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I ignore the comments for weeks, and three or four interesting topics come up.<br /><br />I wrote a comment, but it was too long, so I posted <a href="http://halfwitonblogs.blogspot.com/2009/09/regarding-probability-study-posted-on.html" rel="nofollow">over here</a>. Here is a dramatically abridged version, in an attempt to get you interested enough to click over:<br /><br />A) you may be interested in <a href="http://en.wikipedia.org/wiki/Iowa_gambling_task" rel="nofollow">the Iowa Gambling Task</a> experiments, one observation of which is (I'm paraphrasing here, so read some real research if you want to see what the studies really claim) that people can develop a sense of a probability from a smallish sample, _before_ they consciously recognize that they have such a sense. (Paraphrase 2: drawing from a deck of possible rewards/penalties, someone will start reacting negatively to a "bad" deck before they consciously decide it's a "bad" deck.) <br /><br />B) [not actually chronological] LTG says "I am surprised any mathematician on this blog would expect children to have an intuitive grasp of probability. Adults sure don't...It is well known...that individuals tend...to treat a 99,999 to 1 chance...differently from...a 1 in 100,000 chance..."<br /><br />Mathematicians (especially in education) think about what is and isn't intuitive mathematics a great deal, because what _seems_ intuitive to us is clearly not to our students. <br />So we talk about "number sense" and "proportional reasoning" and "intuitive probability".<br /><br />Thus, while mathematicians cannot help but see that most adults' sense of probability is _solely_ intuitive, and unconnected to any attempts at probabilistic abstraction or language, and (therefore?) usually lacks _nuance_, we cling to the belief (well justified, really) that there is in fact an intuition there. <br /><br />C) I took Pombat's examples to just be illustrations of the way probabilities are bandied about in misleading ways, and not a claim about DNA evidence per se. For more examples of this and other grievous misquantification, John Allen Paulos' <i>Innumeracy</i> (and/or <i>A Mathematician Reads the Newspaper</i>) are excellent reading.<br /><br />D) [cut for length]<br /><br />E) If anyone wants to talk more about math, probability, and math education, I'm very keen on it -- you could say it's my job. Email me!Scholeologisthttps://www.blogger.com/profile/01110359677410211160noreply@blogger.comtag:blogger.com,1999:blog-6762928.post-60832697243493659182009-08-23T22:49:30.994-07:002009-08-23T22:49:30.994-07:00These thirteen or so genetic markers have been sho...These thirteen or so genetic markers have been shown to be almost completely statistically independent in humans. But as you rightly observe, the high confidence of the DNA match hinges completely on that statistical independence. It remains theoretically possible that these genetic markers might be more highly correlated in some small subgroup (say, the residents of some Romanian village). <br /><br />Fortunately, using DNA to establish <i>innocence</i> does not suffer even from this potential minor flaw. If two samples are found not to match, and the laboratory work was not contaminated, then the statistics are irrelevant: there is simply no match.Dr. Strangelovehttps://www.blogger.com/profile/14407042105777411150noreply@blogger.comtag:blogger.com,1999:blog-6762928.post-42567551904636800662009-08-23T19:57:35.810-07:002009-08-23T19:57:35.810-07:00Dr.S.- Let me see if I understand. IF every perso...Dr.S.- Let me see if I understand. IF every person's DNA is different, then the match is 1 in 100 billion. But what if some people (other than identical twins) have large segments of identical DNA. I'm not saying this is true, of course. But isn't it true that the test, then, depends on this not being the case?The Law Talking Guyhttps://www.blogger.com/profile/17886791396468512490noreply@blogger.comtag:blogger.com,1999:blog-6762928.post-7471390038614771822009-08-21T11:07:12.624-07:002009-08-21T11:07:12.624-07:00So just to sum up, DNA matching is not a single te...So just to sum up, DNA matching is not a single test but really two tests (one for each sample) whose results are then compared. Repeating the comparison gains you nothing. And once you see how the test works, it is clear that Pombat's interpretation of the "one-in-a-million" stat is correct.<br /><br />It is worth emphasizing, perhaps, that laboratories compare only short sequences, instead of the entire DNA pattern, because that would be far too difficult. Indeed, the first time anyone sequenced the entire DNA sample of a person it took several years and we called it the "Human Genome Project."Dr. Strangelovehttps://www.blogger.com/profile/14407042105777411150noreply@blogger.comtag:blogger.com,1999:blog-6762928.post-69836726011411538942009-08-21T10:55:44.551-07:002009-08-21T10:55:44.551-07:00Here's how the test works. The lab compares s...Here's how the test works. The lab compares short genetic sequences at about thirteen standard chromosomal sites in each sample. These thirteen or so genetic "markers" are used for DNA comparison because the sequences are statistically independent and easy to distinguish since only a handful of genetic sequences are possible at each of these sites for humans. The essence of the probability calculation is straightforward: If each genetic marker consisted of a single GATC base only, then the chance of a coincidental match would be 1 in 4^13 (67 million). There are usually more than four options for each genetic marker, however, so the actual chance that all 13 sequences would match is more like 1 in 100 billion.<br /><br />Thus as a practical matter, DNA evidence is as close to certainty as one can get in this world. It may only refuted by showing that the incorrect sample was tested, or that the sample was contaminated, or that the laboratory test was botched... Or that one's identical twin did it ;-)Dr. Strangelovehttps://www.blogger.com/profile/14407042105777411150noreply@blogger.comtag:blogger.com,1999:blog-6762928.post-40132394011921978112009-08-21T09:09:15.962-07:002009-08-21T09:09:15.962-07:00I really think the "one in a million stat&quo...I really think the "one in a million stat" does NOT mean "every millionth should be a match" but rather "this test has a one in a million chance of being wrong." <br />Do it twice, and you might increase the certainty dramatically.The Law Talking Guyhttps://www.blogger.com/profile/17886791396468512490noreply@blogger.comtag:blogger.com,1999:blog-6762928.post-65038046953321159932009-08-20T23:40:34.911-07:002009-08-20T23:40:34.911-07:00No need for explanation Dr.S - I took it as the jo...No need for explanation Dr.S - I took it as the joke it was intended as (and smiled, for the record) - you know I'm not overly touchy about that kinda stuff :-) Or maybe it's just that I like language-related humour (all the grammar nerds say "wuh wuh". I watch too much Daily Show).Pombatnoreply@blogger.comtag:blogger.com,1999:blog-6762928.post-59850336231321680142009-08-20T23:16:32.672-07:002009-08-20T23:16:32.672-07:00That last comment was supposed to be a joke, by th...That last comment was supposed to be a joke, by the way. (I am not actually suggesting the English have any worse time of it than anyone else.)Dr. Strangelovehttps://www.blogger.com/profile/14407042105777411150noreply@blogger.comtag:blogger.com,1999:blog-6762928.post-13372442755055054142009-08-20T23:15:46.350-07:002009-08-20T23:15:46.350-07:00The English have an especially acute problem with ...The English have an especially acute problem with this, Pombat, because they have many "maths" to learn. Fortunately we only have to learn one "math" over here :-)Dr. Strangelovehttps://www.blogger.com/profile/14407042105777411150noreply@blogger.comtag:blogger.com,1999:blog-6762928.post-39886447607151182302009-08-20T18:24:42.371-07:002009-08-20T18:24:42.371-07:00Wow! So that's how the police do it! Thanks ...Wow! So <b>that's</b> how the police do it! Thanks RbR!<br /><br />I was talking about just the one in a million stat, and how easily people will believe it. And yep LTG, it could also be translated that way around - it depends on the context in which the one in a million is described/presented.<br /><br />As Dr.S says - we're talking about the simplest terms here. The thing is, we all have an intuitive grasp of things like probability, and the ability to learn about them properly. Sadly though, a large number of people never do, because they're taught maths so badly when they're younger that they're switched off to it / anything that looks remotely mathsy once older (and by older I don't mean that old - 9, 10yrs old maybe), and simply refuse to grasp even the simplest concepts, on the basis that "I'm no good at maths" / "I don't like maths" / "maths is too hard". Which is so incredibly frustrating, because no matter how good a teacher you put in front of them, they just won't learn :-(Pombatnoreply@blogger.comtag:blogger.com,1999:blog-6762928.post-72211216833126596072009-08-20T14:35:51.778-07:002009-08-20T14:35:51.778-07:00"I am surprised any mathematician on this blo..."I am surprised any mathematician on this blog would expect children to have an intuitive grasp of probability. Adults sure don't, beyond the very simplest items."<br /><br />Yes, but we are only talking about those very simplest terms here.Dr. Strangelovehttps://www.blogger.com/profile/14407042105777411150noreply@blogger.comtag:blogger.com,1999:blog-6762928.post-64418792491233214952009-08-20T08:16:08.783-07:002009-08-20T08:16:08.783-07:00Of course any time I talk about evidence, I feel o...Of course any time I talk about evidence, I feel obliged to recount Ambrose Bierce's description:<br /><br />We have no admissible evidence that there ever was such a person as Caesar or Napoleon, since that is all hearsay. But the courts may take judicial notice of the records of other courts, so the evidence that there were witches is is indisputable.The Law Talking Guyhttps://www.blogger.com/profile/17886791396468512490noreply@blogger.comtag:blogger.com,1999:blog-6762928.post-15710598055662810082009-08-20T08:12:38.428-07:002009-08-20T08:12:38.428-07:00If the authors think either interpretation is acce...If the authors think either interpretation is acceptable - i.e., that the children are as likely observing mismatch as probability - then the article in the Times was completely off base.<br /><br />I am surprised any mathematician on this blog would expect children to have an intuitive grasp of probability. Adults sure don't, beyond the very simplest items. It is well known through research on what is often now called mental accounting or behavioral economics (or some branch of psychology) that individuals tend, for example, to treat a 99,999 to 1 chance of a successful lifesaving operation differently from hearing about a 1 in 100,000 chance of death. <br /><br />Similarly, adults will believe that the statement "4 out of 5 dentists recommend this toothpaste" and "1 in 5 dentists would not put that crap in theirm ouths" are contradictory statements.The Law Talking Guyhttps://www.blogger.com/profile/17886791396468512490noreply@blogger.com