Eric The Blue

A poem for Memorial Day

Wilfrid Owen's "Dulce et Decorum Est."

Probability blindness

In the field of mind science, the inability of people to think straight about odds and probability is such a well known phenomenon that the term "probability blindness" has been coined to describe it.  Is it a simple matter of innumeracy?  In "How the Mind Works" Steven Pinker reports the results of some experimental problems in probability.  My favorite concerns a woman, "Linda," 31, bright and outspoken, a college philosophy major interested in social justice and the anti-nuke cause.  Given this description, test subjects are asked to rate the likelihood of the following statements: (1) Linda is a bankteller, and (2) Linda is a bankteller who is active in the feminist movement.  One is tempted to say that (1) is "obviously" more likely, but in tests there is considerable support for (2).  It seems that the mind forms an idea of Linda's character and scans the propositions, searching for "hits."  It is the binary world of computer science.  The first statement is a "miss," the second at least a partial "hit," and thus the tendency to rate it more likely than (1).

So some people are bad at math, others just slow all around: what is so interesting about that?  Try another problem described by Pinter, one almost identical to the coin problem at Mathproblems.info.  A certain invisible medical condition afflicts one person in a thousand.  It can be detected before onset by a test that yields 5% false positives.  If a randomly selected person tests positive, what is the probability she has the condition?

The answer is: just about exactly 2%.  You may satisfy yourself of this, I hope, by considering (as in the coin problem) what results you'd expect with more than a single trial--in this case, say, 100,000.  Since the condition afflicts one person in a thousand, you'd expect 100 to test positive on account of having the condition.  Of the remaining 99,900, five per cent, or 4995, would test positive.  So the ratio of afflicted to all positive results is given by 100/5095, or 0.0196.

According to Pinker, the average wrong answer submitted by Ivy-league test subjects is on the order of 0.5, which is around 25 times the actual risk.  I know that I myself, barred from doing the number grinding, would have guessed a number much higher than 0.02.  Something about the way the problem presents itself to the mind seduces us toward the conclusion that the subject is much more likely to be afflicted than she actually is.  And the error, it seems, is similar in all the problems I have discussed in the last two posts--a failure to recognize, or to weigh, different bits of information.  In the coin problem, people tend to consider only that a "head" result does not eliminate the possibility that the conventional coin has been selected; in the "Linda" problem, people are so sure she is a feminist that any proposition including "feminist" is deemed more likely than one containing only dissonant information; and in the problem relating to the test for an invisible medical condition, people tend to discount the low incidence of the condition in favor of elevating the significance of a quite unreliable test.

Pinker thinks that these tendencies are evidence that our minds, the products of evolution by natural selection, are essentially for information processing of the binary variety.  That lower function still asserts itself, tempting us, at least  on first consideration, toward false conclusions on problems that require subtler reasoning, the deployment of a higher order of processing.  I think it is exciting, in a humbling sort of way, to consider that our lizard minds, unelected monarchs in certain conditions, such as the presence of physical danger, may also explain our fondness for casino gambling and state lotteries.

Geeky probability

Okay, I majored in English, so probably do not qualify for the the geekiness competitions that sometimes break out over at the science blogs.  But my wife will tell you that I work math problems for recreation, more or less the way others might do sodoku or crosswords.  Lately I've been using for a source Mathproblems.info.  Problem 16 posits a box with two coins, one "fair" and the other two-headed, and asks what is the probability that, if a coin is drawn that displays a head, the reverse side of that coin is a head as well.

I can easily satisfy myself that the answer is 2/3 by observing that a single trial can have four equally likely results: one side of the two-headed coin, the other side of the two-headed coin, a head on the "fair" coin, and a tail on the "fair" coin.  So you get a head displayed three times, and, of those three, you are holding the two-headed coin twice.  Hence, 2/3. 

The problem is probably less interesting than the host's hilarious account of the mail generated by this problem and his claim that the answer is 2/3.  It seems like very many people, observing that a result of heads does not preclude the possibility that the "fair" coin has been drawn, insist that the answer must be 1/2.  Of course, 2/3 acknowledges that the "fair" coin may have been drawn, and the host patiently illustrates with an example concerning mislabeled soup cans.  Suppose there are 2 million cans of tomato soup, half correctly labeled but the other half labeled "chicken noodle," and that one store receives 999,999 correctly labeled cans and one bad one, while another store receives the other 1 million, all but one incorrectly labeled.  If you went to one of the stores, opened a can, and found it to be correctly labeled, would you ignore that piece of information when trying to decide which store you most likely were in?  So why ignore an analogous piece of information when trying to decide what coin you are most likely holding?

Nevertheless, one of the host's correspondents, a high-school biology teacher, could not persuade his principal, still less many of his students, that the correct answer to the coin problem is 2/3.  He recruited the host to remonstrate with the principal, which the host did, finally persuading the principal to relent, though in grudging fashion festooned with weasel-words.  Check it out.

In my next post I plan on describing what Steven Pinker, author of How the Mind Works, says about episodes such as this one.  It turns out that the mind scientist, though up-to-speed on the math, has considerable sympathy for the principal.

Some poetry

I thought recently of this poem, by Thomas Hardy, when I came to the passage, near the end of Philip Roth's Everyman, in which the septuagenarian narrator is rebuffed by a fit young female jogger along the Jersey Shore.  It could serve as an epigraph for the book, and is also an antidote to the happy chimera about the serene elderly enjoying the respect that their wisdom elicits from the young.  As Schopenhauer noted, we are like children disporting ourselves before the stage's drawn curtain.  We do not know what will happen to us, which is fortunate.

Here is another poem by Hardy on a similar theme.  My wife sometimes gets annoyed with me for failing to rise to her level of giddiness when something good happens.  Too much internalized Thomas Hardy.

One more note on poetry:  While sorting out some old New York Reviews to recycle, I stumbled across, and reread, an essay by Charles Simic on Elizabeth Bishop.  Simic seems to have something in about one out of every three issues of the Review and he is one of my favorites.  The occasion for the Bishop essay was the publication of a book of poems she never published while alive.  The book is controversial:  Helen Vendler, for one, thinks that if Bishop didn't regard the poems as finished, and did not seek to have them published, then they should not be published.  But I am thankful for the book, if for no other reason than I get to find out what Simic thinks on yet another subject.  In the essay he quotes at length from a poem Bishop wrote when 16 years old.  Here is my favorite passage.

 

 

And so besides my diamond rings
I carry with me but two things:
A blue balloon to lift my eyes
Above all pettiness and lies,
A neat and compact potted plant
To hide from a pursuing aunt.

Bishop was a close observer of the natural world, and an avid reader of the works of Charles Darwin, whom she praised for "the beautiful solid case [he] built up out of his endless, heroic observations. . . his eyes fixed on facts and minute details."

Speaking of Darwin and poets, here is a poem I found by following a link at the blog Pharyngula.

 

Sex, lies, videotape and formulae

Rented and watched, with my wife, "Sex, Lies and Videotape," directed by Steven Soderbergh, and found that I did not like it as much as the first time I saw it, in a theatre in 1989.  The sexual theme now seemed to disguise, rather too thinly, a ready formula.  There is a bad man who, to the dismay of the audience, is succeeding on several fronts, including the sexual, where he has traded in his frigid wife for her very hot sister.  You get to really hate him and wish he'd receive his comeuppance, that the wool will fall from his wife's eyes, that the bad man's more deserving old college friend will become the new object of the wronged woman's affection, and that the rival sisters will be reconciled as the bad man suffers humiliation.

Believe it or not, that is exactly what happens.

Myriad-minded

My wife and I saw a performance of The Merchant of Venice at the new Guthrie here in Minneapolis last weekend.  The production has been the subject of some spirited commentary, in local newspaper editorials and letters to the editor, about the character of Shylock and the question of anti-Semitism.  A rabbi stops short of calling Shakespeare, perhaps the iconic cultural icon, a bigot, but asks rhetorically what the purpose of staging the play could be.  Another theatre-goer answers that the real villains of the play are the hypocritical Christians who, not content to deny the Jew his just suit, torture and humiliate him soon as they gain the upper hand.

The opinions are strong and diverse and I disagree with all of them.  Or maybe it is that I agree with them all only in part.  To claim that Shylock is a victim of the Christian hypocrites, not a vicious caricature of a greedy scheming Jew, requires an ability to close one's eyes and ears for long periods, only to open them again at convenient moments.  Which is a way of allowing that there are such moments, the most famous being the lines in which Shylock asks whether a Jew does not bleed when pricked, laugh when tickled, die when poisoned.   More to the point about the Christian hypocrites, Shylock in the trial scene asserts:

What judgment shall I dread, doing no wrong?
You have among you many a purchased slave,
Which, like your asses and your dogs and mules,
You use in abject and in slavish parts,
Because you bought them. Shall I say to you,
"Let them be free, marry them to your heirs!
Why sweat they under burdens? Let their beds
Be made as soft as yours, and let their palates
Be seasoned with such viands"? You will answer,
"The slaves are ours." So do I answer you.

Perhaps it is that the newspaper prints letters that are all on one side or the other and that Shakespeare is at all times on all sides--thus the phrases, invented by his best critics, such as "negative capability," "myriad-minded," "simultaneousness," all these nicely summarized by Arthur Eastman, in A Short HIstory of Shakespearean Criticism, as "the power to forego the gains derived from a particular vantage point in the interests of an inclusive view."  Shakespeare is more elusive than the letter-writers.

Yet I cannot myself help trying to catch hold of Shakespeare the man.  Little of his biography is known but we do know that, as a mature and prosperous man of the theatre, he more than once used the courts to compel payment of rather small debts.  Stephen Greenblatt, in Will and the World, writes:

For whatever the reason, Shakespeare seems to have treated money--his money, at least--with considerable seriousness.  No one refers to him as a skinflint, but he did not like to waste his substance, and he was clearly determined not to be an easy mark for anyone. In 1604 he was storing more malt in his barn in Stratford than he (or, more to the point, his wife) needed for domestic consumption.  He sold twenty bushels of it to a neighboring apothecary, Philip Rogers, who had a sideline brewing ale.  Rogers's debt, including another two shillings he borrowed from Shakespeare, amounted to a little over two pounds.  When the debtor returned only six shillings, Shakespeare hired a lawyer and took his neighbor to court to recover the remaining thirty-five shillings tenpence and damages.  Thirty-five shillings ten pence was not a trivial sum at the time, but neither was it a king's ransom.  It took energy to pursue the matter, just as it took energy a few years later when Shakespeare once again went to court to recover the six pounds, plus damages, that he said was owed to him by John Addenbrooke.

If "nothing human was alien to him" it might follow that his villains, too, partook of aspects of his character.

The 12 ball problem

Over at Time to Blog Abhay Parekh poses a problem:

The Oddball Problem: You are given n identical looking balls. n-1 of them weigh the same, but one of them is either heavier or lighter than the others (you don't know which). Given a two pan weighing machine what is the minimum number of weighings you need to do to be sure that you have identified that odd ball? You can only use the weighing pan as follows: put some balls in the left pan, some in the right pan and observe one of three possible outcomes: either the left pan is heavy or the right pan is heavy, or they are even.

Since you don't know if the odd ball is heavier or lighter things get a bit tricky. This problem is often posed with 12 balls. Here's a problem worth working on:

Show that for 12 balls you can always identify the oddball in 3 weighings!

It happens that, a few days ago, I was confronted with the 12-ball variety of the problem by the local Big Brothers Big Sisters organization, which every week, before I meet my "little" at a nearby school, sends out to the "bigs" a "suggested activity."  As I worked on the problem during a boring work meeting, and then some more over my lunch break, and yet more at a coffeeshop in the afternoon, I began to question how much Big Brothers Big Sisters knows about the mental development of sixth graders and what they are apt to enjoy.  I was hooked by the difficulty, however, and eventually worked out what I take to be a solution.  Since my "little brother" exhibited no interest, I shall set it out here, before Abhay posts his version of a solution.  If nothing else it is an interesting exercise in "technical" writing.

For ease of exposition, let us think of the balls as being numbered 1 through 12.  Begin by weighing 1, 2, 3 and 4 against 5, 6, 7 and 8.  Suppose they balance.  You now know that the "oddball" is 9, 10, 11 or 12.  Use the second trial to weigh, say, 1, 2 and 3 against 9, 10 and 11.  If the result is another balance, you know the "oddball" is 12, and it is a simple matter to use the third trial to determine whether it is heavier or lighter than the rest.  But suppose 1, 2 and 3 are, for example, lighter than 9, 10 and 11.  That could only mean that 9, 10 or 11 is the "oddball" and heavier than the rest.  On the last trial, weigh 9 against 10.  If they balance, 11 is odd (and heavy).  If they don't balance, the heavier one is the "oddball."

Now we need to consider the case where, on the first trial, 1, 2, 3 and 4 do not balance with 5, 6, 7 and 8.  If, for example, 1, 2, 3 and 4 are lighter than 5, 6, 7 and 8, we know that either 1, 2, 3 or 4 is the "oddball" (and light) or 5, 6, 7 or 8 is the "oddball" (and heavy).  On the second trial, weigh 1, 2, 3 and 5 against 4, 9, 10 and 11.  Suppose that 1, 2, 3 and 5 are lighter. Since 5 cannot be light and 4, 9, 10 and 11 cannot be heavy (refer to result of the first trial), this means that 1, 2 or 3 is the oddball (and light).  Weighing 1 against 2 on the third trial will reveal which of those three is the (light) "oddball."

We have yet to consider the case where 1, 2, 3 and 5 balance 4, 9, 10 and 11 and the case where 1, 2, 3 and 5 are heavier than 4, 9, 10 and 11.  In the former case, we learn that 6, 7 or 8 must be the "oddball" (and heavy).  Use the third trial to weigh 6 against 7, which will suffice to reveal which of those three is heavier than the other twelve.  In the latter case, we learn that either 5 is odd (and heavy) or 4 is odd (and light).  The solution is clear if on the third trial either 4 or 5 is weighed against any one of the ten that we know are not the "oddball": if, for example, 4 balances with 1, then 5 is odd (and heavy), whereas if 4 is lighter than 1 then 4 is the (light) "oddball."

Slip sliding away

Reading Atul Gawande in the New Yorker over the weekend and then starting Philip Roth's Everyman on my morning coffee breaks has made it more difficult than usual to forget about what is happening to us all--the long slide toward dissolution and oblivion.  My sister, 47, was recently diagnosed with multiple sclerosis and my mother is about a year removed from chemotherapy treatments for breast cancer.  A few weeks ago my wife's uncle went to bed one night, no complaints, his evening constitutional completed, and never woke up: in the morning his wife discovered she had been sleeping with a corpse for around six hours.  I think he was 62.

For years nothing seems to happen.  Nevertheless at a zoomed-in level a highly unusual assortment of matter slips toward something less unusual.  Eventually some visible disaster occurs and the whole complicated system implodes. 

I remember an old letter to the editor in which the writer argued that Ted Kaczynski, the unabomber, should be sentenced to spend his days walking around with a ticking bomb on his back without being told when the inevitable explosion would occur.   Okay, I thought, but what should be the penalty for his crimes?