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/ .
=================================================================
