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Math Circle Blog: Problems in Probability

Answers Usually Boil Down to How You Defined the Terms in Your Question

October 2, 2012: “Death is a constant.” Or at least so claimed most of our group, after R had contributed “12 inches per foot” and G “60 seconds per minute” to our list of constants.

At that point, S had argued that “60 seconds per minute” is not a constant: we could easily redefine the term minute. Good point. So we stated our assumption that we will use conventional definitions of the terms “minute” and “foot.”

But what is the conventional definition of death? And what aspects of it are constant? First we eliminated “cause of death” and “age at death” from consideration as constants. We seemed to agree that “the eventuality of death” is constant. But what is death? G brought up organ harvesting, P brought up respirators, and someone else brought up the possibility of future advances toward immortality. Assumptions had to be stated in order to define death as constant. So, as usual in mathematics, the answer boiled down to how you define the terms in your question.

The topic of constants came up because N and C were recapping last week’s session for A and S, who had been absent. We carefully distinguished variables from parameters from constants, and then expressed the solution to our popcorn problem in letter variables: Vr/Vb=B, and BW=the amount of popcorn needed. (The lower case letters were actually subscripts.) Then we moved on to data collection.

Not everyone was interested in discussing data collection methods and ideas. Most were, though, and some hope to collect some data on their own. S pointed out that if we collect data about nose rings, we have to define exactly what we mean by “ring.”

Nor was everyone interested in simple probability questions. We did have a nice discussion about the specious probability assumption that each outcome is equally likely. D and N explained the difference between fair coins and unfair (real) coins in coin tossing events. P and A pointed out that the act of “randomly calling on someone in class, even with your eyes closed,” is not necessarily random. We also discussed the problem of random samples not necessarily being representative of a population. And I achieved my goal, which was to determine that everyone in the group knew what probability was and how to express it mathematically. But, since the collective curiosity of the group was not sufficient to continue with small examples, it was time for something big.

“If an equilateral triangle is inscribed in a circle, what is the probability that a randomly selected chord is longer than a side of the triangle?”
This question* caught everyone’s interest immediately. (It’s a good example of what Bob and Ellen Kaplan call “an accessible mystery.”)

Conjectures were posited: More chords would be shorter than a side than longer. No, wait, more chords would be longer. No, maybe the same number would be shorter as longer. Actually, the numbers wouldn’t be the same, but they’d be close…

Students came to the board to test these conjectures. Some did it on paper. Questions arose: Where are more opportunities for shorter chords? Would it matter whether a human or a computer were filling in the chords? Do the starting points matter?

R suggested shifting the triangle so that the chords all emanated from a vertex, as opposed to a random point on the circle. G realized that “you can keep on drawing thinner and thinner lines,” and pointed out that the circle would be completely shaded with chords if we consider all possibilities. R noticed that “the space inside the triangle is where the longer lines are, and the space outside the triangle is where the shorter lines are.” I heard murmers about area.

P pointed out a complication: “The area containing shorter chords from one vertex of the triangle overlap with the area containing shorter chords from another vertex, and the same with the areas containing longer chords.” Then D had an idea for another way to quantify the shorter and longer chords. He was showing everyone his reasoning with the side of a pencil against a diagram on paper. People were trying to understand when I looked at the clock. Sadly, we had already gone 5 minutes beyond our ending time. Therefore, this problem is to be continued next time, hopefully with D up at the board with another approach.

-- Rodi

*This problem is a classic probability question, not one that I can take credit for creating. I’ll tell you the name of it next time. In the mean time, have fun with it, because I know you’re thinking about it.