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Topic: Summer '96 Institute: Shared Lessons and Activities
Replies: 10   Last Post: Jul 19, 1996 1:30 PM

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Sarah Seastone

Posts: 171
Registered: 12/3/04
Probability - presented by Paul Myers
Posted: Jul 19, 1996 6:30 AM
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Probability and the Cereal Box Problem

On Thursday afternoon we took a break from the lab to participate
in a shared activity presented by Paul Myers. We did two
mini-lessons, both related to probability and statistics, and both
with the aid of a TI-92 calculator hooked up to an overhead
projector.

First Paul handed out index cards with a 9 by 7 grid printed on
one side. Each of us drew our "favorite triangle," connecting
three vertices of the grid. Paul collected the cards, shuffled
them, and then redistributed them, face down. We were instructed
to poke 50 holes randomly in the back of this second card. The
idea was that counting the number of "random" hits that fell
inside the interior of the triangle would be a way to estimate the
area of the triangle. Clearly, the greater the number of pokes,
the better the estimate, but we were all pleasantly surprised with
the accuracy achieved with only 50 pokes. Here's the calculation:

[(number of triangle hits) / 50] * [total area of grid] =
estimated area of triangle

Cereal Box Problem

The second activity Paul presented is known as the Cereal Box
Problem. A cereal company is offering six different prizes in its
cereal boxes. How many boxes would one expect to have to buy, on
average, in order to collect all six prizes?

Rather than giving us a lot of time to compute probabilities and
expected values, Paul asked us to make a quick guess and then
construct a histogram of all our guesses on the board. Our guesses
were substantially lower than those Paul usually finds in his
classroom, due probably to our relative experience in this type of
problem. The mean of our guesses was somewhere around 20.

We then discussed how students might "do" this problem, to test
out their hypotheses. Short of (or in fact better than - see
discussion below) going to the store and buying a lot of cereal,
the best choice is to use dice. We all conducted two trials,
recording how many roles it took to get all 6 numbers on the die.
With these results we built another histogram on the board. Now
the mean was closer to 15.

That's pretty close to the theoretical result. In order to see
this, you need to understand the relationship between the
probability of getting a certain prize and the expected number of
trips it takes to do so. Think of the dice. Since the chance of
getting an odd number is 1/2 on each roll, you expect to have to
roll an average of twice before seeing an odd. In general, the
expected value is the reciprocal of the probability. On the first
trip to the store, you're happy with any of the six prizes since
the probability of getting a desired prize then is 6/6. The
expected value is also 6/6, or 1. On the second day, you need 1 of
5 prizes. The probability of getting a desired prize is 5/6, and
the expected value is 6/5.

The total expected number of trips, then, is the sum: 6/6 + 6/5 +
6/4 + 6/3 + 6/2 + 6/1. This works out to be about 14.7, or 15.

The discussion of this interesting and well-received problem was
quite lively. It focused on assessing the pedagogical value of the
exercise, and on potential extensions in the classroom, and
quickly turned to the issue of how "real life" this problem is.
This is clearly not a new question, and mention was made of the
ongoing and recurring thread on the NCTM-L mailing list. Sarah
Seastone pointed out that reality for a student did not often
coincide with the assumptions of the cereal box problem, and she
asked Paul what further applications he could suggest to make it
even more realistic. We all agreed that most problems in
classrooms, and in mathematics in general, are somewhat contrived,
but that they are not without value because of this. At the least,
the setting and props for the problem are familiar to us.

Steve spoke of the role a problem like this one can have in
helping to develop people's mathematical intuition. Probability
and statistics, he said, are areas of math in which students'
intuition is particularly lacking. By working out a problem like
this one, people can hone their estimation skills by orders of
magnitude, and this is in itself a help in the real world. The
rest of the group agreed, and added that discussing the
discrepancy between the model and reality would be a good exercise
for students. In particular, Ruth Carver suggested an
interdisciplinary unit in which the students could write letters
to the cereal company explaining their calculations and
experimental results.

- Eric Sasson








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