It's well known that compared to some other kinds of reasoning, humans are, on average, very poor at probabilistic reasoning. You can come up with many reasons why this might be so, such as evolutionary rationales, etc. but the fact remains that human intuitions are often wildly inaccurate when it comes to probabilistic judgments. Even professional mathematicians are prone to error in simple probability problems. The other day I was reading a book by a Macarthur genius grant winner. In one of the chapters, dealing with probability, there was an elementary error that author, reviewers, and editor failed to catch.

The most famous example is the Monty Hall problem, which caused a storm of controversy back in the early 1990s when Marilyn vos Savant wrote about it in her Parade column. In this game, you choose one of three curtains. Behind one curtain is a car. Behind the other two are goats. You get the prize behind the curtain you choose. The host, Monty Hall, knows what is behind which curtain. After you've made your choice, he will open the one of the two curtains you haven't chosen which contains a goat. Then he will offer you the chance to switch to the remaining unopened curtain. What should you do? The correct answer is that you should switch, because 2/3 of the time you will win the car this way. When vos Savant wrote her column, many mathematicians around the world wrote in to say how ignorant she was, and that the correct answer was that the probability in either case was 1/2. (Luckily, none of these mathematicians was a probability theorist.)

Leibniz himself, in the days before probability was a worked out theory, found it difficult to calculate the odds of throwing 11 with two dice.

There are bloopers every day, such as in this blog entry:

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I've had people tell me that they are worried about abductions. They say this gently, trying not to criticize my parenting, but they obviously think I'm being too laissez-faire with my kids. And I'm pretty sure they mean stranger abductions, although I know (even if they don't) that custodial interference abductions are the main threat to our kids, and it's not like the parent has to wait to snatch them off the street if that's going to happen. They also raise the specter of the bogeyman child murderer. I've got an answer for that one: the person statistically most likely to kill my children is me. Time they spend walking home is time they spend away from the person most likely to murder them, as well as time spent out of a car (their most likely place to die.)

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It's great as a debating riposte, but aside from the obvious objection that knowledge about specific individuals renders such statistical information much less useful, there is also an error of statistical inference at play: the base rate fallacy. That is, the average child spends a lot less time walking home alone than with their parents.

Base rate problems are also at play in a familiar paradox that I've mentioned before here. You go to get a medical test for a horrible disease that affects 1 in 10,000 people. The test is 99% accurate. 99% of people who have the disease test positive. And 99% of people who don't have the disease test negative. Your test comes back positive. What is the chance you have the disease? Out of 100,000 people, 10 have the disease, and about 10 will show up as positive. But out of the remaining 99,990 people, about 1000 will also show up as positive who DON'T have the disease. So out of every 1000 positive test results, only 10 will have the disease. If you get a positive test result, the chance you have the disease (absent other information) is about 1 in 100.

I recently dealt with a case like this with my doctor. I had a puncture wound and was deciding whether to get a tetanus shot. I'd accidentally gotten two tetanus shots quite close together, about 11 years ago. The second shot caused flu-like symptoms. Once you've had any kind of reaction to a vaccine, you're at increased risk of having a serious reaction. Medical practice is to automatically give tetanus boosters to people who have puncture wounds. The question I wanted to know was, was my risk of dying of a reaction to a tetanus shot HIGHER than my risk of dying from lockjaw? In which case, the shot made no sense. The doctor didn't know and there isn't anywhere to readily get such data to make a quantitative comparison. I did some research. It turns out that people who have had their primary series and one booster of tetanus have very good protection against tetanus until age 50 or so, even if they are not "up to date". There are very, very few cases of tetanus in Canada, so low the risk is barely measurable at this point, particularly among people who had their primary series. Although the numbers were not hard, research articles suggested that the risk of severe reaction among people who've had a reaction in the past was not trivial. So we decided not to get the shot. We tested for antitoxin instead and the test confirmed a high level of immunity even though I'm "not up to date" on my tetanus shots.

In the OJ Simpson case, the fact that OJ abused his wife played an important role. Alan Dershowitz belittled this as irrelevant by citing research that said "only" 0.1% of men who abuse their wives actually murder them. Was Dershowitz' argument sound? No. Because the crucial information in this case is that Nicole Brown actually WAS murdered. Using crime statistics, an article in a 1995 issue of Chance magazine does the math (which I can type up later--actually I read the case in a very good book, Understanding Probability by Henk Tijms), and gives the result that there is a probability of 81% that the husband is the murderer of the wife, given that he previously abused her.

So, my question is first to lawyers: how do absurd uses of statistical inference such as Dershowitz's stand? How does the court evaluate them? Why aren't lawyers trained in basic inductive logic? This is hardly the only example of badly flawed statistical inference I've seen in court. It seems to me that much of the law is about probabilities, and becoming more so with various kinds of well-quantified forensic evidence. Can a judge simply instruct the jury that Dershowitz's argument is mathematically invalid, as much as saying 2+2=34?

And for doctors, I also wonder why, given that many decisions rely crucially on statistical inference, most doctors can't do the basics of probability reasoning. Again, why isn't this a basic part of their curriculum?

[ 01 November 2005: Message edited by: rasmus raven ]

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Originally posted by rasmus raven:
It's well known that compared to some other kinds of reasoning, humans are, on average, very poor at probabilistic reasoning. You can come up with many reasons why this might be so, such as evolutionary rationales, etc. but the fact remains that human intuitions are often wildly inaccurate when it comes to probabilistic judgments.

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Probability does not exist

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... conveys his idea that probability is an expression of the observer's view of the world and as such it has no existence of its own. Although the idea of probability as a measure of the observer's belief that an event will happen had already been conceived by F P Ramsey in 1926, Bruno de Finetti was unaware of Ramsey's work and, moreover, his chief interest was for coherent probability assessments and not for rational decisions; see the obituary by L Daboni [3] for more information. As a consequence of the subjective approach, statistical inference is no longer an empirical process producing opinions from data, but it becomes a logical-psychological process selecting opinions compatible with data among the available ones.

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Originally posted by rasmus raven:
And for doctors, I also wonder why, given that many decisions rely crucially on statistical inference, most doctors can't do the basics of probability reasoning. Again, why isn't this a basic part of their curriculum?

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Biostatistics, or biométrie, is a basic part of the curriculum in a lot of schools - as in, just another first-year or pre-med course that you have to pass to get through the program. As well, epidemiology is a field of study in which doctors learn about probabilities, incidence, prevalence, risk, positive and negative predictive values for diagnostic tests, sensitivities and specificities - it's a core subject in veterinary medicine because we deal more with overall populations (herds) than individuals; in human medicine, epidemiology is more of a specialty, and you won't find an epidemiologist in your local walk-in clinic. A lot of the time, epidemiological probabilities are simply annoying when you are dealing with an individual patient and honestly, speaking from personal experience, they can even be misleading when dealing with individual cats or dogs. Very helpful when vaccinating a herd of cows, though.

There was a review in (I'm pretty sure) this past weekend's G&M of a new book from one of the Canadians known best for working on probability, and while it was fun, some claims were made that I questioned. Now, that may be the reviewer oversimplifying the book, but can someone here explain to me how to think of this example:

Almost btw, reviewer raised the issue of the possible avian-flu pandemic (I couldn't tell whether the author had) and seemed to be pooh-poohing it, on the basis of statistics for deaths from other diseases and causes, which are, of course, normally much higher. (Sorry: I'm putting this badly because I don't have the text here.)

That struck me as unsubtle. Surely the context of highly contagious diseases is a specific context, not the norm, and it's not as though we don't know that such diseases can suddenly, if briefly, wreak exceptional havoc.

Yes, I always assumed that doctors had some brush with probability and statistics in their curriculum. I guess my frustration is that in the average case, it doesn't sink in and they are still so bad at it. Question: when they get test results, are the probabilities spoon-fed to them in the reports (like "probability that the patient has the disease, given the test is positive")? I certainly hope so.

Skdadl, here is the Globe review of Struck by Lightning. I think the problem in some of his examples is the assumption that our worry about them should boil down to "the chance that it will happen to me". Certainly, with 9/11, that becomes an arid analysis. Sometimes the argument IS only about the chances, and that's when sound probabilistic inference is important, and rarely encountered. But often the chances are only part of the discussion.

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Originally posted by rasmus raven:

The most famous example is the Monty Hall problem, which caused a storm of controversy back in the early 1990s when Marilyn vos Savant wrote about it in her Parade column. In this game, you choose one of three curtains. Behind one curtain is a car. Behind the other two are goats. You get the prize behind the curtain you choose. The host, Monty Hall, knows what is behind which curtain. After you've made your choice, he will open the one of the two curtains you haven't chosen which contains a goat. Then he will offer you the chance to switch to the remaining unopened curtain. What should you do? The correct answer is that you should switch, because 2/3 of the time you will win the car this way. When vos Savant wrote her column, many mathematicians around the world wrote in to say how ignorant she was, and that the correct answer was that the probability in either case was 1/2. (Luckily, none of these mathematicians was a probability theorist.)

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This is drift, but I've seen that before and was initially unconvinced that vos Savant was right; the way the explanation was done seemed convoluted or poorly expressed or something. At any rate, I found it deuced hard to grasp. But I just set up a group of sticky notes marked "goat", "goat" and "car" and went through the process of what happens given different choices, and it seems to me way simpler than it's been explained to me in the past. So for anyone who finds this problem a bit weird, here's the deal as I see it:
If you picked the car, switching will get you a goat. But there's only a one third chance of that.
On the other hand there's a two thirds chance you picked a goat, 'cos there's two of them. And if you pick a goat, then Hall will show you the other goat. Switching your choice then *must* get you the car. So, picking a goat and switching automatically gets you the car, and that's what will happen two thirds of the time.

So while on the surface it seems as though the procedure is just reducing the initial three choices to two, making things fifty-fifty, it's actually inverting your choice, turning goats into cars and cars into goats if you do the switch.

Edited to add: Mind you, I'm not sure how much the problem's difficulty has to do with probability. It's just tricky. It's not initially obvious that having picked a goat automatically means that switching gets you a car.

[ 02 November 2005: Message edited by: Rufus Polson ]

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Originally posted by Rufus Polson:
So while on the surface it seems as though the procedure is just reducing the initial three choices to two, making things fifty-fifty, it's actually inverting your choice, turning goats into cars and cars into goats if you do the switch.

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Originally posted by Makwa:
Nope. The chances are 50/50 - whatever your choice (as it is not revealed), a goat is removed from the choices. Therefore there are actually only two choices. Keeping or switching doesn't matter.

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But the there were 3 choices in the beginning. Each door started out with a 1/3 probability of hiding the car prize behind it.

Most medical schools teach epidemiology which includes statistics. It makes up 1/7 th of the licencing exam.

This site which can be reviewed by anyone looks into "evidence based" treatments

We have:

Goat 1
Goat 2
Car

There's only three possible things I can do. Let's go through each.
If I pick Goat 1, Hall shows me Goat 2. If I switch, I have the Car.
If I pick Goat 2, Hall shows me Goat 1. If I switch, I have the Car.
If I pick Car, Hall shows me (one of the Goats, doesn't matter which). If I switch, I have a Goat.

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Originally posted by Fidel:
But the there were 3 choices in the beginning. Each door started out with a 1/3 probability of hiding the car prize behind it.

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[ 02 November 2005: Message edited by: Makwa ]

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Originally posted by rasmus raven:
Stephen, the subjectivist and frequentist interpretations of probability are philosophical differences that make a practical difference in only a few cases, as I understand it.

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In point estimation, and if there's lots of data, then yes (although the techniques are in fact different - I integrate; my frequentist colleagues maximise). But not in hypothesis testing, which is what a trial amounts to. The difference there is fundamental. Frequentists ask: 'what is the probability of observing the data we did, conditional on the hypothesis that the accused is innocent?' - that is, p(x|H). Bayesians ask 'what is the probability that the accused is innocent, conditional on the data we observed?' - that is, p(H|x). To my mind, that second question is much more interesting - we know what the data are, but we don't know the accused's guilt or innocence. A fequentist can't even ask that question, since that s/he can't assign a probability to guilt or innocence (at best, s/he can ask what would happen in repeated samples - which isn't really interesting, unless you're writing a Star Trek episode). But a subjectivist can - I don't know with perfect certainty, so I use a probability to represent that uncertainty. And then after I observe the data, I use Bayes' rule to update those beliefs.

But more generally, I think you're quite right about the use and misuse of even the most basic concepts. All anyone needs to understand is the distinction and relationship between marginal, joint and conditional probabilities (and perhaps Bayes' rule). That's not really that difficult; a half-course in probability theory is more than sufficient.

The really hard part is asking the right question. Once you know what the probability you're looking for is, it's usually pretty easy to figure it out. (Or to hire someone to calculate it).

[Edited to add:]

This was all discussed in terms of a trial, but it also works if H is the hypothesis that a certain drug is effective, or that a patient has a certain disease.

[ 02 November 2005: Message edited by: Stephen Gordon ]

In other words, the house always wins. No?

*Warning: drift coming, driven by passionate conviction*

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Originally posted by BleedingHeart:
Almost all medical research is subjected to statistical analysis. You can't get a paper published without it. There are of course many old treatments which are so well established that nobody is going to do a study on them, we know appendectomy is the treatment for appendicitis for example. There are a number of unproven treatments which have survived because they seem to work.

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Gee, BleedingHeart: You wanna hear my current problems with epidemiology in this country? I am spending my days right now slowly chip chip chipping away to get an institution to bring in a coroner to verify a disputed diagnosis -- Alzheimer's or multi-infarct? (Or maybe CJD, or who knows, eh?)

And, like, everyone thinks I am bananas. I am the mad wife. *pat her on the head* But how the hell does medical science expect ever to describe truly the incidence of diseases like Alzheimer's without actually doing the examinations? Like, how? And, believe me: that is not being done right now in Ontario, not without some mad wives banging fists on desks.

All very well for you to say that all medical research is subjected to statistical analysis. But how much of the raw material is being studied in the first place?

Medicine in Canada is a scandal. Simply a scandal.

One good thing: the new president of U of T is an epidemiologist, an excellent one, and a good writer to boot.

Sorry. End drift.

*/drift*

[ 02 November 2005: Message edited by: skdadl ]

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Originally posted by Mr. Magoo:
Anyway, for those still hurtin' in the head over it, Wikipedia has a decent explanation.

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Based on the blood in the driver's seat in a bus, I got back a report saying that the chance the driver was NOT my client was about 1 in six billion.

The blood in the back seat had allele X; so did my client. The calculation was based on the frequency of that allele in the population of Ontario (I simplify somewhat.)

I called the expert and asked him if the frequency calculation would change if all of the potential perpretrators belonged to a native population which had lived together, and interbred, on the same square mile of land from the year 1796 to date. He said he'd get back to me.

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Originally posted by Makwa:
This is the trick. A 'wrong' choice is always removed between choices. Thus the 'third' choice is illusionary. While it is true that you have 2/3 chance of picking a goat the first time, changing the odds before the second place reduces the outcome to 50/50 no matter what. Keeping the first choice or changing is irrelevant - the odds are still 50/50.

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The unmitigated gall! Only sheer insolence would allow someone who failed mathematics to make the claim that the win probability is raised to 2/3 by switching doors. Allow me to expose the columnist's error: suppose there are one hundred doors, and the contestant chooses for door number one. He then has a 1% probability of having chosen the correct door, and there is a 99% probability that the automobile is concealed behind one of the other ninety-nine doors. The host then proceeds to open all of the doors from 2 to 99. The automobile does not appear behind any of them, and it then becomes apparent that it must be behind either door number one or door number one hundred. According to the columnist's reasoning, door number 100 now acquires a 99% probability of concealing the automobile. This, of course, is pure balderdash. What we actually have here is a new situation consisting of only two possibilities, each one being equally probable.

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This has to be one of the most poorly chosen arguments ever! At any rate, the letter writer is dead wrong. Rufus' explanation is very clear, and just rephrase Rufus' explanation to see what the choice of NOT switching leaves you with:

Goat 1
Goat 2
Car

There's only three possible things I can do. Let's go through each.
If I pick Goat 1, Hall shows me Goat 2. If I don't switch, I have a goat
If I pick Goat 2, Hall shows me Goat 1. If I don't switch, I have a goat.
If I pick Car, Hall shows me (one of the Goats, doesn't matter which). If I don't switch, I have the Car.

By not switching, you have a 1 in 3 chance of winning the car. By switching, you have a 2 in 3 chance of winning the car. It's quite clear.

A professional mathematician wrote to vos Savant:

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As a professional mathematician, it concerns me to see a growing lack of mathematical proficiency among the general public. The probability in question must be 1/2; I caution you in future to steer clear of issues of which you have no understanding.

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A week later the professor (Robert Sachs of George Mason University) wrote to say he was eating humble pie. Mm, mm. Isn't that delicious? Go Marilyn!

Stephen, perhaps I was rash in understating the differences between Bayesians and frequentists. Frequentists lose sleep wondering where those prior probabilities come from anyway! But in terms of the cash value of the difference in research, what differences do you see?

Bleeding heart, yes, clinical trials involve statistics, but the question is how to interpret them, and this is where I think most doctors are weak. A professor of mine cautioned that even among researchers, the prevalence of statisical packages that automate most of the tasks makes many people lazy and quite superficial in their understanding of statistical inference. And on the subject of "evidence-based medicine", I find it depressing to contemplate that this is considered a relatively new trend.

On the subject of the law, one of the most famous cases of misuse of probability was People v. Collins. This case raises issues quite similar to those raised, in more technical form, by DNA evidence.

[edited to add:]

In 1997, the brothers who co-host NPR's Car Talk, one of the greatest radio shows in the world, discussed the Monty Hall problem. Afterwards, they put up a simulation game on their website. Chance News, at the Dartmouth Chance website, reported the results of the plays to date back in 1998:

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The statistics so far for those who play "Let's Make a Deal" on the cartalk website are:

Number of .... Winning Pct

Initially correct: 7941 34.039%

Stickers: 10625 33.459%

Switchers: 12704 66.821%

Total Trials: 23329

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Chance News

[ 03 November 2005: Message edited by: rasmus raven ]

From the same website, here is a sample of some irate professors' letters to MvS.

Also from the same website, Cecil of the Straight Dope takes the problem on.

Finally, if you're interested, you can download this this FREE (GNU Free Document License) introduction to probability. The Monty Hall problem is discussed on p. 144 of the PDF (numbered 136 on the paper pages). Warning, it's a large PDF file (a few hundred pages), so if you have dialup, it will take a while.

[ 03 November 2005: Message edited by: rasmus raven ]

I don't have a good gut feel for how statistical reasoning works, and god knows I've used enough mathematical tools in my time to do statistical analysis where needed - and it's all the more bizarre that I don't have a good feel for statistics considering that radioactive decay behaves this way. It's just very hard to imagine something which behaves completely randomly and yet produces predictable results in the aggregate.

Game show stuff.

RPolson's excellent analogy has simplified the problem for me of the game show, but the 99 out of 100 thing makes me wonder if there isn't a kind of limit (the calculus definition) towards which switching a choice becomes effectively the same as not switching.

Right. Onto DNA stuff. The probabilities always sound impressive, until one realizes what jeff house came up with - that the statistics have a built-in set of unstated assumptions.

I've read some papers (I'll try to find references if I can) that talk about the importance of being careful when you use statistics to describe the error in a measurement. Statistics can be abused in the name of science as well as medicine.

[ 03 November 2005: Message edited by: DrConway ]

[ 03 November 2005: Message edited by: DrConway ]

quote:

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Originally posted by DrConway:

RPolson's excellent analogy has simplified the problem for me of the game show, but the 99 out of 100 thing makes me wonder if there isn't a kind of limit (the calculus definition) towards which switching a choice becomes effectively the same as not switching.

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No, the more choices there are the better the odds get on switching.

Try this take: the only time you win by standing pat is if you have already picked the car. Given the rules if you haven't picked the car the first time and you switch you have 100% chance of winning. If there are n doors the chance of not picking the car first time is 1 - (1/n), so yes, with 100 doors switching gives a 99% chance of winning the car.

BTW the first time I saw this I screwed it up too.

The above analysis assumes that only one door is closed when the decision to switch or not is made. If only one door is opened the odds do converge to a limit.

[ 03 November 2005: Message edited by: jrootham ]

So what's the incidence of medical errors in our hospitals, btw ?. How many doctor Jekyll's do we have doing harm out there?. We need free markets in doctors and clean water in Canada. Because the federal Liberals and provincial conservatives are proving that they cannot be trusted to handle anything for the public good.

[ 03 November 2005: Message edited by: Fidel ]

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One form of the fallacy results from neglecting the a priori odds of a defendant being guilty--i.e., the chance of an individual being guilty absenting specific evidence is the gross incident rate of perpetrators in the general population. When a prosecutor has collected some evidence (for instance a DNA match) and has an expert testify that the probability of finding this evidence if the accused were innocent is tiny, the fallacy occurs if it is concluded that the probability of the accused being innocent must be comparably tiny. The probability of innocence would only necessarily be comparably tiny if the a priori odds of guilt were 1:1--that is, if the probability of innocence is computed with an a priori presumption of guilt, which violates the conventional presumption of innocence.
# Another form of the fallacy results when evidence is compared against a large database. The mere size of the database elevates the likelihood of finding a match by pure chance alone. i.e., DNA evidence is soundest when a match is found after a single directed comparison because the existence of matches against a large database where the test sample is of poor quality (common for recovered evidence) is very likely by mere chance.

[...]

Now consider this case: you win the lottery jackpot. You are then charged with having cheated, for instance with having bribed lottery officials. At the trial, the prosecutor points out that winning the lottery without cheating is extremely unlikely, and that therefore your being innocent must be comparably unlikely. This reasoning is clearly faulty: the prosecutor failed to mention that cheating lottery winners are much more rare than honest winners.

[...]

Consider for instance the case of Sally Clark, who was accused in 1998 of having killed her first child at 11 weeks of age, then conceived another child and allegedly killed it at 8 weeks of age. The defense claimed that these were two cases of sudden infant death syndrome; neither prosecution nor defense offered any other explanations for the deaths. The prosecution had expert witness Sir Roy Meadow testify that the probability of two children in the same family dying from sudden infant death syndrome is about 1 in 73 million. But based on this alone, it is likely that there would be at least one person in the country to whom this has occurred. To provide proper context for this number, the probability of a mother killing one child, conceiving another and killing that one too, should have been estimated and compared to the 1 in 73 million figure, but it was not. Ms. Clark was convicted in 1999, resulting in a press release by the Royal Statistical Society which pointed out the mistake. (See link at end of article.) A higher court later quashed Sally Clark's conviction, on other grounds, on 29 January 2003.

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That page also links to a discussion of People v. Collins.

In Calculated Risks: How to Know When Numbers Deceive You, Gerd Gigerenzer discusses many cases where doctors and lawyers have abused probabilities. He suggests that trying to think in terms of "the probability that so and so is innocent" usually leads to error for most people. Instead, he prefers expressing probabilities in frequencies. Take the following situation (I think this is an example from Gigerenzer, paraphrased in Henk Tijms' book cited above).

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A doctor discovers a lump in a woman's breast during a routine physical. It could be cancer. Without performing any further tests, the probability that the woman has breast cancer is 0.01. A mammogram is a test that, on average, is correctly able to establish whether a tumour is benign or malignant 90% of the time. A positive test result suggests that a tumour is cancerous. What is the probability that the woman has breast cancer if the test result from the mammogram is positive? Results from a psychological study indicate that many doctors think the probability of cancer given a positive is slightly lower than the probability of a positive given cancer, and estimate the former as being about 80%. The actual value for the probability of cancer given the positive test result, however, is only 8.3%.

Gigerenzer argues that it would be clearer to most people to present the information in the following way. Out of 1000 women examined, 10 had breast cancer. Of these 10, nine had a positive mammogram, whereas, out of the 990 healthy women, 99 had a positive mammogram. When the information is presented this way, most doctors can correctly estimate the probabilities.

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[ 03 November 2005: Message edited by: rasmus raven ]

Why do epidemiologists -- or journalists quoting them, anyway -- say that influenza pandemics "tend to come around about every 30 to 35 years"? That is, on how many years of data do they base this generalization?

Unless I've missed something, there was a pandemic in 1918-19, another in 1957, and another in 1968. So far, so good: three in the 20th century, or one every 33 years and change, on the average.

But is that it for data, or are factoring in some knowledge of influenza pandemics in previous centuries?

I'm not well up on epidemiology, or on the theory or interpretation of probability. But based on what I know of earthquakes, floods and such, you'd need data on way more than three occurences of an event with magnitude X to be able to say its average return period is Y years.

1. statistics random: involving or showing random behavior

2. statistics involving probability: involving or subject to probabilistic behavior

3. involving guesswork: involving guess work or conjecture ( formal )

edited to contine:

i should remember more statistical theory but...i can see def's 1 & 2 being related but isn't 3 - guess work/conjecture sort of logically opposed?

[ 09 November 2005: Message edited by: chester the prairie shark ]

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Gee, BleedingHeart: You wanna hear my current problems with epidemiology in this country? I am spending my days right now slowly chip chip chipping away to get an institution to bring in a coroner to verify a disputed diagnosis -- Alzheimer's or multi-infarct? (Or maybe CJD, or who knows, eh?)

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sorry for your loss

Did you ask for an autopsy. Generally if the family asks for an autopsy one is granted. To find the true incidence of various types of dementia would require autopsying a large randome sample of dead Canadians.

Estimated frequency of a meteorite striking an aircraft (short answer: once every 59,000 to 77,000 years)

How many times do you have to shuffle a deck of cards?

Short answer: to shuffle a SINGLE deck of cards to something that is random for cardplaying purposes, using an imperfect riffle shuffle, you need to shuffle the deck 7 times. A perfect riffle shuffle will return the deck to its original order in 8 shuffles. More than one deck requires more shuffling. And for mathematically random distributions, 15 or more riffles shuffles are required.

The United States National Research Council on the forensic use of DNA Good for the lawyers out there.

The 1970 US draft lottery was not, it turned out, truly random.

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In 1970, Congress instituted a random selection process for the military draft. All 366 possible birth dates were placed in plastic capsules in a rotating drum and were selected one by one. The first date drawn from the drum received draft number one and eligible men born on that date were drafted first. In a truly random lottery there should be no relationship between the date and the draft number. However, this dataset suggests that men born later in the year were more likely to be drafted.

While the trend is not at all clear when viewed as a scatterplot of draft number vs. birth date, a series of side-by-side boxplots by month illustrate it clearly. The correlation between draft number and birth date is -0.226, which is significantly different from zero. A further investigation of the lottery revealed that the birthdates were placed in the drum by month and were not thoroughly mixed.

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For those who are interested in learning more about probability, here are my recommendations. None of these books require advanced mathematics, which is necessary for general results in probability theory.

There are no probability books for the truly math phobic, but the following three come close:

An Introduction to Probability and Inductive Logic, by Ian Hacking

Statistics, by Freedman, Pisani, and Purves

Both of these books are "deep, without being technical", as Persi Diaconis writes about the latter book. Statistics is a true classic. The glowing blurbs on the back page are from a who's who of statisticians. As Russell Lyons says in his review, "It reads like a detective novel. This is the best textbook I have come across on any subject." I cannot recommend it too highly. The sections on design of experiments, survey sampling (including polling) are outstanding. If you use this book, don't ignore the notes, either, they are full of interesting extras.

The third book is Richard Isaac's excellent book, The Pleasures of Probability. Like the other two, it is very light on math. In fact, there is probably less math in this book than in the others, but slightly more formalism. The textual exposition is fantastic. He starts with the Monty Hall problem, and uses it and many other fascinating examples to expound the central notions of probability and statistics.

A simple, 100 page introduction to probability is this one by two prominent Soviet mathematicians. Although it requires only pre-calculus high school math, it is rigorous and deep in its approach. Unfortunately, it was written under Stalin and the examples are of manufacturing processes and ballistics targeting. Also, the Soviet bond offering, which apparently was a kind of lottery. Nonetheless, this book is compact, and very good, if you are of a mathematical cast of mind. No exercises, though.

Henk Tijms' Understanding Probability is truly a delight. He uses more math, but most of it is high school math and many of these sections can be skipped without breaking the flow. He has a lot of exercises. He uses tons of real-world examples and famous probability problems to build intuitions about probability, before slamming you with a lot of math in the second half of the book.

Volume 1 of William Feller's Introduction to Probability Theory is likely THE classic in discrete probability, and most probabilists' favourite book. Although most of the book requires little advanced math, sometimes university level calculus is required and he doesn't grade or star the problems, so you're on your own with them. If, like me, you've forgotten most of your calculus, you won't be able to do some of the problems. However, the book is the best for developing your intuitions in probabilistic thinking. It is a math textbook, but as Gian Carlo Rota said, "Feller's treatise is one of the great masterpieces of mathematics of all time."

While not a classic, Kai Lai Chung's Elementary Introduction to the Theory of Probability, with Stochastic Processes and an Introduction to Mathematical Finance is a pedagogically sound undergraduate work that is carefully graded and compendious. It is less entertaining than any of the other works, but it is more workmanlike than most of them.

Finally, something I've taken on long train or plane rides is Frederick Mosteller's Fifty Challenging Problems in Probability with Solutions. When it was published, at least one of the problems did not, in fact, have a solution. For the rest, though, Mosteller's solutions are extremely instructive. A neat little book.

[ 12 November 2005: Message edited by: rasmus raven ]

http://tinyurl.com/cdtdm
http://tinyurl.com/9z2yr
http://tinyurl.com/7myut

I particularly enjoyed these - none of them require any advanced mathematics - they fall in the category I'd describe as "books about mathematics" as opposed to "mathematics books". The topics covered are wide and varied - the misuse of statistics is high on the list of things discussed. "A Mathematician Reads the Newspaper" includes a couple of discussions that are relevant to questions earlier on this thread. For example, what was the correct question to ask if a blood sample found at the scene of the crime matches that of the suspect. [It's not, "What's the probablity that someone else has the same blood type?"]

One other book that I think it definitely worth the read is Against the Gods by Peter Bernstein. It's been a couple of years since I read it but it does not require an extensive mathematics background. Having said that, it's written at a level that even those with solid mathematics and statistics backgrounds can enjoy. (It was actually recommended to me by an actuary.)

Edited to fix problem (?) I'm not showing any sidescroll but I'm guessing it was my link that caused the problem - I've replaced it with three to the individual books.

[ 12 November 2005: Message edited by: abnormal ]

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An article (http://www.salon.com/politics/war_room/2005/10/31/jury/index.html) in Salon.com for October 31 discusses the case Riley v. Taylor (http://caselaw.lp.findlaw.com/scripts/getcase.pl?court=3rd&navby=case&no=989009v3&exact=1). The plaintiff Riley, convicted at trial of first-degree murder, was African-American; at the trial, the prosecution used its peremptory challenges to eliminate all three of the African-Americans on the jury panel. In the same county that year, there were three other first-degree murder trials, and in every one of those cases all of the African-American jurors were struck.

Riley appealed his conviction. A majority of the judges on the appeals court thought that there was evidence that jurors were struck for racial reasons. According to them, a simple calculation indicates that there should have been five African-American jurors amongst the forty-eight that were empanelled in the four cases. However, there were none. To these judges, this was clear evidence of racial motivation in the striking of such jurors.

Judge Alito dissented. He called the majority's analysis simplistic, and stated that although only 10% of the U.S. population is left-handed, five of the last six people elected president of the United States were left-handed. He asked rhetorically whether this indicated bias against right-handers amongst the U.S. electorate.

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The following piece is also interesting:

Do airbags save lives? (Short answer: no. Read the articles for the longer answers.)

Who wants airbags?

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The main difference between our study and the previous studies is the choice of the dataset. We use the NASS CDS database, which is a stratified random sample of crashes nationwide. Previous studies showing beneficial effects of airbags have all used the FARS database, which contains data for all crashes in which a fatality occurred. We can limit our analyses to a subset of the NASS CDS database, choosing only crashes where there was at least one fatality; this should be a random sample of the FARS database. When we perform the analyses on this subsample, we can reproduce the results of previous studies: in accidents in which a fatality occurs, airbags are beneficial. However, for the entire random sample of crashes, they increase rather than decrease the probability of death.

Here is an analogy to help understand this: If you look at people who have cancer, radiation treatment will improve their probability of survival. However, radiation treatment is dangerous and can actually cause cancer. Making everyone in the country have airbags and measuring effectiveness only in the fatality group, is like making everyone have radiation treatment and looking only at the cancer group to check efficacy. Within the cancer group, radiation will be found to be effective, but there will be more deaths on the whole.

This is what seems to be happening with airbags. In a severe accident, airbags can save lives. However, they are inherently dangerous and pose a risk to the occupant. Our analyses show that in lower-speed crashes, the occupant is significantly more likely to die with an airbag than without. This effect is not seen in the analysis using FARS, because this database does not contain information about low-speed crashes without deaths.

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