Though I try my damndest to make all of the material interesting when I teach – that's where the 100,000,000 anecdotes and random facts come in handy – it's inarguable that some topics of academic interest (and curricular necessity) are a bit dry. Teaching statistics is a challenge in this regard. In my view, probability is the easiest thing to talk about. Probability and chance and randomness are fascinating. The problem is that very few people understand how to evaluate risk, odds, and probability correctly. It's very easy to mislead people with definitive-sounding statistics if one is so inclined.

In conversation this is an annoyance. In courtrooms it is a life-or-death issue.

Juries don't understand statistics. Judges don't either. Hell, the lawyers and witnesses citing statistics in court probably don't.

Say you are on trial in Chicago and evidence is presented by the prosecutor indicating that you match DNA found at the scene. Your specific DNA profile is declared "1 in 2,000,000." The prosecutor uses this statistic – and the jury most likely hears it this way without prompting anyway – to imply that your guilt is 1,999,999/2,000,000 certain, or 99.99995% certain. You're going away for a long time.

The problem is that having a 1-in-2M DNA match does not in fact means nothing more than that in any randomly selected sample of 2 million people, 1 will have your profile. In Illinois' population of 13 million, this means six people have that profile. Adding in the bordering states' population within a short drive of the Chicago area, that's four or five more people. Add in the billion people around the world with access to fly into and out of O'Hare Airport on any given day and you have a veritable horde of DNA Twins out there. But limiting it to the Chicago area only, there are, statistically speaking, 10 individuals who match your DNA profile.

That means that it is not 1,999,999/2,000,000 percent likely that you are guilty. It would be more accurate to say that it's about a one in ten chance. And that doesn't even include the rates of false positives and human errors on DNA tests, which are both small but relevant in a large sample.

Here's another (real) example. During the OJ Simpson trial the prosecution unwisely downplayed its physical evidence and instead spent two weeks detailing the football star's lengthy history of violent abuse of his wife.
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Defense attorneys (Yes, they can use the trick too) then told the jury that about 1 in 3000 people who abuse a spouse or partner go on to murder that person. Therefore, they said, the tales of abuse were regrettable and true but totally irrelevant since the odds were so small (0.03%). Unfortunately, the only thing irrelevant is that statistic. The relevant question is not "What percentage of men who abuse a woman will go on to murder her?" but "What percentage of women who are murdered are murdered by the person who abused them?
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" According to FBI Uniform Crime Reporting data, somewhere between 70-80% of women who are murdered and had been abused are murdered by the person who abused them – not exactly a lead pipe cinch, but damning enough to establish that abuse can turn into murder to the jury.

The misleading statistic of 0.03% reflects nothing other than that murder isn't very common under any circumstances. A vanishingly small percentage of any defined category of people will ever commit murder. But at the trial that was not relevant, because the murder had happened. That the abuse happened was also established. So the question of how likely it was to happen is not relevant. It did.

Think of it this way. Say a plane crashes and Boeing is on trial arguing that the cause was not mechanical failure, citing that, "Only 1 in every 100,000 planes will ever have this particular mechanical failure leading to a crash." Probably true, but who cares? The question you want answered is, of planes that do crash, what percentage had that mechanical failure? It might not be a large number, as there are many possible causes of a crash, but it sure as hell will be larger than 1 in 100,000.

Once you're aware of this and begin to notice it, you'll be amazed at how often you encounter this fallacy in everything from advertising to news to your performance evaluations at work to casual conversation. You're welcome.


  • There are many reasons to be frustrated about the Simpson trial. I pretty much assume he did it. That said, I was pretty young when it happened and finally really read about it a couple years ago.

    If you don't think the lead investigator investigating a prominent black athlete accused of murdering his white wife having a history of calling black people niggers, and, when questioned about it *committing perjury* isn't enough doubt to drive a fucking truck through, you're a racist piece of shit. That right there is game set and match for beyond reasonable doubt.

    On tapes the jury mostly wasn't allowed to hear, Fuhrman claimed sometimes cops have to lie. He also took the 5th in front of the judge when asked if he'd ever falsified police reports or planted evidence in the Simpson case.

    I have no sympathy for OJ, but goddamn.

    And LAPD let him continue to work there. I dunno why black people think the pigs are out to get them though.

  • I must admit I do not understand the first example with the genetic profile. Sure, if you go randomly through the population of Chicago until you find the first person who fits the DNA profile the chance of having found the culprit is probably 10%. But that is not how these things are done, right?

    If the question is if this person who had a relationship with the victim and a motive fits the DNA profile, while the other nine people who would hypothetically also fit the profile have no such relationship or motive, then I do not see how fallacious reasoning is involved.

  • @Alex SL: In fact, that is sort of how these things are done. If you are convicted of a crime (or maybe just arrested, depending on jurisdiction), the police keep your DNA on file.

    When the police recover DNA from a crime scene, they run it against their database. Suppose they have 5% of the local population on file. On Ed's figures, that's 1 million people in the greater Chicago area. There's about a 50% chance of that group containing someone with a DNA match, just at random. (In a larger database, say that of the FBI, the chance of a false positive is correspondingly greater.)

    So the police find you and bring you in for questioning, based on DNA evidence alone. That's a prudent thing to do, but the statistical chance you are innocent is about 1/2, not 1/2,000,000. If they can't find any other evidence, you should walk free. But there have been miscarriages of justice where this didn't happen, because the judge, jury, and defence attorneys didn't understand statistics.

  • Another example is the tragic case of Sally Clark in the UK. Two of her children died suddenly, both when they were a few weeks old. It could have been Sudden Infant Death Syndrome (SIDS). But Clark was charged with murder, and the prosecution argued that since the chance of a SIDS death in the general population is 1 in 8400, the chance of two in the same family is 1/8400 times 1/8400, or 1 in 70 million.

    The thing is, simple multiplication only works for the probabilities of independent events. If you flip a fair coin twice, the chances of two heads is 1/2 * 1/2 = 1/4. If you flip the coin, pick it up, and drop it so it falls on the same face as before, then your chance of 2 heads is 1/2, not 1/4.

    Deaths from SIDS are not independent. The babies had genetic and environmental factors in common. Also, if you're making a statistical argument, you need to compare it to the chance of double murders in the same family, which are extremely rare. Statistically speaking, there was about an 80% chance the babies died from natural causes.

    However nobody involved in the murder trial understood this. Clark served 3 years of a life sentence, was freed on appeal, but developed severe psychological problems died four years later from alcohol poisoning.

    Bad statistics can kill. Consider yourself warned.

  • Talisker,

    If that is how things are done, then that is remarkably stupid. I would just have assumed that genetic profiles are used in addition to other at least circumstantial evidence.

  • @Alex SL: As I say, a 50% chance of guilt is perfectly reasonable grounds to bring someone in for questioning. It shouldn't be enough for a conviction. Shouldn't be, but sometimes is, because the legal system is stupid like that.

    It's worth noting that people on a DNA database are disproportionately likely to be poor and/or involved in illegal activities. They are less likely to be able to mount a decent legal defence. The drug dealer who matches DNA found at a murder scene may not be a very nice guy, but his other crimes are irrelevant to whether he committed that particular murder. However police and prosecutors often don't see it that way.

  • What was that Mark Twain quote? "Lies, damned lies and statistics". Perhaps something like "Introduction to statistics"" would be more useful to young people than other math classes?

  • OtherAndrew says:

    Unfortunately, I find that familiarity with statistics is not a one-to me educational event but rather a lifetime habit of mind. I hope I'm wrong but the only people I've ever known or seen getting it right consistently are people who regularly refresh themselves on the concepts.

    A permanent post of "statistics advisor" in the vein of another sort of editor might be the only way to save things.

  • We quant types get two pieces lately: this and the big numbers item. The odds of that happening are …… Never mind.
    How to lie with statistics by Darrell Huff is, of course, the primo work in the field. A bit tongue in cheek but actually worth the read: a journalist, not a pro.
    I was married to a full-blown statistician for over 20 years. This kind of stuff would make her crazy. Takes time and a big, fluid brain to understand this stuff. I've studied it a lot and can spot errors but it's a slog to make it useful and not get mixed up.
    The lazy alternative appears in Ed's examples: cynical and evil might also apply.
    Here's the rule: the odds that the next statistic you hear comes from someone who understands it……..pretty near zero. Add in bias and there you go. Grain of salt time.

  • It's true that probabilistic inferences have many pitfalls for the unwary. However, the Prosecutor's Fallacy has a very close deductive analog, Affirming the Consequent.

    Formally, the Prosecutor's Fallacy is confusing the probability of observing the evidence given innocence, P(E|I), with the probability of being innocent given the evidence, P(I|E). The analogous deductive error is confusing p => q with q => p.

    Related probability fallacies are the Base Rate Fallacy ("Is Linda a farmer or a librarian?") and Berkson's Paradox (in general, independent events are conditionally dependent.)

  • @Tim H — I agree, but then we would have a population that was well educated and could think critically. The Powers That Be would never stand for that, because advertising and political bullshit wouldn't work any more.

  • The distinction I always make when explaining it to people who are not math-intuitive is the difference between the statement "I might win the lottery tonight" and the statement "Somebody might win the lottery tonight." The first is probably not true, the second is probably true.

  • Celebrity trials are the worst. OJ, where I still think the police tried to frame a guilty man and didn't get away with it. Michael Jackson, where the prosecutors spent so much time establishing that he was a total freak they never got around to proving he was a total freak with the alleged victim. And so many other.

    And so many more.

    But the use of statistics happens all the time, everywhere, all around us. How are the measurements for DUI created? Statistics. Why is the speed limit 10mph lower on a curve? Statistics. Weather forecasts? Deciding that Sanders has no chance in Michigan? Should I trust Rotten Tomatoes or Fandango? All of that can be evaluated and used or misused. (Never trust Fandango, by the way.)

    I try to have a rule in life where if someone tells me a statistic and can't explain how it's true, I will doubt that statistic. Sometimes I'll investigate, sometimes I won't, but I try not to believe something solely because there's math involved. It works 80% of the time.

  • Well, Fandango just bought RT, so outside of mom-and-pop type review sites, there isn't a place to go any longer that is more interested in conveying subjective impressions of movies than to sell tickets.

  • Mac the Knife says:

    64 percent of all the world's statistics are made up right there on the spot.
    82.4 percent of people believe 'em whether they're accurate statistics or not.
    I don't know what you believe but I do know there's no doubt
    I need another double shot of something 90 proof
    I got too much to think about

  • This was very educational, Ed. I read "The Drunkard's Walk" re: probability but must remember to wear my skeptic's hat when listening to the media maul statistics. Good mental exercise, even if it makes my lumbego flare up to be reminded of the imcompetence around us.

    Must remember to pay tuition here for today's class.

  • This is why people who potentially have a better grasp of logic and statistics are often struck from certain types of trial juries (especially of the criminal variety)

    I was a potential juror in a murder trial and the prosecution was asking about the up-front details about me and before the R in the word 'engineer' stopped ringing in the courtroom, the word 'Excused' was just about shouted at me.


  • So, big whup, I'm an engineer. I can still be fooled and wrong about numbers stuff..

    I have been making the point (several times in the last 6 months) in response to snark here that FOX News' ratings on their best program pulls between 2 and 3 million and their whole primetime averages 1.8 million per hour.

    The US adult population is about 243 million. Straight up, about 1% of the total population.

    OK, let's narrow it a little. 'Conservatives' depending on how you ask the question constitute about 1/3 to 40% of the adult population.

    That'd be about 80 million to pump it up a little. Again, we are talking small stuff at about 3%

    TO requote my Granny "BB, that's the fly $hit in the pepper…"

    Are there Sociological, Psychological, and/or Communication theories that explain how FNC becomes this Mega media monstrosity that is, for example, able to defame HRC's trustworthiness (per former MI Governor Granholm)?

    One poster here suggested that it is just the 'Magnification of Your Opponent' that is typical in politics.

    In the spirit of Ed's article here, is there another angle from which we should look at it ?


  • TheOtherHank says:

    @BB – I was once in a vehicular manslaughter jury pool and ended up being juror #12 because the defense lawyer had used his last you-go-away chit on the person before me. I could see him die on the inside when I said I was an engineer. My job in the deliberation room was to point out that the defense's expert's testimony established that the driver was going _at_least_ 90 mph when things went wahoonie shaped, not no more than 90 which was the defense wanted us to think.

  • Saying you're a scientist will also almost always get you excused from a jury.
    Though the quickest I ever was excused was when I was asked if I had any experiences with the justice system and I replied we have a LEGAL system, not a justice system – and those are two very different concepts. The prosecuting attys, defense attys, and the judge couldn't get me out of there fast enough.

  • HoosierPoli says:

    bb – Some basic assumptions that send your analysis way off. Their best program pulls in 2 million views; your analysis only works if the people who tune in to the best-rated program are the ONLY people who EVER watch Fox News.

  • @ Hoosierpoli and bb

    Plus you are assuming a standard distribution among different ages, and Fox targets the AARP crowd, who vote at an extremely high rate. Add in in all the hotel lobbies and repair shop waiting rooms ( and Sunday Church sermons echoing what the preacher heard on Fox TV that week) and the Fox propaganda machine ends up with long tentacles.

    After that, add in the life sucking force that is rural radio, where you have three choices for your listening pleasure: Rush Limbaugh and company, the groovin' sounds of seventies light rock or listening to the corn and soybeans grow as you rocket past Dwight and Pontiac and into McLean County. Hell, Noam Chomsky would vote for Trump after kind of brainwashing.

  • I want to add another layer to BB in GA's comment that: "FOX News' ratings on their best program pulls between 2 and 3 million and their whole primetime averages 1.8 million per hour. The US adult population is about 243 million. Straight up, about 1% of the total population."

    However, polling consistently finds that 70% of republicans say they only trust Fox for their news. That would be 70% of the 80 million republicans number BB posited, or 56 million republicans saying they only trust Fox.

    I have tried without success to reconcile the Fox ratings numbers with these temporally consistent polling numbers. I suspect part of the difference is that many people watch Fox away from their home (at work, at the bar, etc), or watch it institutional settings like nursing homes where Fox is on constantly.

  • I have tried without success to reconcile the Fox ratings numbers with these temporally consistent polling numbers.

    First of all there are going to be more than 1.8 million people who watch Fox News. Because that's an average primetime viewership and not everyone watches every night in prime time (Fox news viewers do actually watch things other than Fox News). The most recent GOP debate on Fox News, for example, had about 17 million viewers – a full order of magnitude larger than the primetime average.

    Second of all Fox News has a website. So poll respondents may be reporting that as well.

    Third of all the questions may be poorly designed to promote tribal responses from Republicans who don't watch TV news at all or visit websites, but who know what the "right" answer is and so they give it. If the question is "who do you trust" rather than "who do you watch" then that's got nothing to do with whether they actually watch Fox News on a regular basis. I suspect that a lot of that 56 million would be people who sporadically watch Fox news when they've got nothing better to do, rather than the news junkies who watch it nightly or the folks who are in the habit of watching the news when nothing else is on (the way my father does, or the way my father-in-law will turn on ESPN if he can't find anything else to watch).

  • Anonymous Prof says:

    FWIW, I have a family member (who will not be named,) who is always spewing FOX News talking points.

    When I say, look, these are all FOX talking points, he says, "How DARE you accuse me of being one of those morons who watches FOX News!"

    Then I provide the link to show that his entire argument, point-by-point, is taken from one episode of some FOX news show.

    I suspect that he really doesn't watch FOX. But I guarantee you that all his friends are the kinds of morons who watch FOX. He also gets his news from sites like FreeRepublic, which he insists is "centrist, straight down the middle." (Naturally when I insist that FreeRepublic is actually conservative, and prove it by linking to an article that says the Nazis were a gay death cult devoted to persecuting Christians, he tells me I'm a bigot.)

    So FOX has an impact far beyond its actual audience. Gosh, if only there were a word to describe this situation- maybe we could call it the "echo chamber"?

  • Then there's the people who have learned just enough statistics to be dangerously wrong, and will insist that statistics leads to some wildly unintuitive conclusions. Even when there's plain evidence to the contrary right in front of them.

    For example, I've heard several people explain the Birthday Problem with some variation of, "In any group of 70 people, it's 99.9% likely that someone will have the same birthday as you." I'll go, "OK, leaving aside the math, how many people are you friends with on facebook?" The answer's almost always a significant multiple of 70. "Uh huh, and how many of them have the same birthday as you?" The answer's usually 0 or 1. They'll flail around until I point out that it's the probability that one random pair of those 70 people will have the same birthday, not that a specific one of them will have the same birthday as someone else.

    And then there's the fun times when people try to explain conditional probability using an example where conditional probability doesn't apply. I'm pretty sure the majority of explanations of the Monty Hall problem state the problem incorrectly.

  • But all probability theories eventually fall apart, because that's not the way real people think and make decisions. If they did, casinos would be empty. Put 10,000 people in a casino for a day. Some will go home with more money. Most will go home with less money. But the house will always have more money at the end of the day.

    The reality is that people aren't betting based on probability theories. They're betting that they may be the lucky one.

    I always get a kick out of politicians and policy makers who try to reduce crime by incrementally increasing the penalties. So, the penalty for breaking and entering will go from 7 years to 9 years. That makes sense to a bean counter. But the people breaking and entering aren't doing that sort of cost-benefit analysis. They're not thinking about getting caught.

    It's like the famous Zeno's Paradox that anyone who takes Philosophy 101 learns. Before you can cross a road, you have to cross half the road, but before you do that, you have to cross half that, but before you cross half of half, you have to cross half of that — ad nauseam. So you can never really cross the road.

    I was talking to a well-known philosopher one day and brought that up. He laughed and said it was "bullshit." "That's not how people cross roads. They just start walking and stop when they get to the other side."

    As far as juries, the probabilities may be well and good, but the defense's job is not to argue probability theories, which even many PhDs don't understand. The job of the defense attorney is to sow doubt and cause massive confusion on the part of the jurors.

    Back in the '90s, I was following three pretty large — multi- multi-million dollar — lawsuits. In each of the three cases, the defense just flooded the jury with so many doubts and confusing facts that deliberations went on for weeks. In each case, the verdict came down, mostly in favor of the defendants, on Friday afternoon of a long holiday weekend. The juries were so confused, they just wanted to get the freak out of there.

  • I didn't scan all the posts here so this might repeat, but one of the best intuitive understandings of Statistics comes from an aircraft engineer in World War II who challenged the up armoring of bombers based on where the most shrapnel holes were found. He posited that the returning planes actually sampled were the airframe could withstand penetration. By armoring the vital areas that showed least damage on returning bombers he saved countless airmen's lives.

  • Anonymous Prof – I admire your conviction and persistence. But someone who is persuaded that Free Republic is 'centrist' is only suitable as an organ donor.

    Full disclosure: I used to browse FR as a voyeuristic experience. Then I realized that the experience was degrading my own sensibilities, and abandoned it. It was a modern equivalent to going to Bedlam to laugh at the madmen.

  • One of my more relevant college instructors, years ago, noted that statistics were like a bikini bathing suit- what they reveal is interesting, what they conceal is vital!

  • Do you think this problem with statistics goes deeper down? I really believe many people just have no concept about numbers in general.

    I've been conducting an experiment on my own buying habits and the family budget, and part of that has included using cash and only cash to pay for small purchases. Unlike using a card where the cost and payment are vague, cash is immediate, and when your pocket is out of cash, you're out of money.

    What I'm finding is that people can't quite grasp the idea of using money to pay for anything, and they seem to have lost the connection of how numbers work. For example; my day took me past a small local chain that happens to carry a brand of tea I like. I realized I was running low on that particular tea, so I stopped in and grabbed a box. The cost was $1.99. I handed over two dollar bills. The cashier was highly annoyed for several reasons: 1) they'd already hit the Credit button (for a $2 purchase) and had to go back and start over, and 2) I swear this is a direct quote, "What am I supposed to do with *this*?" (meaning the two dollar bills). Gee, how about putting it in the cash drawer and giving me back a penny?!?!

    It's not just this one example; in the month or so that I've been running around paying cash for minor purchases, I've found that people have really lost the ability to deal with the concept of numbers. If you try to pay for $60 worth of groceries with 60 actually dollars, they're just downright confused. If they can't do simple arthimatic, what makes you think they can grasp statistics?

    P.S. On the good-news front, I managed to pay a 25-cent library fine with a quarter and the good folks at the library understood what I was trying to accomplish! Hooray for public libraries!

  • Katydid – an actual exchange the other day, while I was in a small local shop exchanging an item that cost $39.99 with one that cost $89.99.

    Clerk: I'll just take this one back and charge you the difference.
    Me: Sure, sounds good.
    Clerk: hmm, somebody walked off with the calculator, hang on let me go find it.
    Me: It's fifty bucks.
    Clerk: hey, anyone know where the calculator is?
    Me: No, for real, $89.99 minus $39.99 is $50.
    [someone finally shows up with the wayward calculator, clerk punches it in]
    Clerk: Damn, you're good!

    Yeah, go me, can subtract $39.99 from $89.99 in my head. Math genius ovah heah.

  • EJ, that reminds me of when I first moved to California for my Real Job after grad school and my mother and I were buying me a bed. Their computers were down and they couldn't figure out the sales tax, so they ended up not charging us for it.

    We were shocked that they couldn't figure it out on their own or that they didn't have little charts of the amount of sales tax on certain dollar amounts.

  • EJ and Ursula; yup, that's exactly what I'm finding, too. I am so very much not a math genius, yet I'm constantly astounded by the people around me who seem to have absolutely no grasp of numbers whatsoever.

    Had this discussion just this weekend when i went to help a friend who has a small family farm. She sells eggs to the public. The eggs are packed in the conventional egg cartons that hold a dozen (in fact, the egg cartons are donated by various people from egg cartons sold in the supermarket).

    Customer: "I need ten eggs for Easter."
    Me: "The eggs are sold by the dozen; pick the carton you want."
    Customer: "But I need more than a dozen; I need TEN!"
    Me: "A dozen is twelve."
    Customer: "????"
    Me: "Twelve is two more than ten."
    Customer: "Whatever. Do you take credit?"
    Me: "Yes, but we charge you the 3% processing fee the credit card company charges us. It's cheaper for you if you just dig in your wallet and find the $3 the eggs cost."
    Customer (offended): "You mean pay MONEY?"
    Me: (thinking to myself) "credit is also money…money that you pay extra to use…)

  • Posts like this teach us how to think. In comparison to other media that mostly teach us what to think. Fallacies and Biases are very important and I find this blog to be one of the greatest resources online. Check my first attempt to understand and explain fallacies: http://www.illogicalfallacy.com Any feedback would be highly appreciated

  • @Katydid: Do you live in Ky LOL … my husband went to a local butcher shop we have been patronizing for years, and told the new clerk he wanted "three quarters of a pound of hamburger". The woman literally did not know what that was……Finally the store owner had to wait on him. Innumeracy is rampant anymore….

  • @evodevo; nope, don't live in Kentucky, but am not shocked at your story.

    I am no math genius, and this is why I'm so dismayed that there are grown people who have no idea that "dozen" means "12", can't make change for $2 when the bill is $1.99, don't know how to figure 3/4 of 1 pound, have no idea how to figure taxes onto a purchase.

    As a pre-teen, I started babysitting for the princely sum of 75 cents an hour. I had it figured out what I'd make if the parents were gone an hour, three hours, four-and-a-half hours, etc. I stress again that I am not the Rainman of math, so it dismays me that so many adults can't figure out the slightest things having to do with math.

    Yet another anecdote; we put our oldest (now grown) in a private kindergarten when the time came, because the local public school had half-day (2-hour-and-15-minute) kindergarten. They ran two sessions; morning and afternoon. This was hell for working parents who had to provide car service between the school and whatever daycare they could find to fill the working day (which is strangely *not* two-hours-and-15-minutes a day, go figure!).

    Since the school day was so short, the kids were coming home with 2 – 3 hours a day of homework, which is simply absurd for a 5-year-old. The problem was that such a short school day leaves no time for the teachers to actually teach anything–the kids get off the bus, take off their coats, sit down, then get back up again, put their coats back on, and line up to get on the bus. Therefore there was a great hue-and-cry for all-day (6-hour) kindergarten.

    The kindergarten program consisted of 2 classrooms for morning and the same 2 classrooms for afternoon kindergarten (for a total of 4 classes). Grades 1 – 5 were 4 classrooms per grade. One of the rantings against all-day kindergarten was, "They will have to *double the size of the school* to fit in all the extra kids." Uhm, no; they had to build 2 extra classrooms (for a total of 4). You'd be appalled how many people didn't understand that, either.

  • Death from SIDS does not mean death from natural causes. In fact, the original SIDS cases in the medical literature were almost surely murders. Now that parents are putting infants to bed on their backs, SIDS deaths have dropped dramatically, which increases the chances that they are now murders. Unexplained infant deaths in the same family are hardly independent if the infants have been murdered.

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