Regarding probability study posted on TheCitizens

Yow. I ignore the comments for weeks, and three or four interesting topics come up.

A) you may be interested in the Iowa Gambling Task 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.)

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..."

This, and I really do not mean to offend, is because you are not a mathematician, particularly not a mathematician who has to teach non-mathematicians.

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. And one big idea that's come out of this is the impetus to connect the language and structure of mathematics to the structures and intuition that "normal" people already have, rather than treat math as abstract and foreign.

So we talk about "number sense" and "proportional reasoning" and "intuitive probability" and part of what we mean is a non-linguistic concept of those things. The theory is that people have a sense of probability, both in the cases of extremely rare events and relatively common events, but:
a) your intuition can be misled, and a high-level, abstract argument lacks potency (e.g., because you've heard about miraculous recoveries from coma, your intuition is that is a reasonably likely event, and a doctor telling you it's a one in a million chance doesn't dissuade you.)
b) it is difficult to connect your intuition, in the many cases when it is right, to the abstract claims of probability, if they're couched in examples not relevant to your experience or in language you don't have other intuitive connections with (most people troubled by the Monty Hall problem didn't watch episode after episode of Let's Make a Deal; most people have never thought of what "1 in 100,000" means, in any concrete way, and usually don't have a reason to.)

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.

(I should point out that numbers are themselves language, as I'm using the phrase. I doubt that caused any confusion, but you can never be too careful on the interweb.)

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' Innumeracy (and/or A Mathematician Reads the Newspaper) are excellent reading.

D) RbR's argument (and I know I shouldn't say this, but I just can't help it, so please forgive me) assumes facts not in evidence. A claim about the sensitivity of a forensic test is just that, independent of other factors. If a jury is deciding guilt or innocence, they are (or should be) weighing all the factors RbR suggests (which surely the prosecution would present) in their "reasonable doubt" analysis. But those other factors don't affect the accuracy of the test.

RbR's description also bears some resemblance to an example in Innumeracy, about a case in Los Angeles in 1964. (I'm not saying RbR is guilty of the probabilistic fallacy in that case, just that his description is similar to the set-up for the fallacy, and the story is a fun one, so I'm going to relate it.) Brutally summarized, the argument was: a pair of lawbreakers were seen with several "distinguishing characteristics" (blonde with pony tail with black man with beard in a yellow car). The defendants matched those characteristics. In this case, there was nothing else linking the defendants to the crime, but surely the chances that a couple would match all those characteristics were so small that this must be the couple in question, right?

So it was decided. Until the appeal, in which the defense pointed out that the relevant probability was not whether a couple would have those characteristics, but rather whether, out of the ~2M couples in LA, there was _another_ couple with those characteristics. (The probability that there was one such couple was actually a certainty, since the defendants were living proof.)

It turns out (and no one would suggest that this is intuitive) the likelihood of another such couple (given the prosecution's estimate of the relevant probabilities) is about 8%, which provided reasonable enough doubt for the California supreme court
to overturn the original verdict.

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!