringing in non-ABET-accredited degree holders was not good, even though the work they did was not, strictly speaking, engineering. People with B.A.s, even in management, tended to get lost in the technology. It was just easier to train engineers to be managers than to train managers in technology.

I was thinking about this in the context of the Ricci decision. As articulated by Justice Kennedy, the test of "business necessity" for purposes of avoiding disparate impact liability is whether the employment standard is "manifestly job related". But what if the evidence for job relatedness is only statistical? Ricci's exam avoided that problem because its questions were obviously specific to firefighting. But what if someone objected to the disparate impact of the AFOQT (and you know there is one) on the grounds that pilots and engineers did not actually have to solve the kinds of problems on test in their day-to-day jobs? The military could presumably point to the correlation between AFOQT scores and performance, but would that satisfy the courts?

I think the survey mentioned above wisely chose to rate broad categories of ability rather than narrow ones. I suppose that, say, mathematical computation and reasoning are measured by the ability to solve a second-order differential equation, but while professional engineers only seldom face the second problem, the first is indispensable to success in the profession.

Understanding Inequality

Our pastor told the following story in church this week:

An elementary school had what was called a "hunger banquet" intended to "raise awareness of world hunger," [or something]. They took the students and divided them among three tables. The students at the first table were served plenty of pastries and fruit. Those at the second table received a doughnut and a juice box. The third table had only saltine crackers and water.

The teachers/experimenters solicited feedback from the students, and reported two reactions in particular. A student at the fruit-and-pastry table said she enjoyed her food, but felt bad for the students at the other tables. Meanwhile, a student at the saltine table asserted, "Everybody should get the same food."

What to make of this?

I would speculate that few if any of the students would assert that all students should get the same grades. Many would like to receive, and may think they deserve, higher grades than they actually do, but few would quarrel with the proposition that grades ought to reflect what a student learns in class and demonstrates on tests and homework.

So which -- school or the table experiment -- is a more appropriate metaphor for understanding inequality?

This is not an attempt to assert some kind of metaphysical justice in inequality. On the contrary, even academic performance is front loaded with all kinds of differentials well beyond an individual student's control. To start with, nobody deserves credit for having been born with intelligence, or good health, or energy. No student deserves credit for having attentive parents who set high expectations, who solicit feedback from teachers and monitor progress, and who create a home environment conducive to education. I would argue that these have far more to do with what a student learns that his own disembodied sovereign will. But at the end of the day, some students have learned more than others, and nobody thinks that grades oughtn't reflect these differentials.

Likewise for economic inequality. We should get little personal credit for having been lucky enough to be born to a people who, for reasons not fully accounted, evolved the ability to build the civilization we have inherited. A civilization that protects property and enforces contracts. A civilization that nurtures discovery, industry, cooperation, and enterprise. A civilization that, by the hard work and self-discipline of our forbears, left us huge reserves of capital: physical, social, moral and intellectual. This is the world to which we Americans have been born.

But there is no fruit-and-pastry-dispenser-in-the-sky who sets more food before some than others. The people of some nations grow more food, produce more value, generate more wealth than the people of other nations. These productivity differentials drive the differentials in living standards. This doesn't require that the less fortunate suffer a character flaw. On the contrary, a third-world subsistence farmer may labor much harder than his American counterpart, but his tools -- physical, mental and social -- are much, much weaker.

It is on this ground that the fruit-and-cracker lesson comes up short. The students contributed nothing to the meal in front of them, but life doesn't usually work like that. I can think of several improvements to the exercise. For instance, what if the students at the "rich" table had to slice the fruit and bake the pastries? Yes, they would get better food, but they would have to work for it. What if all the ingredients were distributed equally, but only the rich table was given appliances and utensils. Or only the rich table received the recipe. Or maybe (not that a public school would or should actually do this) the "poor" table had to work with the short bus riders who set about wrecking the work of others. Now we would be closer to teaching the real causes of wealth and poverty.

As it happened, the topic came up again during Sunday School. Our class had signed up to "do Christmas" for a poor family in the city (the real city, not Φ's lily-white little burg) through a charity that puts churches in touch with such families that apply to it. We generally thought this meant gifts for the children and Christmasy food items; we pitched in with a ham, for instance. Anyway, for reasons not here important, we were asked to also consider a second family. Two things stood out about this family's application (I mean, other than an evident history of sexual incontinence, which almost goes without saying). First, the children were older. And second, under "needs", the family had listed some expensive electronics (XBox) and -- get this -- "gift cards."

Let me take a stab at what was off-putting about this. In theory, gift cards are a way of avoiding the risk that the gifts we bring are things the family doesn't want or can't use, and they don't have quite the level of condescension as cash. But some of the families in our class are going through a lean stretch; as one mother pointed out, she didn't buy much in the way of new stuff for her own children. And even I don't have an XBox. But also, while I'm happy to bring a little Christmas cheer to young children who can't help what their parents are, teenagers really ought to begin to feel the sting of their circumstances. Maybe, it will help them evaluate their choices in life a little more wisely.

So how do you draw the line between "helping" and "enabling"?

As Ross once pointed out, the reason rich people give millions of dollars to rich universities is because real charity is hard work. At least a university will put your name on the building. But you can drop millions of dollars into poverty with nothing to show for it but more poverty. Really helping people requires knowing them up close. Knowing, for instance, that they have a car that needs a working solenoid so that they can get to work, or which child has worn out her shoes, or that they're behind in their utility payment this winter because of a layoff, even though they keep the thermostat way down. It's about knowing enough not to be gamed, and that's not even getting into the developmental issue of how to get someone beyond needing help. It means getting close to people from whom we are seperated, not just by race or income or class, but by a whole range of moral and character factors that govern how we make important decisions. It means getting to know some deeply, deeply dysfunctional people with whom we have almost no common frame of reference.

The Way Forward

Ross Douthat recommends Jim Manzi's piece in the "Whither Conservatism" vein:

[T]he challenge in front of conservatives is clear: How do we continue to increase the market orientation of the American economy while helping more Americans to participate in it more equally?

Here are two ideas among many.

First, improve K-12 schools.

That's it!?! That's the great Jim Manzi Plan for restoring the middle class!?! The same idea politicians have been promising for the last 25 years!?! Isn't this like the plan for earning $1M without paying taxes: "Step one: get $1M"?

To be fair, Manzi's prescriptions -- choice and accountability -- are good ones. But conservatives and Republicans have groped around -- unsuccessfully, and for some time -- for the political leverage to actually implement these kind of reforms, and I can't help thinking that their effect will be marginal.

Better yet would be to admit that the kind of college preparatory academic work to which our high schools have defaulted have essentially negative utility for those students at the bottom half of the cognitive bell curve. These students should be learning specific job skills appropriate to the kinds of work for which their abilities suit them.

But the flip side of this problem is that improving (or rather, redirecting) education will accomplish nothing if there are no jobs to which they can apply this education. The inescapable reality is that the only way to improve the economic standing of the bottom half of our people is to re-invigorate the industries in which they are likely to be productive. And this, in turn, will require making changes to the legal and regulatory environment for such industries of exactly the kind that the SWPL crowd is likely to oppose.

To give credit where it is due, Manzi's second recommendation -- using immigration policy to improve our nation's human capital -- is much better, and when Congress returns to immigration reform, as it most surely will, Republicans will have an opportunity to offer amendments that highlight the Democrats' refusal to do this. I can think of at least two:

-- Require that guest workers or their employers post a bond indemnifying the public fisc against their consumption of public services (healthcare, education, law enforcement) over and above what they will pay in taxes.

-- Require that all immigrants possess the human capital, and earning power, to pay more into the treasury than they and their families extract in benefits.

Of course, these amendments undermine reform's unstated purpose: to lower the nation's human capital through mass unskilled immigration. But they will give Republicans a language for opposing immigration beyond mere nationalism.

UPDATE: Jim Manzi comments (OMG, Jim Manzi commented on my no-account little blog! I swoon!):

Thanks for reading my post closely. I agree that (1) the education thing is easier said than done ([p]olitically), and (2) it's impact would be significant, but not decisive. Nonetheless, I think it is a big thing worth doing.

As I indicated, I was giving a couple of ideas, not a comprehensive plan for renewing America.

On "Inscrimination"

Via Ilkka, a post at L'Hote defending affirmative action:

[I]t's the same thing that Ward Connerly and other affirmative action foes want. A truly open and free and fair society, with neither preference or discrimination for those outside the norm. A high school, a culture, where difference really doesn't matter.

But life, I'm afraid, is not that simple. These old ways of discrimination have a way of outliving anyone who would openly endorse them. And the stark question remains: what if denying this preference really does ensure that these people will have no opportunity to excel at all? Affirmative action foes want to keep the conversation centered purely on principle. But what about the consequences? What if ending widespread affirmative action means that black college attendance rates become extremely low-- as evidence from the California public education system seems to suggest they might? Can't even the most hardened anti-AA warrior concede that there are practical public disadvantages to having college attendance-- and effectively, participation in the American middle class -- be drawn along racial lines? I know that many would say quite openly that they don't care if any black people at all go to college, as long as the selection is based on equitable and fair criteria. For myself, I think the existence of a permanent black underclass has been a major detriment to a just and secure American society, and I can't imagine a way in which ending affirmative action-- without some concurrent effort to ameliorate black poverty and joblessness-- could have a positive effect on that reality.

Freddie skips over a lot of detail here:

How much, and at what costs? Contra Freddie, a color-blind society would not exclude blacks from the middle class. It might focus them into fields in which they enjoy competitive advantages (marketing and sales) over those in which they do not (genome sequencing), but which still allow them to make middle-class money. It might also place them in schools (Cal State) where they stand a good chance of actually graduating with a degree in a substantive field, as opposed to schools (U.C.) where the odds are against completing any but the most worthless programs (i.e. "Black Studies").

But let's stipulate that, given the I.Q. distributions, and assuming that U.C. imposes an appropriate I.Q. cut-off (115, let's say), that the black share of seats in the U.C. system falls to 3.6%.* Even if we further stipulate that a U.C. seat automatically assures higher earnings than a Cal State seat, this scenario isn't exactly the exclusion of blacks from the middle class.

But what share is Freddie proposing? It's population quota of 13%? That is a non-trivial cost, not only in opportunity denied to whites (and Asians), but also in reduced performance, productivity, and competitiveness as the same affirmative action at U.C. percolates through the entire professional world.

Freddie may honestly think we can bear this cost; he should be given the chance to prove it. But having admitted the principle of racial quotas, does he honestly think we will limit them to blacks? Indeed, Hispanics are already standing in line to receive their "share" of the affirmative action pie, and if the census projections are to be believed, the beneficiaries of this pie will exceed 50% of the population in about 30 years. Let Freddie prove that cost to be non-trivial.

What is the distribution of costs and benefits? Affirmative action didn't keep Chelsea Clinton out of Columbia. On the contrary, affirmative action comes at the expense of the most vulnerable whites members of the non-preferred races, those whose backgrounds do not include the money or connections that buy entrance to the Ivy League. Their talent and hard work are all they have. In contrast, affirmative action accrues to the benefit of the most well-placed blacks: Berkeley's quota would have done well at Riverside; Riverside's quota would have made it at Long Beach, etc. Contra Freddie, affirmative action does nothing for the black underclass.

Are the benefits real? Let's remove the stipulation of my first point. In the real world, the Berkeley admissions quota does NOT go on to graduate. We know from controlled studies that there is an optimum pace at which a student assimilates new material; exceed that pace, and the student doesn't learn more, he learns less. If the body of knowledge is cumulative, this puts quotas at a crippling disadvantage. And let's not pretend that "it is better to have tried and failed"; again, in the real world, the college drop-out is not only out the wages and work-experience that he missed while in college, but he's now encumbered with non-dischargeable education debt.

Whither the Law? Freddie is counting on some omnibenevolent entity that will approve of discrimination he likes, while squelching discrimination he doesn't like. But that is not what the Civil Rights Act summons forth. What it actually requires is NO RACIAL DISCRIMINATION in education and hiring! Never mind that this was not and is not the goal of the majority of Civil Rights activists; it was the goal that America actually bought. That Freddie wants to change that goal is certainly his right, but he must change the law.

Repeal the Civil Rights Act, Freddie! Allow firms and colleges to hire and admit by their own lights, whatever they are, and remove this intrusive federal oversight into private and state-level concerns. And sell it to the public.

Alternatively, let the law reflect your true intentions. Specify that it is okay to discriminate against whites (and Asians) in favor of blacks (and Hispanics) at Berkeley, but NOT okay to do the reverse at BJU.** Sell that to the public.

But your omnibenevolent entity doing that from which legislation shrinks is tyrannical, and in any case, the Supreme Court has signalled an end to its patience with willful violations of the law as written. So pick a course. But your time is running out.

*To the uninitiated, here is how I arrived at 3.6%: The mean black IQ is 85. the mean non-black IQ is 100 (it's probably lower given the increasing Hispanic share of the population, but let's assume worst case). The standard deviation of IQ for all races is 15 and the distributions are normal. Find a table of Z-scores on the internet, and discover that the percentage of blacks with an IQ over 115 is about 4% (with rounding), while the share of non-blacks with over-115 IQs is 16%.

Now we assume that the black share of the population is 13% (with rounding); ergo, the non-black share is 87%. Let's define our probabilities thus:

P(B) = Probability of being black = .13.

P(N) = Probability of being non-black = .87.

P(S|B) = Probability of being smart (IQ > 115) given being black = .04.

P(S|N) = Probability of being smart given being non-black = .16.

We can now apply the law of total probability and Bayes' Theorem to calculate the non-quota black fraction of U.C. admitees:

P(S) = Probability of being smart = P(S|B)P(B) + P(S|N)P(N) = (.13)(.04) + (.87)(.16) = .14.

P(B|S) = Probability of being black given being smart = P(S|B)P(B)/P(S) = (.13)(.04)/.14 = .036 = 3.6%.

**Yes, I know: Bob Jones University has abolished its racially discriminatory regulations. But you get my point.

UPDATE: In the comments, Bobvis links to a study showing that, indeed, the primary victims of affirmative action are Asians. I have corrected my above implication to the contrary.