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[ap-stat] Teaching probability (long message)
Posted:
Nov 29, 2005 11:50 AM
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The recent questions about poker hands and Texas Hold'em have me just
concerned enough to post anew a message I've posted a couple of times
before, at the risk of sounding like a broken record. My concern is
solely that teachers new to AP Statistics may not realize that such
questions lie outside the curriculum; I don't meant to discourage teachers
from teaching some combinatorics, nor from discussing the topic on the
listserv.
It used to be that a lot of college statistics courses were really
"probability and statistics", and would include a hefty dose of the first
before beginning the second. I know of one university course in which
probability is intentionally taught first not because it prepares students
for statistics, but because it typically gives students a lot of trouble
and thus weeds out early the students who will (supposedly) have a tough
time with statistics, saving them future agony.
I don't agree with this practice, as students who have trouble with
combinatorics problems often have no trouble with applied statistics. And
many colleges have come to that conclusion and now teach introductory
statistics without requiring a full probability course. This is reflected
in the AP Statistics curriculum, which includes only as much probability
as usefully serves the statistics that we teach. Below is a list of some
probability topics that AP Statistics students should know. I don't claim
it's exhaustive--you should read the course syllabus carefully to see such
a list--but I think it hits most of the big topics. If you look at old AP
exams, both free-response and the released multiple-choice questions,
you'll get a good feel for what's expected of students in the way of
probability. I wouldn't trust non-College-Board course prep guides too
much, though; I've found that some of them seem to misjudge the relative
importance of some topics, notably probability.
My list of big probability ideas, in the order I teach them. These don't
include random variables:
-- Be able to draw a two-factor contingency table containing either data
counts (from which probability estimates may be made) or probabilities.
Know that joint probabilities are at row-column intersections, that
marginal (i.e., unconditional) probabilities are the sums of rows and
columns, and that the total of these for all columns or all rows (or all
joint cells) must equal 1. Use the table to estimate conditional
probabilities or (less usefully) probabilities of unions.
-- Know the definition of conditional probability, P(B|A)=P(A^B)/P(A), and
its consequence, P(A^B) = P(A)*P(B|A).
-- Independence means that P(B|A)=P(B). This occurs when events don't
have an impact on one another, and should be questioned when there's the
possibility that the outcome of one event does have an impact on the
other. A consequence of independence (not the definition) is that P(A and
B) = P(A)*P(B).
-- Be able to draw a probability tree properly and know that the
probabilties in it are unconditional, then conditional, then (on the
"leaves"), joint probabilities. Be able to use it to solve problems
involving conditional probabilities. These include "Bayes's Theorem"
problems, so called because their solutions implicitly invoke Bayes's
Theorem; you don't actually need to teach or name Bayes's Theorem
explicitly.
That's roughly four or five class days. Towards the end of the school
year there always seem to be a fair number of teachers who feel rushed, so
I think it's good to point out to new teachers now a place where time may
be saved. The syllabus shouldn't feel packed and you shouldn't feel
rushed to teach the whole course in a year. So it's reasonable to ask
yourself whether you're spending too long on probability. Most
importantly, notice what is not in my list above: license plates, couples
at round-table dinner parties, and poker hands. That is, no
combinatorics. The only combinatorial element I know of in the entire
curriculum is the binomial coefficient, and that can be skipped
altogether, so long as students are comfortable using their calculators to
compute binomial probabilities directly.
And here's another suggestion for how to teach the topics above. Instead
of doing so with playing cards, dice, and balls in urns (real or just in
problem statements), consider getting simple data sets that include two
categorical factors with two to five levels each, and some interesting
interplay between them, such as U.S. region and smoking habits. Ask the
obvious questions: what region has people most likely to smoke? Is it
true among the chain-smokers as well as the light smokers? Where is the
highest concentration of smokers? Did you use the same numbers to answer
that question as the first question? (One should be A|B, the other B|A).
Is smoking independent of the region of the U.S. a person is from? Two or
three data sets like this could cover all the relevant topics in
probability, and they're pretty easy to find.
I used to feel like I had to make up fake data sets (which I hate doing)
to get good examples of independence since it's so rare in real data. I
thought of two things I could do with real data to still teach the topic.
One is this. I would tell students that independence means P(B|A)=P(B),
without telling them that a consequence is P(A^B)=P(A)*P(B). Then I would
give them some real data, but only marginal counts. I would then ask them
to fill in the joint cell counts assuming that the factors are
independent. They would have to play with the numbers and figure out how
uniquely to make it work out right. We would then talk about the
multiplication "rule" of independence. Then I would show them the real
joint counts and ask whether they thought the factors were independent.
They'd say no, and we'd talk about the fact that smoking rates are higher
in the southeast than in the northeast, for example.
The other thing I do is this. I'd give them a smallish sample data set in
which the numbers didn't produce exact independence but were "close", and
again I'd ask them whether they thought the factors were independent.
Many of them, going strictly by the numbers, would say no. Some others
would say "almost" or "close". That would give us a chance to foreshadow
the future idea of sampling variability, and how the population might have
independence even though the sample doesn't fit the equations perfectly.
And that's it. If I ever had a student who felt lost on the AP exam
because we hadn't counted license plate permutations, she never told me
about it.
(I have had success teaching probability this way myself and I do
encourage other teachers to try it, especially if you feel a need to save
time somewhere in the syllabus. But the person who knows a class best is
always their own teacher, so you are the best judge of whether this
approach will work for you and your students. If others have their
preferred ways to teach probability, I expect I'm not alone in being
interested in them sharing their strategies on the listserv.)
--Floyd
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