Sunday, October 12, 2008

Answering some of the other questions

kw: book reviews, nonfiction, science, engineering, estimating

In Middle School and High School, many American kids (my son included) partake of the Science Olympiad. One contest is "Fermi Questions", named for Enrico Fermi. They are questions you answer just by making successive deductions based on things you already know or can estimate. For example:

How many Jelly Beans will fit in a one-liter jar?
  1. 90
  2. 200
  3. 350
  4. 500

The students figure out an answer, and reply with the closest choice from the list. In this case, one might reason like this:
  • A typical jelly bean is 1.5 cm long and 1 cm in diameter.
  • Its "rectangular volume" is thus 1.5 cubic centimeters.
  • Jelly beans don't pack tightly in the jar, so they might fill only 80% of the space.
  • 1000 divided by 1.5 is 667.
  • 80% of that is 533.
  • Use answer #4: 500.
The two "facts you know" are that a liter contains 1000 cubic centimeters, and the size of jelly beans that you remember eating in the past. Of course, if the contest moderator actually shows a jar of jelly beans at the beginning, you might see that they are of a different size than the ones you've had before, so you have to adjust your figures.

These questions and the "back of the envelope" figuration used to solve them are based on Dr. Fermi's famous predilection for estimating things to see if a more rigorous calculation made sense, or just to find something out. In an anecdote I've read a number of times, the first atomic bomb test was attended by Dr. Fermi and others who were in an open-topped trench a mile away or so. Fermi spent his time prior to the blast by tearing a sheet of notebook paper into small pieces. When the blast went off (everyone was told to duck down just beforehand), he counted off a few seconds and threw the bits above the trench just as the shock wave passed over. Then he climbed up and observed the scattered paper bits, did a quick calculation, and announced the yield of the blast, how many kilotons it was (around ten as I recall).

Further from such shades of Armageddon, this sort of thing runs in my family. I remember we learned as kids to carry a small ruler on trips. We make lots of long car journeys, multi-state affairs, and with four boys in the car, my parents needed lots of creativity to keep us from going nuts. From time to time, Dad would say, "How far off is that bridge?" In our younger years, I (the oldest) and one or two of the younger ones (whoever was paying attention) would guess and guess. Dad would have noted the "question spot" on the odometer, so when we came to the bridge, he'd tell us the distance. It was typically a mile or two. Eventually we learned to hold up the ruler at arm's length and measure the apparent width of one lane (or measure all the lanes and divide), then reason like this:
  • I could hold a ruler 2 feet (24 inches) from my eye (it varied with my age).
  • A car lane on a highway is 20 feet wide.
  • Suppose I measure 1/4 inch. The ratio of 1/4 to 24 is 96.
  • 20 x 96 is just under 2,000...OK, it is 1,920 feet.
Of course, if the bridge is 2 miles away, that's 10,560 feet; divided by 20 is 528. 24 divided by 528 is 1/22 of an inch, which is why we needed to measure a 4-lane bridge to get any accuracy; its apparent width would be 4/22 or 0.18 inch. It is just too hard to "eyeball" a 22d of an inch!

Then one day, Dad asked, "How much water do you think is in that pond we're about to pass?" That is a story for another day. But it is just the kind of story with which Dr. Graham Tattersall opens Geekspeak: How Life + Mathematics = Happiness. With a background like mine, I found this one of the most enjoyable books I've read in a long time. The author's Dad was a lot like mine, but asked a broader range of questions, a tradition which he follows in asking, and answering, 26 interesting questions, and following each with a shorter, related one in a Geek Speak section.

For example, the first chapter introduces sampling statistics by asking the reader to open a dictionary to a few random pages, to first count all the defined words you know the meaning of without excessive memory-bludgeoning, then to count all the words defined on those pages. Suppose you know 60% of the words on a total of five pages. Somewhere in the introduction you'll find the "population" of the dictionary. The American Heritage Dictionary I just grabbed proudly states on the cover: "55,000 entries". If my "hit rate" in this test were 60%, then I could state that my total vocabulary is about 33,000 words.

(However, in this case it is 100%, because I've read the entire dictionary and used it as the basis of a computerized spelling dictionary of my own. I also added words found by scanning an old Funk & Wagnalls I have on hand. I'd have to go to the Oxford English Dictionary (OED) or one of the large "unabridged" volumes to find one in which I don't know all the words.)

Anyway, the "Speak Geek" tidbit that follows this chapter compares some statistics about word lengths found in books by Jane Austen and Ian McEwan. The general feeling many people have that Austen uses lots of long words is borne out: her average word length is ten letters, and there are plenty of 15- to 20-letter stumpers to be found in her writing. McEwan, by contrast, averages seven letters, and very few exceed 15 letters.

In an chapter about the "surprise content" of a message, he estimates the information content of the message "you are to die in a moment". Most of us would find that the surprise of our lives. Considering that the chances of a person under, say forty years of age, dying today, is one in a few million, the message is a one-in-several-million chance. For an older person, the odds change considerably, until you get to ages like ninety, at which the odds are one in a few thousand, or worse.

OK, compare that with the odds of winning one of the bigger lotteries. The known odds of being the sole winner of a PowerBall lottery in the Eastern U.S. are one in 81 million. If you are exactly forty, your chances of dying in the coming week are one in three million per day, divided by seven days, or one in 420,000. Rounding a bit, 80 million divided by 400 thousand is 200. You are 200 times more likely to die than win at PowerBall...unless you buy 200 tickets!

The "Speak Geek" section after this chapter notes that the text of the Holy Bible takes up just 1/150th of a data CD. There are, I happen to know, 26 commonly-used English translations of the Bible. A CD can store them all in less than one-fifth of its area. You might want that CD in case you'd prefer something better than the lottery to spend your money on.

The mathematical figuration is not confined to strictly numerical pursuits. The author analyzes lonely-hearts ads, compares the words used by those that seem to enjoy the most success, and comes up with the ultimate dating ad:
Tall, attractive professional fox, 49, intelligent, sociable and arts-loving, wishes to meet compatible vixen for friendship and relationship.
Geeks aren't only about numbers, you know.

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