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Answer To “Not Metcalfe’s Paradox” Puzzle

As usual, the puzzle's been answered well by commenters. Jason points out that this is the Monty Hall Paradox and – what I shoulda checked for – the puzzle and answer are in wikipedia.

Here's the puzzle for those who missed it:

You are a guest on a game show. There are three closed doors; behind one of them is a car you want to own; behind the other two are goats you don't want despite the fact that they don't burn gas.

You have to pick a door. After you do that, the host will pick a door behind which there is a goat (he knows what's where and has to follow the rules). You then get to decide whether you should be awarded what's behind the door you picked initially or what's behind the door that neither of you picked.

The questions are:

Does it matter which strategy you pick?

If so, which strategy is favored?

What is the quantitative advantage, if any, of the favored strategy?

For extra credit: why?

The answer:

You want to pick what's behind the door that was picked by neither you nor the host. If you stick with your first choice door, your odds of winning are 1 in 3. If you switch to the remaining door after you and the host pick, you increase your odds to 2 out of 3 of ending up with the car.

But this is counterintuitive. I had to be beaten into accepting it but it's right. Here's why:

Obviously, if you pick a door at random (you have no information so your choice IS random), your odds of picking a door with a car behind it are 1 in 3; that part's easy. If you stick with the door you picked, you will win one third of the time and lose two thirds of the time.

Now suppose you follow the switching strategy. One third of the time you will have picked the door with the car initially. In this case you'll lose, however, the other two thirds of the time you'll win. That's because you are actually getting help from the game show host when you follow this strategy and he DOES have information!

Let's look more closely at what happens when you pick a goat door initially (which you will do two thirds of the time): In that case there are two doors left, one with a goat and one with a car. The host MUST pick the door with the goat (see rules above). That leaves only the door with the car which you then get to drive home. Whenever you pick a goat door first, you WILL win with the switching strategy thanks to the host eliminates the remaining goat door. Since your odds of picking a goat door are 2 in 3, you will win two thirds of the time with the switching strategy. QED.

If you don't believe me, check many diagrams in wikipedia.

NOT Metcalfe’s Paradox – A Puzzle

Bob Metcalfe posed this puzzle at an unconference we were recently at. He says, however, he didn't invent it. Most people, including me, get the answer wrong; in fact, I got the answer wrong twice – once by not listening and once by not thinking.

Here's the puzzle:

You are a guest on a game show. There are three closed doors; behind one of them is a car you want to own; behind the other two are goats you don't want despite the fact that they don't burn gas.

You have to pick a door. After you do that, the host will pick a door behind which there is a goat (he knows what's where and has to follow the rules). You then get to decide whether you should be awarded what's behind the door you picked initially or what's behind the door that neither of you picked.

The questions, smart reader, are:

  1. Does it matter which strategy you pick?
  2. If so, which strategy is favored?
  3. What is the quantitative advantage, if any, of the favored strategy?
  4. For extra credit: why?

Answers to Probability Puzzle

As usual, smart readers knocked the cover off the ball almost immediately. Some day I’ll stump you.

First question taken from Randomness by Deborah J Bennett:

“If a test to detect a disease whose prevalence is one in a thousand has a false positive rate of 5%, what is the chance that a person found to have a positive result actually has the disease, assuming you know nothing about the person’s symptoms or signs?”

First to answer correctly was Matt Crawford:

“Assume that the test is performed on everyone regardless of symptoms of the disease. Then out of every thousand people who receive the test, one has the disease and 999 do not. Further, assume that the test has no false negatives: anyone who actually has the disease gets a positive result. Then 1 out of every thousand tests are true positives. The remaining 999 should be negative results, but the 5% false positive rate means that 49.95 (so round to 50) of these people will receive false positive results. Then out of our 1000 tests, 51 return positive results. But only one of these is a true positive, so the chance that a positive test identified someone who actually has the disease is 1/51 or about 2%.”

You might quibble that 5% false positives means 50 false positive out of a population of 1000 (plus one correct positive) but this is close enough. It’s fair to make the assumption that there are no false negatives since this isn’t stated in the question (and otherwise you’d be unable to answer) but Aswath is right to point out this should have been specified.

Second question: “what percentage of the physicians, residents, and fourth year medical students at a prominent medical school who were asked this question got it right?”

jb guessed that 80% of those tested would give the tempting wrong answer of 95%. Actually, only 19% gave the right answer but only 50% said 95%. jb, you would’ve nailed it if you hadn’t given more detail in your answer than called for. Rob’s an optimist and hoped that 80% would get it right because their care is so important and getting into medical school requires critical thinking. He’s dead right that it’s scary that so many get it wrong.

Interesting answers to third question: “why is it critically important that doctors be able to get this one right? Give one example.” Most not about doctors, though. This type of bad thinking does cover lots of ground.

Matt Crawford cites the Red Cross using an HIV test on donated blood which is known to have a high incidence of false positives and speculates that many donors are probably panicked by the result. “However, the Red Cross continues to use the same test, probably because it combines low cost with very low false negative rate. In this case it may be justified to trade a high false positive rate for a low false negative rate, because a false positive merely requires a second test but a false negative would spread HIV through transfusions.”

Curtis Carmack says: “the medical profession as a whole has given insufficient thought to how to address the false positive issue with patients, leading to much more angst than is necessary when patients receive a positive test result -- invariably late on Friday -- and have to wait at least a couple of days to ask questions about it. ;-)”

Dennis Shanley posts: “This directly effects the overall cost of health care in a huge way. Assume that it costs $10,000 to cure a patient who presents positive. Not an unlikely assumption. Assume further that the 50 false positive patients do not exhibit negative effects as a result of their treatment that require further medical treatment and they do not litigate as a result of the unnecessary treatment. This is a highly improbable assumption made for the sake of simplicity.

“The true cost to cure 1 patient is $10,000.

“The cost to cure that one patient and treat the 50 false positives is $510,000.”

Aswath writes: “Suppose now we are told that the false positive predominantly affects a biological group - gender or a racial group. Will that decision stand reason? Let us assume that the situation is internment during WWII in US. A nation has to live with the effects of a callous operation decision to accept a large false positive.”

Otmar: “There is another interesting application for this kind of statistics: The beloved war on terror. The chance of a random person to be a terrorist is hopefully less than 1/1000. Imagine you manage to build some automated system which somehow claims to spot suspicious behavior, known faces, or miscreants by some other clever scheme.

These systems all have a non-negligible error-rate. If you're really lucky, you might push that one down to less than 1%.

“Now do the math again, assuming a 1/100000 terrorist-rate and 1% false positives. No wonder I read that one trial for such a system got terminated.”

The point is that you must weight the costs of being right and the costs of being wrong both for the positive and the negative case. Back to medicine, suppose your doctor is one of the benighted 81%. He or she tests you using the test in the first question and you come up positive. Let’s suppose that the disease is always fatal if not treated and there’s a treatment available but it has a 25% chance of killing you itself. If the doctor believes that there’s a 95% chance you have the disease, the dangerous treatment is clearly justified; but, since the true likelihood is less than 2%, the treatment is more dangerous than your untreated prognosis. Always a good idea to get a second opinion AND check your doctor’s math.

A Probability Puzzle

From Randomness by Deborah J Bennett:

“If a test to detect a disease whose prevalence is one in a thousand has a false positive rate of 5%, what is the chance that a person found to have a positive result actually has the disease, assuming you know nothing about the person’s symptoms or signs?”

For extra credit: what percentage of the physicians, residents, and fourth year medical students at a prominent medical school who were asked this question got it right?

Extra, extra credit: why is it critically important that doctors be able to get this one right? Give one example.

This is an honor system non-open book test.

Answers in comments, please. Will highlight correct answers in a subsequent post. Hat tip to Nassim Taleb in Fooled by Randomness for citing Bennett’s test.

Answers here.

Ubiquitous Power for Traveling Nerds – The Cigar Lighter Inverter

Inverter Reader Omar nailed it: “The one power source that I think is consistent around the world is the 12v power (cigar lighter in older vehicles) in most vehicles.” It’s an incredibly ubiquitous standard. It shows up in boats as well as cars; but, for some weird reason, airplane seats with power use a different plug.

So the answer to my question of which one device does a nerd who has been forbidden to bring his twenty-piece power adapter kit need to take instead to assure that his or her electronic toys can be topped off with electrons is – drum roll – a twelve volt inverter! There are lots around; I’m not recommending any particular brand. You can see that the one pictured here hasn’t even been freed from its bubble-wrap; but note that it does come with an adapter for airplane seats.

Note that this inverter has a 120 volt out so the power supplies for every one of my North American toys plug into it: cell phone, PC, camera, GPS, even my battery recharger. If you’re based elsewhere, you can get an inverter with the right output voltage for your toys . You could buy a separate cigar lighter adapter for each toy and you would use energy more efficiently because you wouldn’t be converting from DC to AC and back again; but that’s too much junk.

One of these inverters saved a vacation for us (and a little marital stress). Mary booked a stay at a place in Belize near ruins and the rain forest; she’d already put up with a week on a sailboat off Belize so this was “her” half of the vacation. At the time I was regularly working on my novel hackoff.com: an historic murder mystery set in the Internet bubble and rubble. After a long bumpy drive down proto-roads we arrived at our home-for-a-week. It looked lovely lit by its kerosene lanterns. Whoops, they’re not just for atmosphere. No electricity!

But no problem. We had a car for exploring. I had my inverters (one plus a backup) because we’d been on a boat. Plenty of juice to recharge my PC after a writing bout and ruins around to recharge me.

The amount of power you can actually get out of an inverter depends both on the size of the inverter AND the voltage available at the source. Sometimes lousy boat batteries cause inverters to trip themselves offline. The solution can be to charge only when the alternator and engine are running but this isn’t fool-proof. Spikes knock inverters offline also. However, the more electronics a boat has, the better the power is likely to be. I’ve also had inverters refuse to work on planes.

The inverter pictured here, for example, is not recommended for PCs with screens larger than 15 inches or TVs. The next size up can handle all laptops, small TVs, and even cordless tool chargers.

BTW, I’ve never had to take a taxi or rent a car JUST to charge my batteries – but it could happen.

If anyone knows the history of how cigar lighters got so standardized, I’d love to hear it. It’s not that there’s only one possible device that could light a cigar. Part of the story must be the prevalence of 12 volt car batteries.

Readers Ben Metcalfe and Michal Altair Valasek both suggested USB-based power attachments. USB certainly is universal and it is fascinating to see it used as a power source even when no data connection to the host PC is needed. Doesn’t solve the problem of how to get power into the PC in the first place but, perhaps, you have a cigar lighter adapter just for that.

Power for Traveling Nerds – A Puzzle

We can’t get power wirelessly – yet. The outlets available vary from country to county; perhaps not with quite as much variety as phone adapters but you still need a pretty good kit to make sure you can connect to wall sockets everywhere. Moreover, of course, the power that come out of the wall isn’t all the same either. Voltages as well as the frequency of AC power differ from place to place. Some electronics can handle these differences; some can’t.

You’re a nerd so you’ve got a pack full of stuff that needs to be recharged or even plugged in to work: computer, of course, mobile phone(s), camera, iPod, GPS, video recorder etc.

Here’s the challenge:

You’re not allowed to take your adapter collection on the next trip; you’re not allowed to buy, borrow, or otherwise acquire adapters as you travel nor will you be buying all new toys which work on local power. You can’t bring a solar charger or a crank driven one or a small windmill (if such are available). You can’t take wall plates off or stick probes into them. What ONE device is essential to preserve your ability to use your toys?

I’ll answer tomorrow but I’m sure some of you will beat me to it.

Now on Kindle!

hackoff.com: An historic murder mystery set in the Internet bubble and rubble

CEO Tom Evslin's insider account of the Internet bubble and its aftermath. "This novel is a surveillance video of the seeds of the current economic collapse."

The Interpreter's Tale

Hacker Dom Montain is in Barcelona in Evslin's Kindle-edition long short story. Why? and why are the pickpockets stealing mobile phones?

Need A Kindle?

Kindle: Amazon's Wireless Reading Device

Not quite as good as a real book IMHO but a lot lighter than a trip worth of books. Also better than a cell phone for mobile web access - and that's free!

Recent Reads - Click title to order from Amazon


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