Mathematical Musings Part II

Things I've said in class, part II:

• If you rewrite the problem first, you only have two embedded functions, rather than three, which will give you a shorter chain, but it’s not like we’re trying to lock up our bicycle to a pole, so we really don’t need a long chain. (on use of the chain rule.
• Since I went to A&M for two years, I feel like I’ve earned the right to tell Aggie jokes, although I don’t get half of them.
• Is it going to rain today? I don’t know. That all depends on the weather.
• Sin and Cosine are considered to model Harmonic motion. Think about the two graphs on the same axis, offset from each other by Pi halves. They look like two playful dolphins harmoniously frolicking in the peaceful sea. . . . or they don’t.
• So the graph of tangent x reveals that it is monotonic increasing, but is it one-to-one? Heck no. Not only is each y-value two-timing with x-values, but three-timing, four-timing, infinity-timing. She’s seeing so many different x-values, she has developed quite a reputation.
• I don’t know where my Hall pass is. Here, take this rubber band. If anyone asks, say it belongs to Mr. Korpi.
• Now we have to shift the graph of cosine to the right by 1.333 seconds. If we had happened up on the system 1.333 seconds earlier, the phase shift would have been zero, and the equation would have been easier to find. The moral of the story: DON’T BE TARDY.
• It’s my mousetrap. (when a student asked what an old computer mouse tied into a loop laying on the floor was.)
• So the problem says, “ A person of weight W is standing on an oak board on the incline.” Does any body in here weigh W?
• To a topologist, a coffee mug, a doughnut, and a trombone are the same thing.
• I’m here every morning at 6:00 if you need to come in for help. I love the smell of mathematics in the morning.
• Now for this problem, the information is hidden within the words. We have to make our own juice here, and ‘squeeze’ the information out.
• So how are we going to tell our calculator to do that? Do we speak into it like a cell-phone?
• You need to be wise in the use of your calculator. Don’t use it as a hammer. It will work, but is greatly missing the true powerful potential and purpose of its original design.
• Do it in your head! What if you were stranded on an island with only your textbook, without a calculator and pencil.
• You there! No mental loitering.
• We can classify 1^st degree polynomials as straight curves. These are very simple to graph, very hard to throw, and even harder to hit.
• A frown is nothing more than a parabola turned upside-down.
• The Ambiguous case is for Side Side Angle. Since it requires the most work, requiring you to set up a quadratic formula, it has become known as the ‘pain in the Angle Side Side’ case.
• The first thing you do when you approach a problem like this is to introduce yourself. Get comfortable with the task at hand. Say, “Hello problem.” It’s a good way to break the ice.
• Yes, you may go to the bathroom. Just don’t do it here.
• P=e^(rt). The shampoo equation. Used by more bankers than the other lesser equations.
• We count in base 10 because our number system evolved from counting on our fingers. This is the same reason why chickens count in base four.
• I really like the new windows in my classroom. I have a great view of the neighboring apartments. Last year I never had the opportunity to watch so many people take their dogs for a poop.
• And we give this ratio a very specific name. Not a name like Fred, George, or Biff, but rather a choice four-letter word starting with ‘S.’ We call it Sine.
• Some days you’re the Radical, some day you’re the Radicand.
• Some days, especially when it’s overcastted, I just like to wear a little Vitamin C. (defending his bright yellow shirt to students.
• Was that ironic, oxymoronic, or just moronic?
• It is easier to adverbialize an adjective than it is the adjectivize an adverb.
• Just because you haven’t met anyone like me before doesn’t mean that I don’t exist.
• As scary as reading your Halloween stories in the dark may be, any terror is negated by how silly the idea of reading them with a flashlight is.
• Now for number diez y dos, also known as doce.
• The test will be similar to the review . . . Not congruent.
• Better late than never or early or on time.
• I’m not really that odd. My eccentricity is closer to zero than one.
• When I was in school, we used to separate all multiple items in a list with commas. For instance, the sentence, “Tom, Dick, and Harry went to the store,” had commas everywhere. Now a days, you don’t need a comma between Dick and Harry. That just doesn’t look right to me. Without the comma, it just looks like Dick and Harry are a single unit.-----(pause, then biggest laughter of the year.)
• I don’t know when you’ll ever use this stuff, but if you never learn it, the answer is never.
• Sure, Einstein failed high school math, but it wasn’t because he failed to understand math.
• An integral part of Calculus is differentiating between the derivative and the integral.
• Negative numbers are so depressing. Be positive, I always say.
• The difference between implicit and explicit is this. If I said, “Gee, you know, I kind of wish that the grades were better on the test. I really think y’all could have done better and I expect more from you. It really want the grades better . . . “ , that would be implicit. But if I said, “STUDY, DAMMIT.” That would be explicit.

Dan the Man does it again!

Yesterday, I talked about Daniel Bernoulli's breakthrough idea called Diminishing Marginal Returns. It is helpful to read it before reading on. Today we'll extend the idea to Bernoulli's idea of quantifiable risk management.

In the NFL, a Fair Game is defined as one that does not involve the New England Patriots and illicit video cameras. In probability theory, a fair game is defined in more general terms: the expected outcomes are the same for each participant. For instance , to start and NFL game, the referee flips a "fair" coin to determine who gets the option of possession or choice of field. Presumably, each time is just as likely to win the coin toss, as the probability of each option, Heads or Tails, is the same: 1/2.

If we assume Daniel Bernoulli's assumption of Diminishing Marginal Utility, an individual who defined his utility of wealth according the the natural logarithm will actually REJECT a fair game. Bernoulli, himself used the following example to illustrate his point.

Suppose two individuals with $100 (or 100 Ducats, as Bernoulli actually used), each offer up $50 to participate in a fair game. What was this intriguing game which cost $50 to play? A simple coin toss, winner takes all money in the pot, $100. So now, each player has a 50-50 chance of walking away with only $50 if they lose, or $150 if they win. Got it?

Now we need to know a little about what Expected Value is and how to calculate it. Expected value is one of the most important concepts in probability theory. It is the sum of the probability of each possible outcome in a situation multiplied by the outcome value (or payoff.) We interpret it as the amount of money one would "expect" to end up with as the result of a fair game that is played many times. This important number can help an individual decide if he wants to play the game at all.

Back to our $50 roll of the dice. The Expected Value of this game is $100, and is calculated as:

$50 x 0.50 + $150 x 0.50 = $100

Now since each player started out with $100 to begin with, they can only hope to "break even" in the long run if they play this game many, many times. As a result, this is also the expected value of NOT playing the game, so it would not make sense for them to play the game, unless of course the just like the activity itself.

But what is the Expected UTILITY of the game? We find it by adding the utility value (the natural log value) of the $50 payoff times the probability of it occurring to the utility value of the $150 payoff time the probability of IT occurring. The table bellow shows the results.

The utility of NOT playing the game is the natural log of $100, or ln(100) = 4.605

So, although the Expected value of playing or not playing is the same, the Expected utility is actually higher for NOT playing (4.605) than it is for playing (4.461). This means the player, according the Bernoulli and irrefutable math, will reject the game even though it is fair. Stated more intuitively, since he only has the chance of breaking even in the long run, it is not worth the time or effort to play the game. This is the same reason someone would reject digging a giant hole in the ground all morning with a shovel only to fill it back up with the same dirt later that afternoon. The blisters just aren't worth it unless we planted a tree, or buried treasure, or struck oil.

This fine example of opting out of game means that we, as humans, have an aversion to risk. It is a nice innate safety feature, if you think about it. In fact, Bernoulli himself called this risk aversion as "nature's admonition to avoid the dice."

Of course, one can still play with dice without any risk at all--"Yahtzee!"

Ducats Matter!!? Yes, but less and less . . .

Things today are more different than they have ever been. Today's point-and-click information age is in sharp contrast to life, let's say, during the Black Death of the 14th century. In fact, it has been claimed that the Daily New York Times Newspaper contains more information in it than the average individual living in the 18th century came across in a lifetime.

We has seen three major cultural shifts in our recent history. The agricultural, industrial, and technological revolutions have guaranteed one thing: we have more choices now than our ancestors did. For much of recorded history, people did what their parents did: Millers milled, Tanners tanned, Smiths smithed, Cartwrights Cartwrighted, and Mathers mathed. Life was dictated by social structures and physical geography. Few people had the desire or resources to rise about their caste or to travel long distances over rough terrain without the latest Nike "All Conditions Gear." In essence, beyond deciding to skin a rodent before eating it or just eating it with the fur, very little was left to personal liberty. But today, we are free to choose. We are filled with choices, dilemmas, conundrums that our ancestors never had to face. Today, there are a bewildering array of options that we have to choose from. In fact, I panic each time I enter a Starbuck's or Subway. Order a simple "coffee" or a "sub sandwich" and people look at you like you're from a rodent-eating era." Should it be paper or plastic? Plastic or cash? Nike or Reebok? Pizza Hut or Mr. Gatti's? Pepperoni or Sausage? Math or Rotting Cabbage? Essentially, our destiny is in our own hands. Social structures have become semi-permeable membranes through which we can waver between at will. Physical geography is now a course we take in college, not something we blame our pitiful future on. But with all these choices, how do we decide??

The answer, thanks to the Mathers of Olde, lies in mathematics. In 1738, a Dutch polymath, Daniel Bernoulli, son of Johann Bernoulli, and a member of the most prodigious math clan of all time (8 Bernoullis made significant contributions to mathematics, which is almost as many people as MY family has made to Health Science . . . as subjects) came up with the answer.

Bernoulli stated that the wisdom, W, of any decision could be calculated by multiplying the probability that the decision will give us what we want, D, by the utility, or usefulness of getting what we want, U.

To him, decision making was a quantifiable experience. Everything was a calculated risk:

W = DU

The first variable, D, is pretty easy to calculate: How likely is it that I will get an A on the math test if I don't study? What are my odds of getting a good job if I'm lazy and stupid? Would I remember to breathe if I was in control of my autonomic nervous system? With a little effort, realistic foresight, and rich parents, we can pretty much figure out how our efforts will actually result in giving us what we desire.

The downside to this, however, is that we cannot accurately determine how we will feel once we get what we desire. Sure, my rich dad got me this "cush" job, but now I'm realizing that I never had to get dressed up in a suit to play video games all day before, and why do I have to play them in this giant, lonely private office. The feeling of novelty usually fades once we have achieved what we desired. The honeymoon will end eventually, usually very quickly. So how can we predict how we will actually feel in the future after our decision is made? Bernoulli reasoned that when projecting the future, our imagination fills in the gaps of future possibilities by substituting realities from the present. In other words, what we objectively want, is not what we subjectively experience. We are quite good at fooling ourselves.

Bernoulli's brilliance lies, not in his mathematics, but in his psychology of the human mind. We all want more once we have what we have. Wealth may be measured in dollars, but utility is measured in how much satisfaction, pleasure, or comfort those dollars can buy. In this sense, wealth does not matter. What is important is utility: how much pleasure can I get out of the things that money can buy.

This is why the beggar is no more happy than the millionaire. This is why King Solomon was not better off than his servant. When asked how much money is enough, billionaire J.P. Morgan stated, "just a little bit more." To make wise choices, we must optimize our pleasure. Bernoulli came up with a quantifiable way to convert dollars to utility--Brillitant! His idea was called "Diminishing Marginal Utility." He thought that each successive dollar provides less pleasure than the one before it; therefore, an individual (who has spent some of his money on someone to do his calculations) can simply (let's say by a flat fee) calculate his utility of any given dollar by adjusting the amount of dollars he has already spent.

He wrote:
the determination of the value of an item must not be based on its price, but rather on the utility it yields. The price of the item is dependent only on the thing itself and is equal for everyone; the utility, however, is dependent on the particular circumstances of the person making the estimate. Thus there is no doubt that a gain of one thousand ducats is more significant to a pauper than to a rich man though both gain the same amount. The utility resulting from any small increase in wealth will be inversely proportionate to the quantity of goods previously possessed.

So the thousand DUCATS has more value, or more utility to the pauper than the rich man. But is he any more happy?

We can graph utility against wealth, with utility on a vertical axis and wealth on a horizontal axis. This graph would rise from left to right, meaning that utility increases with wealth. This translates to the indisputable idea that we like more money than less money. But the graph increases at a DECREASING rate, which is why the curve is concave down. In calculus, Bernoulli was stating that the first derivative is positive while the second derivative is negative.

Bernoulli precisely stated that a change in utility with respect to wealth equals the natural logarithm of the sum of initial wealth plus an increment of wealth divided by initial wealth. For instance, let's say you have $100. What is the utility in gaining another $20? According to Bernoulli, it would be:

ln((100+20)/100) = 0.182

So now you have $120. The next gain of $20 will be worth LESS to you:

ln((120+20)/120) = 0.154

This same gain of $20 is almost insignificant to you if you start with a large amount, say $1 million:

ln((1000000+20)/1000000) = 0.0000199998

At this alarming diminished utility value, it might not even be worth your effort to bend over and pick up a $20 bill if you found it on the ground. In this case, it's probably best to leave it there for the pauper.

Today, we look at Bernoulli's idea and say, "Duuuuuh! Of course we like more money." WE also realize that as we spend more and more time eating at an all-you-can-eat dinner buffet, that we will begin to feel increasingly less satisfied at each return trip to load our plates. His ideas, especially the his ability to reduce his ideas to calculations is what is remarkable, for they have profound implications in the theory of risk management, as I will show you tomorrow.

Until then, enjoy choosing your choices.

Got Beer?!?

The beginning of November signifies not only the end of daylight savings time, but the beginning of the holiday "-fests" that spring up around town. This week, thousands of men dust of their Lederhosen, put on their thick hiking socks, and venture down to the local Wurstfest, a 10-day festival that celebrates sausage, or "wurst" in German.

The festival includes something for all.

For the lovers of dance, there are three main areas where one can polka dance 'til they puke: the Wursthalle, the Grosse zelt (big tent), und the Kleine zelt (diminutive tent.) The most recognizable song (and by far the easiest to "dance" to) is the the "Chicken Dance," which can be heard playing in at least one tent at any given time. All the other songs are Polka Dances that basically all sound the same: a blaring trumpet, and accordion, and a giant tuba. In fact, the songs sound just like Tejano music, except with much sillier hats. But what if you don't know how to polka? Or you say you're too chicken even to dance the jig named after you? Not a problem, because there's plenty of . . . . beer! Hurray liquid courage!

And LOTS of beer there is. Under the guise of a "sausage festival," the true celebrant of Wurstfest is overpriced ale, currently going for $4/12 oz cup, or $6/12 oz. for an import beer. Macho, proud, broke men stumble through the grounds carrying towering stacks of empty cups, signifying how many 12 ounce "shots" they have already consumed and how much money they've siphoned through their kidneys. Some individuals herald their total lack of self control by parading around with a pitcher in their hands, drinking directly from it, as they walk back and forth between the beer booth and the bathroom. These big drinkers try desperately to conceal the true level of their drunkenness beneath their absurd "rubber chicken" hat. Good think silly hats aren't against the law, or many people would get arrested for public ridiculation.

If food is more your style, then there's plenty to be consumed at Wurstfest. Aside from the obvious sausage (including many different varieties), you can get pretty much anything you want as long as you are willing to pay too much for it and eat it off of a stick. The most popular food item is the Kartuffel-Puffers, or potato pancakes. So many of these things are produced that they mix the batter in washing machines (but at least you know they are using clean batter.) These fried treats are like giant hash browns served with, what else?!? . . . a small side of apple sauce of course!. Other delicacies include pork chops on-a-stick, turkey legs off-a-stick-but-on-a-bone, and plenty of sauerkraut. But after a few beers, it all begins to taste the same--greasy and vinegary. The only solution to the heartburn is do douse the fire with more beer.

If you like crafts, there is a craft booth where you can buy something that was crocheted by someone. Although no one would ever waste valuable beer money on an commemorative afghan, the booth is an interesting booth to meander by while you are gorging yourself on sausage and drowning yourself with beer. I don't know how many times I've said with a distended stomach, "OOOOOOOh, Niiice Doiley!. Does it come on a sticK?"

Got kids? Bring them along--it's never to late to set a good example for them as to why they should never drink! Besides, who else is going to eat the apple sauce that comes with your potato pancake? Seriously, though, they DO have many rides for kids, including a giant ferris wheel and the carousel. Most kids are ready for this kind of entertainment, as it has been 4 incredibly long weeks since the county fair left town. Kids are also another good set of hands when you have too many beers to carry, and they can save your place at the limited tables when you get up to polka dance to that waltz.

If you like rubbing elbows, then Wursfest is definitely the place for you. Virtually anyone who's anyone makes a showing. People come from miles and miles for the big event. It often gets so crowded that rubbing elbows becomes more than just an expression as people pack in like sardines throughout the grounds, doubly so in the restrooms. This is helpful in keeping all the drunk people in an upright position. The event is run by the "Opas," who are the official elected elite of New Braunfels. These little "Opas" run around in their green and red vests greeting the patrons, thanking them for their money, before they retire to their private drinking quarters.

Yep, Wurstfest is fun for the whole family. It's a great excuse to get dressed up in clothing you wouldn't even wear at Halloween, overindulge in unhealthy food, drink more beer than you'll ever remember, dance to the sound of blaring trumpets, and blow a whole week's paycheck.