Hearing What We Want to Hear
October 23, 2012: After 3 weeks of work, we finished Bertrand’s Paradox with a discussion of why it’s a paradox. (You can get different correct answers, both theoretically and experimentally, depending upon how you define the term “random.”) This led to debate about whether humans, or even computers, could ever truly generate randomness. Can nature even do it? So far, the consensus seems to be “no,” but I think that people are still thinking about and hoping for it. We also talked, again, about how important it is to define the terms in a question, and to look for assumptions so that we don’t just hear what we want to hear.
Along these lines I told the legend of Pythia, the Oracle of Delphi, telling a great general that “a great army will be defeated.” He had asked her whether he should go to war against a particular foe.* Her proclamation led him to enter battle with confidence. Sadly, the great army that was defeated was his own. P had heard this story before, and was pretty sure that the general was Alexander the Great. I wasn’t sure who it was, and promised to look it up. I checked, and found that it seems P was right, that Alexander did consult the Oracle, with very interesting results. The interaction I mentioned, though, seems to be attributable to King Croesus of Lydia (http://www.delphic-oracle.info/delphic-oracle.asp).
I then posed some life-or-death (hypothetical) choices to the group. Unfortunately, we didn’t have an Oracle to consult. All we had were some statistics, which we used in a game of “Would You Rather:”
“You are exploring a land populated by hydrophobic vicious animals. You are safely wading in a one-foot deep stream when you come to a fork in it. Each branch leads to a different pond. Each pond has a helicopter on the other side of it that can transport you to safety. A sign at the fork tells you that the pond on one side has an average depth of 5 feet, and the other is 7. Oh, and did I mention that you can’t swim?”
At this point, all of the kids were calling out questions before I even asked The Question. (As I’ve mentioned before, problem-solving skills are developed better when questions are intentionally phrased vaguely.) Their clarifying questions yielded them such information as this: each pond has 6 sections of uniform depth, there is no other way to get to the helicopter, your height is between 5 and 6 feet, you can hold your breath for one second, and you will therefore die if you don’t reach the helicopter. The kids then supplied the question: which pond should you cross? C’s conjecture was to choose the 7-foot-deep pond, since surely there’s a trick. R suggested listing out every combination of 6 numbers to determine which average had more “safe” distributions. The group soon realized that each pond had an infinite number of depth distributions. Finally we voted (I don’t remember the outcome), and then the group demanded to know the distribution and arrangements of depths. I put it on the board and they saw that they were dead.
I suggested doing another one, and offered to give them a measure of dispersion rather than of central tendency. The kids protested that this would not help. G asked for both numbers anyway, just to see, so I gave a scenario with both ponds having an average depth of 4, but different ranges (6 and 4). The group realized that this was still no safety insurance, but chose the range of 4 and then asked for the actual distribution. I showed them that this time they lived. I asked them if there was a way to rearrange that pond to insure death, and they found one. We wrapped up with the thought that sometimes summary statistics are not enough to draw conclusions – sometimes you need the actual distribution. I sent them home with this question: Can you lie with statistics?
Since our sixth and final class of this session was postponed due to Hurricane Sandy, I’d like to give you a few other questions to think about until our make-up session (Thursday, December 6, 3:30pm):
- What is the definition of “random?”
- How can you lie with statistics?
- If you could have only one summary statistic, in what situations is it better to have a measure of central tendency (vs. dispersion)? …vice versa?
- Why did I use the word “seems” twice in the second paragraph of this essay?
- (For parents – bad language warning) Is the statistics quote attributed to Mark Twain true? (http://www.twainquotes.com/Statistics.html)
Of course, there is no homework in Math Circle, so these optional questions are only to discuss for fun. When I see your problem-solving kids in December, we will face the ultimate mathematical “Would-You-Rather” challenge. Thanks for sharing them with me.
*Thanks to Ellen Kaplan for this anecdote and its connection to Bertram’s Paradox.