Friday, April 20, 2007

*** Coincidences, Chaos and All that Math Jazz by Edward Burger

An entertaining book on complicated topics that most of us slept through during high school and college. The authors do a great job of making the mathematically abstract graspable by our limited brains. Perhaps you'll gain a new appreciation of math from this book. Too bad these guys weren't around to teach us math.

Next time you find yourself in a room with at least 45 people, brag about your power to predict amazing coincidences and announce that in this small group at least 2 people share the same birthday…Why is that you can be so confident?... It turns out that there is a 95% chance of a birthday overlap… To compute the chance that all 45 don’t have the same birthday, we multiply together the individual chances of 365/366 x 364/366 x… 323/366 x 322/366. Notice that each of these numbers is a fraction less than 1, and if we multiply a list of fractions less than 1, the product is extremely small… Roughly speaking the average of these fractions is around .935, so an estimate of this product is .935 to the 44th power, which is approximately 5%. So there’s about a 5% chance that all the birthdays are different. P17-8

On average, what percentage of Americans have 1 testicle and 1 ovary?
Surprise, the average American, has 1 of each since ½ the population has either 2 testicles or 2 ovaries. The average of this population is 1 of each. P 43

Over the past decade, on average, there have been 183 deaths per year in commercial airline accidents (in the US), which is about 1 death every 2 days or one death per 3.4 billion passenger air miles… Auto deaths are 34 times more frequent per passenger air mile. P56 I have to disagree with this reasoning. There’s a major fallacy in this argument. Let’s normalize these statistics by looking at vehicle operation by hour instead of passenger air mile Let’s say that the average plane carries about 110 people, and that average car carries 1.1 people (most trips are solo). Let’s also say that the avg speed of the auto trip is only 35 miles per hour, while the plane’s avg speed is 350 mph. And assuming that if the plane has a fatal crash, only ½ of the passengers die and if the car has a fatal accident, all 1.1 passengers die. So to convert the bogus statistic of fatalities per passenger air miles to vehicle hours, we must factor that plane goes 10 times as fast, and has 100 times as many passengers multiplied by 1/2 for the passengers that die in the plane crash, so each hour it’s passenger air mile figure is 500 times higher than the car. So the fatality rate per hourly vehicle operation is actually 34/500 for the car. Each hour in the car is actually 16 times safer than an hour in the plane. Hour for hour planes are 16 times deadlier – not safer. If you had a 16 hour drive, and the plane trip would cover that distance in 1 hour, it is a push. Shorter distances should be driven, not flown!

Given that there are about 500,000 HIV positive people in the US population (280M), what is the probability that a person who has a positive test result is actually HIV positive when the test is 99% accurate – that is a false positive or negative occurs only 1% of the time? Again if you tested positive, would you guess that you’re odds of actually HIV were 99%? Well, you’d be relieved that your actual chance is only a meager 17.7%! Here’s why…
Number of people who don’t have HIV= 280M -500K=279.5M
Of the 500K HIV victims, 1% or 5000 will test negative, and 495K will test positive.
Of the 279.5M who are negative, 1% will test positive, or a whopping 2.795M.
The total number of positive cases will be 2.8M. Of these only 495K actually the disease, The odds having HIV after the test are therefore 495k/2.8M or 17.7%. p59

Take an ordinary piece of copier paper. Fold it in half. Now do that again, and again. You’ll probably have to stop after 7 folds. But let’s imagine if you could keep folding it roughly 50 times. How thick is that folded sheet? An inch? A foot? A yard?
The folded paper you imagined would extend past sun and would be over 128M miles!... Here’s how. After 9 folds you’re sheet would have 512 ( 2 to the 9th ) layers. 500 is the number of sheets in a 2 inch thick ream. Bear in mind that each fold doubles the size. The 10th fold now goes to 4 inches. The 11th, to 8 inches. The 15th becomes 11 feet. The 20 is 350 feet. If you keep doing this 51 times, you’ll get to 128,000,000 miles. P87

Offer someone million dollars in $1 bills on this condition. That they alone – without mechanical aid – must carry them away all at once. Are you at serious risk of losing your money, even if you offer it to Arnold Schwarzenegger? … Remember our ream of paper which had 500 sheets? Well each ream weighs about 4 pounds. And each sheet could fit 5 $1 bills. So a 4 pound ream equates to $2500. $1M divided by $2500 equals 400. So we’d need 400 reams of paper each weighing 4 pounds each. The total would be 1600 pounds! No one is going anywhere with your dough! P83

I can’t go into the entire details of this section, but you have to read pages 90-99 to see how it is possible to stack cards without glue or tape so that they can be extend beyond the edge of a table without falling. Yes, you can stack 4 cards, and have the last card completely extend beyond the table. But we’re talking about extending the cards 1 mile beyond the edge of the table, and then having you sit on the last card, and still not having the cards fall off. It sounds impossible… but read the section if you have doubts (I have my doubts about you if you don’t doubt this!). It is possible.

Say we throw 10 ping pong balls into a barrel; then we pull one out. Now we throw 10 more in and pull only 1 out. We all agree that there are now 18 balls in the barrel. Do it again, and we are left with 27; again, and we have 36; again, and see 45; and so forth. What if you did this forever (infinitely). How many balls would be in the barrel? Did you say infinitely many? Well couldn’t be more wrong I’m afraid. The barrel would be empty… How the heck can that be? To help us understand this, let’s modify the scenario just a touch. Suppose we start with all of the balls in the barrel instead, and they are all numbered. Let’s say you remove a ball each time you have ½ of your remaining time. So if you have a minute. The first one is removed, then you wait 30s, and remove the 2nd. 15s later, the 3rd. 7.5s after that the 4th. Pretty soon you’ll be moving faster than the speed of sound. And soon after that, faster than the speed of light. We’ll ignore these limits, since this is the world of pure math – not the real universe. When the 60s are done, and infinitely many ½ life intervals are performed, the barrel will be empty. Our intuition on comparative sizes of collections – especially infinite ones – must be based soley on 1 to 1 pairings. Here each halftime interval is paired with a numbered ball. The first ball with the 1st halftime. The 37th ball with the 37th halftime. P243 The fact that we put in 10 balls when we take one out is just a trick to throw you off. When you deal with infinity, nothing is intuitive anymore. Because you’ll say when you get to the next to last ball, you’ll have to put in 10 more, so how can the number ever go down? Well just stop thinking like that OK!? That’s not how infinity works. But is how your finite brain works I’m afraid.


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