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The Monty Hall problem and counter-intuitive teaching

[ 0 ] December 3, 2009 |

The Monty Hall problem is a well-known thought experiment in probability analysis. The problem is fairly simple, but for reasons that aren’t well understood the right answer is sufficiently counter-intuitive that a very large majority of people get it wrong on their first attempt. More interestingly, I’ve found that students often resist the validity of the correct answer, even when the problem is analyzed in some detail. The problem:

Suppose you’re on a game show and you’re given the choice of three doors. Behind one door is a car; behind the others, goats. The car and the goats were placed randomly behind the doors before the show. The rules of the game show are as follows: After you have chosen a door, the door remains closed for the time being. The game show host, Monty Hall, who knows what is behind the doors, now has to open one of the two remaining doors, and the door he opens must have a goat behind it. If both remaining doors have goats behind them, he chooses one randomly. After Monty Hall opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or to switch to the last remaining door. Imagine that you chose Door 1 and the host opens Door 3, which has a goat. He then asks you “Do you want to switch to Door Number 2?” Is it to your advantage to change your choice?

(Taken from the wiki page if you want to look up the answer).

Here are some strategies I’ve used for explaining the solution to students who resist accepting it.

(1) Redescribing what the offer to switch gives you, i.e., by switching you are in effect choosing two doors instead of one, and thus doubling your odds of success.

(2) Recharacterization via different quantities, i.e., what if there are one hundred doors and 99 goats and a car, and Monty Hall is required to show you 98 goats after you choose a door?

(3) Explicitly working out all the potential iterations, i.e., if you choose Door 1 and there’s a car behind it then X, but if there’s a goat behind it then Y etc.

(4) Empirical testing. Have the student run the experiment and observe the results.

These are listed in descending order of abstraction, and probably not coincidentally ascending order of pedagogical effectiveness. (Occasionally there will be a holdout even after (4). This person is invariably male and almost certainly a future litigator.

Anyway, teaching the problem is a fun way to get students to think about the limits of common sense intuition, which is a much-cited source of wisdom for legal interpretation in general, and statutory interpretation in particular. It’s also a good way to get people to think about how people tend to cling to intuitively correct answers, even in the face of demonstrations that their intuitions are wrong.

Update: Thanks for the comments, and especially to J.W. Hamner’s variation on explanation (1) and Vardibidian’s card trick for explanation (2) — I’m going to use those.

Lemuel Pitkin and Mike Schilling emphasize that it’s crucial that the rules of the game require Hall to reveal a goat after the initial choice, and that without this caveat the situation is different. Just for the heck of it, gaming that out: The contestant doesn’t know what if any post-choice decision rule constrains Hall, or even if Hall knows what’s behind the doors. What should the contestant do?

Possibility (A) Hall doesn’t know what’s behind the doors.

Possibility (B) Hall knows and is indifferent to whether you win or lose.

Possibility (C) Hall knows and wants you to win.

Possibility (D) Hall knows and wants you to lose.

If (A), then the revealing of a goat moves the odds to 50/50 for the remaining doors, and therefore switching neither helps nor hurts.

However, this is where “pure” game theory needs some richer sociological context. In our culture it would be a very strange game show in which the host didn’t know what was behind the doors, and therefore might accidentally reveal the car. So as a practical matter the contestant can probably rule out (A) as an actual possibility. In any case, the sum probabilities created by (B), (C), and (D) remain dispositive, since (A) would leave the contestant indifferent to switching or staying, i.e., whether you estimate the odds of (A) being the case as 1% or 99% makes no difference — the only thing that matters are the odds governing the other possibilities.

Moving right along, if (B) is the case, then as long as you assume he’s not going to choose to show you a car, which given the rules of game shows is a pretty safe assumption, we’re right back to the classic description of the problem, and you double your odds of winning by switching.

If (C) is the case, then deciding whether to switch comes down to your estimate of Hall’s assumptions regarding your mental state. Maybe you’ve chosen the goat, and because he wants you to win he’s giving you additonal information that, if you both understand the probability structure and that he wants you to win, tells you to switch. But here’s a disturbing possibility: maybe you’ve chosen the car and he wants you to win, but he’s showing you a goat precisely because he believes that if he does so he’ll encourage you not to switch, because like most people you’d get the probabilities wrong under the classic assumptions of the game’s rules, and it’s more likely you’ll choose to stay than switch because of endowment effects or sheer stubborness. Remember you don’t know the rules — you don’t know whether he even has to open one of the other doors. The analysis is the same for (D) but reversed.

Ultimately if you don’t know the rules of game, you have to make two separate judgments: what are the probabilities that (B), (C) and (D) are the case, and what are the probabilities within each of those possibilities? Those two estimates then determine whether to switch or stay, since (A) leaves you indifferent.

Imperial naivete

[ 0 ] December 2, 2009 |

St. Ignatius of Georgetown bestows his benediction on Obama’s escalation of the war in Afghanistan, but, being a liberal columnist at the liberal Washington Post, he regrets that the announced plan fails to commit the nation explicitly to perpetual war:

Obama thinks that setting deadlines will force the Afghans to get their act togetherat last. That strikes me as the most dubious premise of his strategy. He is telling his adversary that he will start leaving on a date certain, and telling his ally to be ready to take over then, or else. That’s the weak link in an otherwise admirable decision — the idea that we strengthen our hand by announcing in advance that we plan to fold it.

Of course one would have to be an idiot to imagine that Obama’s announced strategy of employing a Surge(tm) with a “date certain” for withdrawal is what it pretends to be. The plan as presented is obviously for public consumption: the real plan will have to be either:

(1) To abandon Afghanistan, as the Bush administration eventually abandoned Iraq, but only, as in Iraq, after a face-saving military triumph over the current wave of civil insurgency, aka the declare victory and leave option; or

(2) Perpetual occupation.

The most Orwellian moment last night was Obama’s proclamation that, unlike previous empires, “we do not seek to occupy other nations.”

We will not claim another nation’s resources or target other peoples because their faith or ethnicity is different from ours. What we have fought for – and what we continue to fight for – is a better future for our children and grandchildren, and we believe that their lives will be better if other peoples’ children and grandchildren can live in freedom and access opportunity.

As a country, we are not as young – and perhaps not as innocent – as we were when Roosevelt was President. Yet we are still heirs to a noble struggle for freedom. Now we must summon all of our might and moral suasion to meet the challenges of a new age.

Stirring sentiments indeed. He might want to repeat them in Oslo next week, when he picks up his Nobel Peace Prize. It certainly beats “We should invade other countries when it gets good results.”


[ 0 ] December 1, 2009 |

Stephen Walt:

Americans have come to believe that spending government revenues on U.S. citizens here at home is usually a bad thing and should be viewed wth suspicion, but spending billions on vast social engineering projects overseas is the hallmark of patriotism and should never be questioned. This position makes no sense, but it is hard to think of a prominent U.S. leader who is making an explicit case for doing somewhat less abroad so that we can afford to build a better future here at home. Debates about foreign policy, grand strategy, and military engagement — including the current debate over Obama’s decision to add another 30,000-plus troops in Afghanistan — tend to occur in isolation from a discussion of other priorities, as if there were no tradeoffs between what we do for others and what we are able to do for Americans here at home.

Via Yglesias

When is non-consensual sex rape?

[ 0 ] December 1, 2009 |

One of the keys to interpreting reactions to the arrest of Roman Polanski is understanding that, culturally speaking, a lot of sexual assaults aren’t considered crimes by the men or boys who commit them, and to a lesser extent by the women and girls who are assaulted. Consider this letter to a nationally syndicated advice columnist, and especially the columnist’s response.

The writer is confused about whether she was raped, because even though she told a man “many times” that she didn’t want to have sex with him, and he went ahead and had sex with her anyway, she wasn’t “kicking and fighting him off.” In the formal legal sense, the facts as described are unambiguous. Practically, of course, things are a lot more complicated, as the columnist’s response reveals.

The columnist seems to be drawing a distinction between rape and “sex that shouldn’t happen,” with the latter category including sexual assaults between acquaintences when one or both parties are intoxicated. How else are we to understand her otherwise bizarre advice that the raped woman talk to the man who raped her “in order to determine what happened?” The woman’s letter indicates no uncertainty at all about the fact that she was forced to have sex against her will despite making it very clear that she didn’t want to have sex. She just wants to know if this constitutes rape.

The answer, again culturally rather than formally legally speaking, is that this type of rape isn’t “really” rape, because the victim is to blame for putting herself in a compromised situation, i.e., being intoxicated in the presence of a man while having a vagina (the second factor was apparently supefluous in the case of the versatile Mr. Polanski).

These kinds of factors are what makes Polanski’s sodomizing of a 13-year-old girl something Anne Applebaum etc. consider a “far from straightforward” situation. It would be nice to think this is a generational thing, and that young people today are getting a clear message that rape is rape, but given both columns of this sort and the response to Polanski’s arrest the evidence seems mixed.

UPDATE [by SL]: See also Amanda Hess.

Derek Jeter: Sportsman of the Year

[ 0 ] November 30, 2009 |

The moment that this blog has been dreading since its creation has come to pass.

In all semi-seriousness, the hero worship athletes elicit is a subject worth studying. As I noted in the Tiger Woods post below, there’s a deep and widespread desire to see supremely accomplished athletes as generally admirable human beings, even though if anything there’s probably something of a negative correlation between the two things. For one thing, while it’s not necessary to be deeply selfish, or egomaniacal, or a narcissistic perfectionist, or a child of parents in the grip of grandiose manias, or some combination thereof, to get to the top of any sport or other competitive enterprise, it often helps quite a bit, as anyone who has had much contact with such people can attest. (In this regard I recommend Gary Smith’s portrait of the young Tiger Woods, “The Chosen,” from the December 23, 1996 Sports Illustrated issue which named Woods Sportsman of the Year. Another excellent essay on the subject in general is David Foster Wallace’s portrait of Michael Joyce, an obscure professional tennis player).

Of course the highest levels of achievement always require those who achieve them to have certain admirable qualities, such as a willingness to work extremely hard in the pursuit of initially distant goals. But it’s too easy to extrapolate from that fact all sorts of false conclusions, such as that the people who reach the top of a field have done so primarily because they have worked harder than other people. In a loose sense this is true (for example every major league baseball player or PGA golfer has undoubtedly worked very hard to get where he is), but there is no good reason to believe that Derek Jeter is a superstar while Joe Smith has just been granted his unconditional release from Pittsburgh’s AAA affiliate because Jeter works appreciably harder than Smith, or “wants it more,” or whatever other cliche sportswriters like to deploy when celebrating Jeter’s greatness.

This is a point that has more general ideological significance. It’s an article of faith in this country that rich people are rich primarily because they work harder than other people. This is the kind of belief that can and is maintained in the face of all evidence to the contrary, because people want to believe it — just as they want to believe that being the best golfer or shortstop in the world is primarily a matter of working harder at golf or baseball than everybody else.

Another parallel is that a lot of people believe that a high batting average and a high marginal tax bracket are both good proxies for moral election. This is one of those ideas that is sufficiently idiotic that it usually won’t be said in so many words — hardly anyone, after all, will actually say “I think the fact that Derek Jeter is a great baseball player indicates he’s a morally admirable person,” but anyone who has ever been stuck in a conversation with an Ayn Rand fan knows this line of thinking can be found well beyond the world of sports.

Brett Favre and the hype machine

[ 0 ] November 30, 2009 |

Speaking of the culture of celebrity and media saturation, an ironic aspect of the ridiculous levels of worshipful coverage that Brett Favre has gotten over the years is that it has made it eas(ier) to overlook that he’s in the midst of one of the most amazing seasons in NFL history. His 24 TD passes, three interceptions, 69% completion percentage, and 270 yards per game passing add up to by far the highest quarterback rating of his career, and one of the highest in history. He’s doing this at the age of 40, and today he tied Jim Marshall’s record for consecutive NFL starts by a non-kicker (282).

Another aspect of this story I like is that last August all the football insider types were certain that Favre’s flirtation with the Vikings would be, if consummated with a contract, harmful to team chemistry and other similarly mysterious alembics, and that indeed the whole soap opera of his second un-retirement was going to harm his “legacy.”

The Chosen One

[ 0 ] November 29, 2009 |

The Tiger Woods incident provides an interesting glimpse into the world of celebrity image making, and the corporate and media interests that enable it. Woods got into a minor car accident early Friday morning after he was apparently attacked by his enraged wife. She seems to have smashed in the back window of his SUV with a couple of golf clubs as he tried to flee their home at 2:30 AM. Woods was found lying in the street drifting in and out of consciousness and suffering from facial lacerations, raising questions regarding whether the window was the only thing his wife connected with. Woods is refusing to talk to the police, which isn’t surprising, given that a truthful account of the proceedings would probably require his wife to be charged with committing domestic violence.

He did however release this statement on his website, which is a kind of negative masterpiece of botched public relations.

Absurdly, Woods is issuing a fulsome apology to the world in general, while at the same time claiming all that happened is that he got into a fender bender just beyond his driveway. Even more ineptly, he addresses the “many false, malicious and unfounded rumors that are circulating” about him. By doing so, he’s practically requiring the mainstream media to report on, and ask him about, a National Enquirer story claiming that he is having an affair — a story that to this point the more respectable media have refused to even mention, let alone question him about.

The most ridiculous feature of the statement is his whining plea for “privacy.” Tiger Woods has become a billionaire by marketing himself so assidiously that he’s now the most recognizable athlete, and indeed one of the most recognizable people, in the world. His vast wealth (less than 10% of which has been earned directly through his athletic achievements) is a product of making himself into a kind of human logo, that corporations pay him immense amounts to attach to their products. They find it profitable to do so because of the preposterous yet very widespread idea that athletic excellence somehow reflects well on a person’s character and general value as a human being. Tiger Woods alleged adultery has nothing to do with his ability to excel on the golf course, but has everything to do with his ability to market himself as some kind of exemplary person, whose putative preferences in regard to cars and accounting firms and watches should influence your view of these products, and the corporations that produce them.

On one level I do feel sorry for Woods, in that his father was a certifiable lunatic, whose ambitions in regard to his son went far beyond turning him into the greatest golfer in the world. Consider this quote from Earl Woods, from a 1996 Sports Illustrated profile, written when Woods was all of 21 years old, and had yet to win a major golf tournament, let alone transform the course of human history:

Tiger will win because of God’s mind. Can’t you see the pattern? Earl Woods asks. Can’t you see the signs? “Tiger will do more than any other man in history to change the course of humanity,” Earl says.

Sports history, Mr. Woods? Do you mean more than Joe Louis and Jackie Robinson, more than Muhammad Ali and Arthur Ashe? “More than any of them because he’s more charismatic, more educated, more prepared for this than anyone.”

Anyone, Mr. Woods? Your son will have more impact than Nelson Mandela, more than Gandhi, more than Buddha?

“Yes, because he has a larger forum than any of them. Because he’s playing a sport that’s international. Because he’s qualified through his ethnicity to accomplish miracles. He’s the bridge between the East and the West. There is no limit because he has the guidance. I don’t know yet exactly what form this will take. But he is the Chosen One. He’ll have the power to impact nations. Not people. Nations. The world is just getting a taste of his power.”

The craziest part of all this is that Eldrick “Tiger” Woods probably on some level believes it — and very little in his life experience within a media-saturated and celebrity-crazed culture has contradicted this belief.


[ 0 ] November 28, 2009 |

Dave linked to this outburst below. It’s worth noting that when American Thinker Robin from Berkeley describes what she calls the “wilding of Sarah Palin,” she fails to mention that the original “wilding” — the infamous rape of the Central Park jogger in 1989, resulted in the wrongful convictions of six teenage boys, who collectively ended up spending several decades in prison for a crime they didn’t commit.

There’s a bathroom on the right

[ 0 ] November 25, 2009 |

A few years ago I ran into the concept of a mondegreen, which is usually defined as a mis-heard line in a song. The most commonly cited examples include “there’s a bathroom on the right,” as a mis-hearing of CCR’s “there’s a bad moon on the rise” and “s’cuse me while I kiss this guy” rather than Jimi Hendrix’s original “s’cuse me while I kiss the sky.”

Thanks to the wonders of wikipedia, I’ve learned that the original definition, formulated by Sylvia Wright, is actually narrower and more interesting: “The point about what I shall hereafter call mondegreens, since no one else has thought up a word for them, is that they are better than the original“.

It doesn’t seem to me that either of the common examples given above qualify. I humbly submit the following as instances from from my own personal history of mis-hearing song lyrics:

Rod Stewart, Maggie Mae:

I suppose I could collect my books and go back to school
Or steal my daddy’s cue and make a living out of playing pool.

The correct lyric is “fool.” “Pool” deploys a clever pun, and a much more arresting image of the feckless yet suddenly intriguing father.

[Correction: Jim in comments points out that in fact "pool" is the real lyric, and that my subsequent interpretation is the actual mondogreen, except it wouldn't be one by the original definition. As Emily Litella used to say . . . never mind].

Speaking of which, The Kinks, Father Christmas:

When I was small I believed in Santa
Though I knew there was no dad.

Instead of the canonical “though I knew it was my dad.” The mis-hearing adds a level of wistful pathos to the proceedings.

Next up, Neil Young, Helpless:

There is a town in north Ontario
With dream comfort memory to spare

The correct line is “With dream comfort memory despair.”

I’m of two minds about this one, as the correct version is more disturbingly surreal, while the mis-hearing has a certain homey charm.

Anyway, I like Wright’s original definition much more than the contemporary (mis)understanding of what she had in mind, which is rather ironic as Alanis Morrisette did not observe.

Obama’s war

[ 0 ] November 24, 2009 |

An unfortunate aspect of the nature of politics is that principled opposition to disastrous and/or immoral policies tends to either disappear or at least lose much of its intensity when such policies are adopted by politicians one supports.

Certainly over the last year we’ve seen this among what passes for the political left in this country, in regard to the wars in Iraq and Afghanistan.

It’s true that Obama inherited these wars. He was elected to end them.

Yet today it’s being reported that, after nearly doubling the US military presence in Afghanistan earlier this year, he has decided to increase that number by 50%, at a direct cost of one million dollars per soldier. The indirect costs are incalcuable.

The administration’s plan contains “off-ramps,” points starting next June at which Obama could decide to continue the flow of troops, halt the deployments and adopt a more limited strategy or “begin looking very quickly at exiting” the country, depending on political and military progress, one defense official said.

“We have to start showing progress within six months on the political side or military side or that’s it,” the U.S. defense official said.

In short, the next six months will be crucial.

If you haven’t yet seen this recent Frontline program on the current situation in that country it’s worth your time.

Gatsby on Madison Avenue

[ 0 ] November 17, 2009 |

My wife and I started watching Mad Men on DVD about three weeks ago, and are now thoroughly addicted. We’re about midway through the second season, and just viewed the episode that features Mr. and Mrs. Donald Draper on a family picnic, during which Don tosses a beer can into the woods and Betsy cleans off the blanket on which they ate by simply tossing all the wrappers, napkins etc. onto the grass.

This scene reminded me of how when I was a child in the late 60s and early 70s there was what in retrospect seems like an intense anti-littering campaign, featuring among other things this famous PSA, that everyone of a certain age remembers.

I don’t know anything about the genesis or ultimate efficacy of that campaign, but if a scene from a period piece TV show counts as compelling evidence, it seems to have worked.

Seriously, did lots of Americans — including privileged people highly conscious of what was considered socially correct upper class public behavior — just use the outdoors as a wastebasket 50 years ago? Anyone know of any studies of the issue?

This is teh awesome

[ 0 ] November 16, 2009 |

From the Jacksonville-Jets game:

“It included a Jets defense trying to allow a touchdown to give Jacksonville the lead and the Jaguars refusing to score it. And it had nothing to do with either team trying to lose.

Jaguars running Maurice Jones-Drew, one of the stars of the game, intentionally dropped to his knee before the end zone after a 9-yard gain to set up the winning field goal.

Mindful of what some fans care about most, Jones-Drew said, “Tell my fantasy owners I’m sorry.”

He did it to run down the clock because the Jets had no timeouts and the Jaguars did not want the Jets to get the ball back after what was to be a 21-yard field goal by Josh Scobee.

Jones-Drew already had one touchdown and 123 rushing yards. Another score by him would have meant valuable points to fans who drafted him for their fantasy leagues.

“I’d rather take a win any day,” Jones-Drew said.

But when he was told not to score, Jones-Drew said he was surprised at first. “I’m like, a knee? What do you mean?” he said. “I took a deep breath and took a knee.”

What made it even stranger was the play before that. On first-and-10 from the Jets’ 14 with two minutes left, Jets Coach Rex Ryan told his defenders to let Jacksonville score. Ryan figured it would give the Jaguars a 6-point lead but leave his team time to take the ensuing kickoff and drive the field for a game-winning touchdown.

Because not all of his defenders got the message, Jones-Drew was tackled at the 10 by Marques Douglas and Sione Pouha after a 4-yard run.

“We couldn’t even get that right,” Ryan said.

That forced Ryan to use his last timeout. Jones-Drew dropped to a knee at the 1-yard line on the next play, and after that quarterback David Garrard knelt on the next two downs to waste more time and set up the winning kick.”

Meanwhile in related news professional moron Andy Reid kicks a field goal on fourth and one from the San Diego one while down 14-0, and then another one on fourth and one from the San Diego seven while down 21-6.

UPDATE [by SL]: On the other hand, Belichick’s disdain for the conventional wisdom on punting has served him well, but (with all respect for Manning and granting that they got screwed on the call) that seemed to be pushing it to an indefensible extreme.

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