Hi Don, On "mutual exclusive" and "independence" --
Perhaps this may be useful (or not?): One can think of the two in terms of a coin flip, getting "heads" is mutually exclusive to getting a "tails" on any flip of the coin; yet, given that you know that a heads was obtained, you automatically know you don't have tails on that particular trial, i.e., extreme dependence, as I think you [Don] put it. if there are any flaws in my analogy, please do point them out, i'm interested to know -- it was quick and dirty. p > My take on this systematic and nearly unanimous confusion between > "independent" and "mutually exclusive" as descriptors is this: > > (1) The first definition encountered is invariably of "mutually > exclusive" (or, as Moore and others put it, "disjoint"). This notion is > readily apprehended in its surface meaning, and since the surface > meaning works adequately for the first few problems, one seldom has > occasion to correct one's misunderstandings (or perhaps complete lack of > understandings) about the deeper implications. > > (2) Later (and it's always later, both in my experience and in several > textbooks that I just consulted) the idea of "independence" is defined. > Without thinking about it too much, this may seem to be merely a > synonym, as Radford points out. And even WITH thinking a little more > deeply, it may still seem to be a synonym. I suspect the internal > logic, never exhumed so one can get a good look at it and FIX it (either > on one's own or with an instructor's help), runs more or less like this: > (i) "A and B are mutually exclusive" ==> > (ii) "A and B have nothing to do with each other" > (reasonable enough, on the surface) ==> > (iii) "A and B are mutually independent" > (well, if they have nothing to do with each other...) ==> > (iv) "A and B are independent events". > > When one lays it out like that (if that does indeed reflect whatever > ratiocination may have been going on), it is easier to see (but by no > means immediately clear) that > + "exclusive" does not, really, mean the same as "nothing to do with > each other" (so the linkage (i) -> (ii) is faulty); > + "nothing to do with each other" in the sense of exclusivity or > disjunction does not mean the same thing as "mutually independent" > (although there is a sense of the former that could be synonymous, at > least approximately, with the latter) (so the linkage (ii) -> (iii) is, > or may be, faulty, and in any case the linkage (i) -> (iii) is faulty); > + "independent", at least in the sense intended in the probabilistic or > statistical context, implies "mutually", so that the adverb either is > redundant or signifies a reading of "independent" that is not the same > in (iii) as in (iv) (so the linkage (iii) -> (iv) may be faulty). > > I confess this is all conjecture, and I have no empirical data (except > for the ubiquity of the confusion!) to support it. Still ... > > I have (I think!) found it useful, in attempting to counter this > problem, to point out that "mutually exclusive" is not at all like > "independence": on the contrary, for A and B to be mutually exclusive > is for the presence of A to deny the possibility of the presence of B; > if they were independent, the presence of A would be irrelevant to the > presence of B (or, more anthropomorphically, A wouldn't care whether B > existed or not). Mutual exclusion is therefore, really, an extreme form > of dependence. Students seem to accept this argument, although it's > often clear that some of them don't like it. (If that dislike is made > overt, my response is "You don't have to like it; you just have to use > it.") > > Comments? -- Don. > > On Mon, 1 Mar 2004, Radford Neal wrote: > > > Bruce Weaver <[EMAIL PROTECTED]> wrote: > > > > >... I've noticed that undergrad introductory stats students have a > > >habit of treating terms we give them as meaningless labels. Take > > >"independent" and "mutually exclusive", for example. I think the > > >reason so many students confuse them is that they do not think about > > >what the terms mean. > > > > I think that can't explain this rather striking phenomenon. In my > > experience, they don't confuse these two terms, but rather think that > > they are synonyms. A substantial fraction of them continue to think > > this regardless of how many warnings about this specific point they > > are given in the textbook and lectures. I think part of the > > explanation is that the common meaning of "independent" can indeed be > > taken to by synonymous with "mutually exclusive" - thinking about it > > (without actually reading the technical definition) doesn't help. > > This doesn't really explain the persistence of this confusion, > > however. Is there something deeper, such as a failure to appreciate > > that a term can have a technical meaning that is not the same as > > common usage? Or is it just that they don't go to lectures and don't > > read the textbook, and hence never encounter the attempts to correct > > them on this point? > > ------------------------------------------------------------ > Donald F. Burrill [EMAIL PROTECTED] > 56 Sebbins Pond Drive, Bedford, NH 03110 (603) 626-0816 > . > . > ================================================================= > Instructions for joining and leaving this list, remarks about the > problem of INAPPROPRIATE MESSAGES, and archives are available at: > . http://jse.stat.ncsu.edu/ . > ================================================================= > . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
