Subscribe via RSS Feed

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.


ChiComs Under the Bed!

[ 0 ] December 3, 2009 |

Al Kamen, via Jason Sigger:

This month marks the 10th anniversary of the Clinton administration’s cavalier handover of the Panama Canal — leaving an alleged front for the Chinese Red Army in control of the strategic passage — despite the strong misgivings of some top foreign policy experts.

“If we do nothing, I can guarantee you that within a decade, a communist Chinese regime that hates democracy and sees America as its primary enemy will dominate the tiny country of Panama, and thus dominate the Panama Canal, one of the world’s most important strategic points,” Rep. Dana Rohrabacher (R-Calif.) told a House subcommittee on Dec. 7, 1999, as it debated the handover.

Retired Adm. Thomas Moorer warned that China could sneak missiles into Panama and use it as a launchpad for attacking the United States. And former defense secretary Caspar W. Weinberger wrote that fall that Panama’s contract with Hong Kong-based Hutchison Whampoa to control ports at both ends of the passage was “the biggest threat to the canal.”

Is that a money back guarantee, Dana?


[ 1 ] December 3, 2009 |

Edroso, on one of the most prominent bigots on the “Democratic” side of the aisle in Albany:

I will say that Hiram Monserrate surprised me, in that I don’t see how he manages to be so perfectly disgusting all the time. From his beginnings as a deranged cop to his (as a councilman) Willets Point double-cross to his involvement with the Albany “Gang of Three” shakedown artists and Coup to his assault on his girlfriend, this guy seems almost consciously determined to set new standards of repulsiveness. Maybe he’s a government experiment of some kind.

And what’s even worse is that when it comes to determining the most loathsome member of the Democratic caucus he has serious competition.

a more positive example.

World Cup 2010 seeds

[ 0 ] December 3, 2009 |

The seedings for Friday’s group stage draw were released yesterday.

There were some fairly significant changes from past practice. For the past three, four, or five? World Cups, a combination of past performances in the World Cup (either two or more often three tournaments back) with an index based on current, one year, and two years past FIFA rankings. For 2010, it’s the October FIFA ranking (only) combined with geography. The hosts, as is practice, are also seeded.

Teams are divided into four “pots”; each group will be populated with one team from each pot. Here they are:

Pot 1 (seeds): South Africa, Brazil, Spain, Netherlands, Italy, Germany, Argentina, England

Pot 2 (Asia, Oceania and North/Central America): Representing Asia: Japan, South Korea, North Korea, Australia; Oceania: New Zealand; CONCACAF: United States, Mexico, Honduras

Pot 3 (Africa and South America): Africa: Ivory Coast, Ghana, Cameroon, Nigeria, Algeria; S. America: Paraguay, Chile, Uruguay

Pot 4 (Unseeded Europe): France, Portugal, Slovenia, Switzerland, Greece, Serbia, Denmark, Slovakia

The logic is the “best” eight sides are kept apart in the group stage, and no two teams from the same confederation will meet in the group stage (thus making my dream match of South v North Korea highly unlikely) except for Europe — there will be five groups with two European teams.

Hence, by selecting what FIFA believe to be the top eight sides, even though those top eight sides are not directly related to their own sketchy monthly rankings, and ensuring that those eight sides are placed in eight different group, the odds are significantly enhanced that those eight will make it through to the knock-out stages.

This doesn’t always happen of course; France finishing last in its group in 2002 is a clear memory, but all eight seeded sides did progress in 2006, but it does sharply reduce the odds of, say, an Algeria v North Korea quarter final (but imagine the TV ratings back in Pyongyang).

There are, as usual when it comes to FIFA, some idiosyncrasies. Neither France nor Portugal are seeded, even though they both are (currently) ranked higher than England. I don’t think France are all that any longer, but Portugal did knock England out at the quarter finals of both the 2006 WC and the 2004 European Championships. BBC Radio 5 Live suggested this morning that the French are being punished for the Ireland tie. What has not gone reported is that the seeds were based on the October, not November, rankings, in order to mitigate any built in advantage that teams involved in playoffs (as opposed to friendlies) during the month of November might have enjoyed.

Which is a different way of saying “FIFA sleight of hand”. The only two teams in the top seven in November are Portugal (5th) and France (7th). Neither Argentina nor England would have been seeded.

What does this mean for the USA tomorrow? It’s going to be grim, regardless; put the Confederations Cup performance away (which was uneven in any event). There are several best / worst case scenarios out there in blogosphere, but before we get too depressed, The Times has this worst case scenario for England:

a worst-case scenario would still involve them being drawn in the same group as France, Ivory Coast and the United States.

How sweet of the English media to suggest that the USA are in their own personal group of death. For the US, placing the CONCACAF and Asian and Oceania teams in the same pot means that we can not draw any of them — this screws us as it’s the weakest of the four pots; while ruling out the North v South Korea match, this also rules out the USA v North Korea match (remember France 1998 against Iran? I’d rather I didn’t as well).

Prost Amerika suggests these best / worst cases:

Best Case Scenario: Argentina, USA, Algeria, Switzerland.
Worst Case Scenario: Spain, USA, Ivory Coast, France.

I’d rather draw South Africa from the first pot, but that would rule out Algeria from the third. It’s a worthy trade off I think, so this is my best / worst case scenario:

Best: South Africa, USA, Paraguay, Greece (or Slovenia, or Switzerland . . . )
Worst: Brazil, USA, Ivory Coast, Portugal.

When it comes to drawing from the first pot, outside of the hosts they are all scary; when it comes to drawing the least dangerous of the European pot, there are several that are equally preferable to Portugal, France, or Denmark.

I will be discussing the resulting draw at some point this weekend, possibly even tomorrow evening (UK time).


[ 0 ] December 3, 2009 |

Thers has a more extended rant on the subject, but this is rich:

Republican senators feel burned by Al Franken — and not by his old jokes.

The Republicans are steamed at Franken because partisans on the left are using a measure he sponsored to paint them as rapist sympathizers — and because Franken isn’t doing much to stop them.

The band of Republican crybabies includes John Cornyn, whom you might remember was one of the people willing to stall Eric Holder’s confirmation hearings because the prospective attorney general didn’t promise to excuse Bush-era torturers from prosecution. So John Cornyn appears not only to be perfectly fine with torture in the service of boneheaded military adventures, but he also seems to believe that it’s Al Franken’s job to troll the blogosphere and defend him from accusations that he’s perfectly fine with torture and rape in the service of boneheaded military adventures. No word yet on whether Franken is supposed to ask us to stop referring to his friends as “box turtles”.

We’ll Dance Around Like Ol’ John Brown on the Long End of a Rope

[ 0 ] December 3, 2009 |

I dunno; I’m as contemptuous of the antebellum American South as anyone, but I’m still not convinced that John Brown deserves a pardon. You can be sympathetic with Brown’s ends, and even recognize that his actions played a significant role in emancipation, while being reluctant to embrace the idea that private citizens ought to be able to murder other private citizens for the conduct of activity which is perfectly legal. I appreciate that pardons are often used to recognize conduct which is laudable without being legal, but it’s difficult to forgive Brown the murder of Hayward Shepherd, who was not a slaveholder, and who likely did not support slavery. I think that there’s rather a better case for the pardon of Nat Turner, on the grounds that the institution of slavery freed its subjects of any ethical responsibility to the constituted legal authority.

In any case, I doubt that Brown would approve of even a 150 year posthumous pardon.

I Am Shocked, Shocked to Learn of the Necessity of Additional Troops!!!

[ 0 ] December 3, 2009 |

Donny Rumsfeld, not quite lying:

In his speech at West Point last night, Obama claimed that before he took office, “commanders in Afghanistan repeatedly asked for support to deal with the reemergence of the Taliban, but these reinforcements did not arrive.”

Rumsfeld is now strongly denying that claim, calling it a “bald misstatement.”

“I am not aware of a single request of that nature between 2001 and 2006,” Rumsfeld said. “If any such requests occurred, ‘repeated’ or not, the White House should promptly make them public. The President’s assertion does a disservice to the truth and, in particular, to the thousands of men and women in uniform who have fought, served and sacrificed in Afghanistan.”

There is a sense in which Donald Rumsfeld is telling the truth. The sense in which he is telling the truth is structured as follows: Donald Rumsfeld either intimidated or outright fired anyone in the military brass who tried to make a formal request for additional troops, in either Afghanistan or Iraq. He made it clear from a very early point in his tenure that he would view such requests as acknowledgments of defeat, both in terms of the wars in question and for his project of military transformation. As Bradley Graham details, when the idea of reinforcement was mooted or when informal requests were made, Rumsfeld brusquely interrogated the generals in questions until the topic was dropped. Sending more troops was not something that Rumsfeld was prepared to entertain, and he was careful to surround himself with people who made certain that the topic was never seriously raised.

So yes, Don Rumsfeld is telling the “Truth.” Virtually everyone understands the worthlessness of this “Truth;” even wingnuts, enamored of the post-Rumsfeld surge, are reluctant to man this particular barricade. Recognition of Donald Rumsfeld’s incompetence is perhaps the last truly bipartisan consensus in modern American politics.

Keep Waitin’

[ 0 ] December 2, 2009 |

I must remind everyone once again that Peter Beinart, like so may centrist contrarians, was wrong about many things other than the Iraq War. You may remember his claim that the New York courts refusing to end the state’s marriage discrimination was the best thing ever to happen to same-sex marriage rights in New York. Well, as we were reminded again today, this was and remains egregiously false. (Depressingly but not surprisingly, 8 Senate Democrats hopped on the bigotry train.)

In a rational world, the comparison of New York and Maine on the one hand with Massachusetts, Vermont, Connecutcuit, and Iowa on the other would put the countermobilization myth to bed once and for all. Alas, if you’re pitching articles contrarianism sells a lot better than rationality.

Deep Thought

[ 0 ] December 2, 2009 |

I am absolutely shocked that, despite a near-total lack of precedent, a wealthy professional athlete has engaged in sexual relations with persons to whom he is not married, and I hope that cable news will devote more time to these remarkably surprising and important revelations.

Instant Reaction: Bloggingheads Style

[ 0 ] December 2, 2009 |

Last night, I diavlogged with Matt Duss on the subject of the Obama speech:

As this suggests, I’m pretty ambivalent about the escalation.

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.”

Blair: The Biggest Villain?

[ 1 ] December 1, 2009 |

Tom Ricks:

As a British naval historian friend I know once noted, the time when the British government could have helped — and perhaps stopped the war — was back in the winter of 2002-2003. Real friends speak up when a friend is making a big mistake. Instead, Tony Blair may have destroyed the “special relationship” by supporting the invasion when he should have opposed it. My friend said he believes Blair should be confined right now in the Tower of London.


1. I wonder if Blair really could have stood and said “No.” I always kind of suspected that Blair pursued the Iraq War with the enthusiasm he did because he believed that he couldn’t stop it if he wanted, and a) wanted to be part of the action, and b) wanted to maintain the “special relationship.” This isn’t to say that Blair privately opposed the war, just that his primary motivations were about the relationship more than conviction about the wisdom of the invasion. But I really don’t know.

2. If Blair had said “no,” would the neocons have spewed the same vitriol towards Britain that the sprayed at France? I would have loved to see a book explaining how the United Kingdom is our enemy, and in fact has always been our enemy; it makes even more sense than France.