Posts Tagged ‘MATH AND MIKE’

NO, IT CAN’T BE ANYTHING

March 18, 2019

A panda walks into a bar and eats shoots and leaves.  Lynne Truss’ book with that title showed how punctuation matters in a sentence.  In both instances, the panda had a meal.  What isn’t clear is whether the meal was plant based or whether a firearm was involved.

Punctuation matters.  Words do, too. They matter greatly in science, where miscommunications occur with the public with common words.  The word “theory” in general usage means a guess.  In science, a theory is a statement of what one believes based on a compilation of facts. Gravity is a theory.  So is relativity.  So is evolution.  Our understanding may be incomplete, but we are hardly guessing at what is occurring, and a great deal of our daily lives are made easier because of theories. Newtonian mechanics got us to the Moon, but we need Einstein’s relativity to calculate Mercury’s orbit accurately.

Two or more sides to a story don’t mean all sides have equal weight. They do on a die, but not the sum on a pair of dice. The numbers 1-6 come up with equal probability for a die.  There are 11 possibilities with the sum of two dice, but the probabilities are very different for each, from 1/36 for 2 (or 12) to 1/6 for 7.

There is uncertainty in scientific results.  Unfortunately, the lay public views “uncertainty” differently.  In general usage means one isn’t sure and in fact may be guessing.  Malpractice lawyers love to misuse these words, “Were you uncertain?”  If one answers “A little,” then the next comment may be, “So, you really didn’t know what was going on, did you?” putting words in one’s mouth and treating the uncertainty of a diagnosis as a character flaw and a substandard physician.  I’ve been there. When I practiced neurology, I had many instances where I was uncertain of the diagnosis, and frequently the patients, through having been told by someone else or not listening to me, felt that I had no idea what was going on.  Neurology is one of the most difficult specialties in all of medicine, but I was usually considering several diagnoses.  Also, the fact that I could not cure a person with a severe brain injury didn’t mean I was uncertain of what was going on.

We demand temperature predictions to the nearest degree and rainfall’s beginning to the nearest minute despite inability to correctly predict these regularly.  A temperature range would be a far better forecast.

Uncertainty in science is vastly different from how the public perceives it, and it is one reason many phenomena with a high degree of confidence (another important word) are not believed, because of such uncertainty: “they really don’t know for sure.” The difference is that uncertainty is usually quantified in science.  If we say we are 95% confident of a result, that means if we ran one hundred simulations or saw this particular phenomena one hundred times, 95 of them would contain the value we were measuring.  We wouldn’t know which 95, but it is far from the “anything can happen,” approach, and it doesn’t mean that 5% of the time we don’t have a clue.  Consider “95% certain there is a fracture in your hand,” a probability, which when studied was far less.   It doesn’t mean that there is a 95% probability the interval is right; it either is or it isn’t, and that makes no probabilistic sence.

If one tosses a fair coin four times, one would expect it to come up heads twice.  This is the expected value, 50% probability of heads each time*4=2.  But a priori, we are uncertain. It may come up heads all four times with probability 6.25%, one-half multiplied by itself four times.  Or, it may come up three heads 1/4 of the time, two heads 3/8 of the time, one head 1/4 of the time, and no heads 1/16 of the time.

If somebody told me I would have to pay them a dollar for every time exactly two heads occurred, because that is the expected value, and I would have to pay them a dollar every time it came up some other number, I would take that bet in a heartbeat.  Am I certain of winning?  No, but the probability—future oriented—of my winning is 62.5%, and that is solid. I am uncertain what will exactly happen, but I am highly certain what the probabilities are and my expected gain. Casinos don’t take money from everybody; they occasionally lose big, but over time, they win, and furthermore, they have a very good idea of the range of their winnings.

With 10 coin tosses, there is a 1.1% probability that there will be 9 or 10 heads.  The expected number, 5, has slightly less than a quarter probability of occurring, no longer 3/8.  Notice that extreme events still occur but with much lower probability with a few more attempts.

Toss a coin 20 times and the likelihood of 90% heads or more is on the order of 1 in 5000, not 4.5%, and the probability of 50%, or 10 heads, is less, about 1 in 6.  The likelihood of exactly half, the expected value, diminishes, but the variability decreases much faster, and more and more of the outcomes cluster closely around 50%, even if they are not 50% exactly.

It’s like weather and climate.  There are many who say if we can’t predict the weather accurately, how can we possibly predict climate?  It’s because climate is made up of many weather events over a long period of time, where exact averages are not likely to occur very often, but the variability around those averages is much less.  Indeed, extreme values will be far less likely unless the system itself changes.  The issue for science is to try to predict as accurately as possible, but science recognizes that there is always a certain degree of uncertainty—not that we have no idea what is going on, but exact predictions of many phenomena may be impossible. Instead, there is an interval, the “plus or minus,” stating the range where the true value of the parameter of concern is believed to lie.  We will never know that true, exact value, but we are very confident in its interval.

Uncertainty doesn’t mean “it can be anything.”  No, 100 consecutive heads cannot occur with any sensible probability. Indeed, even 75 or more heads has probability 0.0000002, the likelihood of guessing a second chosen at random in the past two months.  It’s only about a 1 in 6 chance there will be 55 or more heads.

I have long argued in climate scenarios that those who believe there is no significant global warming occurring must offer a confidence interval of what they think the temperature will be in 10, 50, or 100 years.  The interval would be expected to contain zero, no change.  It is not enough to say the current data are wrong. What is the margin of error?  What is the confidence?  It can’t be 100%, for that would be saying one could look at thousands of variables and know exactly how they would behave.

Uncertainty is reality. We embrace it in science, do not consider it a sign of weakness but a strong statement of “we could be wrong, but this is how wrong we can reasonably expect to be.”

BLESSED NERD

January 13, 2018

I didn’t know there was a “nerd” icon, but I sure recognized it on my post.  I wasn’t surprised.  I’m blessed, really.  Blessed that I can see not only the beauty in nature that others see, but additionally another way, too, that most others don’t. I see it in understanding what is happening and why it is happening.

The post was a picture I had taken from the top of Spencer Butte showing the clouds rising from the valley floor.  A week prior, we had an inversion, where a cold air mass filled the valley floor, and as one ascended, it became warmer, not colder.  The normal pattern is cooling with height, as anybody knows who has traveled into the mountains on a hot summer day.  I took a picture of the scene below, then I googled the Salem weather sounding, which was the closest sounding to me.  It’s easy to find these things online for those who are curious.  I just typed in uwyo sounding, and two taps later, a map of the US appeared, with a bunch of three letters all over the US, airport call signs for various cities.  People know many if they fly regularly.  Salem (SLE) is one of two in Oregon; the other is MFD (Medford).

Salem’s temperature was about 7 C (45 F) at the valley floor, and it became progressively colder up to the freezing level of about 1900 m (6000 feet), a normal pattern, although I didn’t bother to look at the “Lifted Index,” which is a description of how strong the tendency is for warm air to rise.  We can determine that, too.  A week earlier, Salem was 0 C, and at 800 m or 2600 feet, it was 13 C or 55 F.  That’s a classic inversion.  I posted the picture and the weather sounding.

Fog layer in Springfield, Oregon with smoke rising and then reaching warm layer where the temperature of the smoke is less than the temperature of the layer, and it can no longer rise any further. Mt. Jefferson in the distance.

It earned me “nerd of the day,” to which I simply say, “I’m blessed to find things fascinating that are lost on nearly everybody else.”  The individual who placed the icon knows I am a weather junkie but has never expressed any interest in much more detailed forecasts than he gets from his Weather Channel app, which he broadcasts to everybody near him.  It’s taken me a while, but I now just stay silent.  He’s not interested in weather models or much else I say.  Seeing a Rex Block (a high pressure system north of a low pressure system, which blocks normal flow of west to east air) or an Omega Block, and knowing the weather is going to be very unchangeable days before it is announced, is interesting.  It’s also good practice to learn to curb my tongue.

Omega block over SW US. Low pressure systems force upper level winds northward, producing a stable high pressure system in the SW US.  Numbers represent the dekameters above sea level where half the atmosphere is above and half below.  Higher numbers mean higher pressure and more stable, dry, warm air.

Rex Block over the eastern Pacific.  High pressure (notice the barbs moving clockwise) is over Vancouver Island with low pressure (counterclockwise flow) is off the southern California coast.  The upper level winds from the Pacific are directed northward to SE Alaska and then turn southward and enter the US in Montana.  These last for several days and produce often stagnant weather.

I’ve had a lot of these moments.  I understand why solar eclipses occur, and indeed, I think the mathematics of an eclipse is every bit as beautiful as the eclipse itself.  Most would disagree, and I feel a little sorry for them, because I get to appreciate both the natural beauty and the mathematical music of the spheres.  The two interact; they are not mutually exclusive.  Before the Libyan eclipse in 2006, a senior editor at one of the astronomy magazines gave a talk about eclipses, not mentioning a word about the Saros Cycle. I asked him later, alone, why he didn’t bring it up.

“Nobody is going to be interested in that.”  Maybe he needed to make it interesting.  He was the editor, after all, not me.  Maybe nobody knew that such beauty existed.

In 2007, at Big Bend National Park in Texas, I was hiking on the South Rim Trail, when out near the edge of a steep cliff, 2500 feet above the valley floor, I looked ahead to see something that looked like smoke.  I got closer and realized it was water vapor, condensing, right in front of me, as south winds from the North American Monsoon brought moisture-rich air up against the walls, where the air was forced to rise and in doing so cooled and condensed into clouds (for the Lifted Index was negative and hot air was going to rise, not layer out) right in front of me.  This is called orographic lift.  I have seen orographic lift from a distance, watching cloud tops develop on mountains, eventually leading to thunderstorms, but I had never before seen it right in front of me.

I sent a picture to the Weather Channel, but this wasn’t a powerful storm, a great sunset, or any one of a number of non-nerdy things.  I never heard back.

Orographic Lift, Big Bend National Park, June 2007. The moist air is condensing right in front of me.

When I was a first year medical student, I was allowed to see a C-Section in a Denver hospital.  When asked afterwards what my impression was, I said it was interesting, and all I could think of were the enzymatic reactions that were closing the ductus arteriosus, the shunt between the pulmonary artery and the aorta, that needs to close so that de-oxygenated blood can go directly to the lungs for the baby’s initial breaths.  Knowing this stuff to me makes life more interesting.  I am able to appreciate both the sheer beauty of what I am seeing with the knowledge of knowing why it is.  Or, in the case of orographic lift, I find beauty where most would not.  That’s being blessed.

I don’t think too many amateur astronomers saw the Saturn-24 Sgr occultation in 1989. That’s nerdy stuff.  Saturn passed in front of a star (Saturn’s being closer to us, so it is possible), and as it did so, the star appeared to pass through Saturn’s rings.  That was remarkable.  From the Earth, with a moderate size telescope, I was treated to an hour long show of exactly how thick Saturn’s rings were, and believe me, they are very different for each layer.  Finally, the star was visible between the globe of Saturn and the rings, very odd appearing, before it gradually blinked out behind the globe, the gradual loss being proof of Saturn’s atmosphere. (When the Moon occults a star, it happens suddenly, because there is no lunar atmosphere).  This was a top 5 astronomical event for me, and I’ve spent a lot of time observing.

I get made fun off a lot, and when I taught, whether it was my being enthusiastic about the Rule of 72 for doubling time of money or population*, proving why the quadratic formula is what it is*^, understanding the age of a tree by its diameter**, the distance of an object if I know its height, or why the Sun sets earliest in early December rather than on the solstice, where the full Moon is going to rise*** and why or how to tell clock time using the Big Dipper.****

It’s a remarkable world around us, worth exploring, worth understanding, worth finding answers to the many questions we have about it. Nerds are blessed.  So there.

*Rule of 72: The doubling time of money in years is 72/interest rate in per cent.  9% rate doubles in 72/9 or 8 years.  It has to do with P=Poe^rt. P is twice Po so 2=e^rt.  ln both sides is ln 2=rt, so t=ln2/r, and if we use per cent, this is 69.3/r, close enough to 72, which is evenly divisible by 2,3,4,6,8,9,12,18,24, and 36.

*^ax^2+bx+c=0; x^2+(b/a)x=-(c/a); complete the square, x^2+(b/2a)x+b^2/4a^2=-(c/a)+(b^2/4a^2); [x+(b/2a)]^2=(1/2a)(b^2-4ac), and x=(1/2a)(-b+/- sqrt(b^2-4ac)

**For a Douglas fir, about 5 years per inch of diameter at breast height (DBH).

***Directly behind where the Sun set, basically.

****Let the pointer stars be the hour hand and Polaris the center.  Every two hours, the clock moves counterclockwise 1 hour.  Over a month, this changes, but for typical outdoor camping experiences, it works well.  A quarter turn is 6 hours, and American cowboys knew this and when it was time to relieve or be relieved. If one is Down Under, sorry!

OH WOW, I CAN’T BELIEVE I DID THAT. THANKS.

November 24, 2017

I’ve probably written about this before; if so, I apologize.  I happened to think about the incident when I awoke this morning.  I slept well, which is a blessing as I approach the last year of my sixties.  Most of us my age and older don’t sleep as well as we’d like: through the night without awakening, and awakening refreshed.  It’s a good night if I’m up once, fall back asleep, and awaken on my own, without one of the cats standing on me, yowling (that was this morning, at 4:47), or barfing up a hair ball.

Anyway, for some reason, a dream maybe, I awoke thinking of one day when I volunteered  in the calculus class at Sabino High School in Tucson.  I occasionally helped out there, if for some reason the college algebra or geometry classes where I usually worked didn’t require my presence.  I liked Dave, who taught calculus; indeed, years ago, when email was new, Dave was the only faculty member with one, which is how I contacted him and began to help.

Anyway, that particular day he began by discussing the behavior of the function y=x^2, a parabola, near where x=1 and y=1.  Specifically, he was looking at the rate of change or the slope of the function as it approached the point.

What we learn by the slope is how fast the curve is changing at that point.  People get this concept wrong all the time.  If we hear the rate of growth of population is slowing, some people think the actual number is going down.  It isn’t.  It is still rising, not just as fast.  This is extremely important to know, and worth repeating: If the rate of increase is slowing, the actual number is still increasing, not falling.

Dave started by showing the slope where x=0.99 and y=0.99*0.99, or 0.9801.  The slope then was (1-0.9801)/(1-0.99), 1.99/0.01, or 1.99.  No problem. Then he let x=0.999 and, with a calculator, squared it, 0.999^2, which was 0.998001. The slope was now 0.001999/0.001, or 1.999.  He continued, saying that as x got closer and closer to 1, y would get closer and closer to 2, and the limit; that is, if we could take take x as 0.99999999— to an infinite number of 9s—and here he paused….

Notice the pattern for y—first is .81 then .9801, then .998001,then .99980001.

“You’d need a big calculator to calculate y, but the slope would be 2 at the limit.”  he stated.

I sort of blurted out without thinking, “You don’t need a calculator to get y.”  The words just appeared, I swear.  Everyone in the class turned towards me.

Dave looked at me, held the marker out in one hand, and said, “come up and write it down.”   He wasn’t at all angry.  We had known each other for several years at this point. As I walked up to the board, I asked, “How many 9s are there in your number?” I have bad astigmatism.

“Eight.”  So, I am multiplying 0.99999999 by 0.99999999.

Without a calculator.

“It is one less 9, followed by an 8, followed by the same number of 0s as you have nines, followed by 1.”  That would be 0.9999999800000001.  Take that, Texas Instruments.

I then turned to the class.  “Last year, I was here when this problem was discussed. You”—-I pointed to Dave—-“said that there was probably a pattern, and you were absolutely right.  I found it in a few minutes. The pattern is one fewer nine, an 8, the same number of 0s, and a one,” repeating myself.

Dave is good: he knows what he has taught, what a student should know, makes the student think and find answers to his questions, because they have all the information available.

I could have added that Dave, as a good teacher, didn’t take my coming to the board as showing him up; indeed, he knew that I was modeling exactly the behavior he wanted in his students.

I’ve been down that road before.  A dozen years earlier, I was in grad school getting my Masters in Statistics.  I had several professors.  One was absolutely brilliant, able to teach an entire difficult upper level graduate course in linear models without consulting notes.  There may have been one time where she made an error that another student caught, but that was it.  She was brilliant.  I’m sure I mentioned that in my post-course evaluation.

What I didn’t mention, because there were only 7 of us in the class, and I knew I would be identified, is that had she more patience with those students like me (it was the only B I got in grad school, and I worked hard to get it) who were not as brilliant, she would have been a life-changing teacher, the top of the heap.  The best teachers have patience with those who don’t have their skills.  Occasionally, I approach that.  Dave was there. So was my advisor.

By doing that, my advisor forced me to use the new tools I had learned, to make mistakes, to figure everything out, and learn that way.  It was painful, but it was learning.  It was education, and it worked.

I’ve never gotten to my advisor’s or Dave’s level: substitutes don’t have a close relationship in one meeting, and in my brief for-profit so-called university teaching experience, students wanted everything handed to them.  But when I tutor today in the advanced math room at the community college, I occasionally encounter material far beyond what I know.  Sometimes, I try to help anyway.  And as I go through the problem with the student, asking him or her at each step how they got there, there is often a pause.  The student suddenly says, “Oh, I see what I did wrong. Oh wow, I can’t believe I did that.  Thanks.”  And walks away.  I’m still wondering what the answer was.

It just dawned on me that maybe my advisor sometimes didn’t know the answer, either, at least when I asked the question.  But he knew me well enough to know that I was capable of finding it.

Oh wow.  I can’t believe I did that. Thanks.

TONS OF RAIN AND A TINY CHANGE OF pH

September 5, 2017

“Tons of rain” was the grossest underestimate I heard of the amount of water accumulating during Hurricane Harvey.  A ton of rain is not very much, roughly about 0.02 inches on a moderate sized roof of one house.

What about 11 trillion gallons of water?  The media used the number to say how much water fell on southeast Texas.  The problem is first, too many don’t know how big a trillion is and second, a trillion is not a term often used with water.

A trillion, 1,000,000,000,000, 1 x 10 ^12, is a term used to describe both the national debt and the Gross National Product.  It is roughly the number of days the Earth has existed.  It is about the number of seconds in 31,700 years.

The term we use to describe a lot of water here is an acre-foot (1 acre covered with one foot of water), and while perhaps archaic, it is useful.  An acre-foot of water is roughly 326,000 gallons, what an average family of 4 uses in a year.  Lake Shasta, the largest reservoir in California, has a capacity of 4.5 million acre feet.  Eight Shastas would have flooded Houston, or 11 trillion gallons of water would cover all of New York State a foot deep.  If the catchment area described were 10,000 sq miles, it would have covered it to a depth of 5 feet, basically what people needed to know and could see from the pictures.

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Per cent is a useful term, but often misleading.  I inadvertently misled people before the recent eclipse.  What I should have said in the talks I gave prior to the event was that there were two kinds of eclipses (there are more, but I will keep things simpler), partial and total.  They are very different experiences.  Had I said that, mentioning that where I lived was in the partial zone, I might have persuaded more people to go to see totality.

Instead, I and many others said the eclipse would be 99.4% (or 99%) and most people figured, reasonably enough, that they would see almost the full event from their house.  I have since had several people tell me that they wish they had gone to totality.  One poignant comment was that 0.6% made all the difference in the world, since it was still sunlight and not totality.  If the eclipse isn’t total, it is partial.  It may be a little darker, a little more of the Sun will be covered if it is a deeper partial, but it is not total.  Next time, if there is a next time for me, I won’t make that mistake.

Per cent shouldn’t be used when counts are a better measure.  I have said in 2001 that the per cent of domestic flights not hijacked was 99.999996%.  Counts matter, especially when the counts should be zero.  When I was medical director of a hospital, we had a surgeon operate on the wrong side of the head.  Actually, we had three wrong side cases that I knew about—one was the wrong knee, and the other was the wrong side of the colon.  With the craniotomy, the OR head said that 99.9% of the time they did it right.  No, I retorted, we did 99.99% of them right, and that wasn’t the issue. There should be zero wrong side cases;  99.9% of landings done right means a plane crash every other day at O’Hare.

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Not every measurement is interval or ratio, meaning that the difference between 10 and 20 is not the same as the distance between 20 and 30.  Most of us realize that with temperature, that 110 is not twice as hot as 55.  That is because the Fahrenheit scale has an artificial zero that is 459 Fahrenheit degrees above absolute zero.  Therefore 55 degrees is 514 above absolute zero and 110 is 569 above absolute zero, a difference of 10.7%.  With Celsius, these numbers would be 273 above absolute zero, 13 and 43, respectively, making the temperatures 287 and 316, or 10.1%, really the same, given the rounding in the conversion from one to the other.  It’s important to recognize what is ratio data, meaning that multiples make sense, and what isn’t.  Money is ratio data, as are height and weight.  Others are ratio data, but they are used in ways where one has to be careful.  Height is ratio data, but the Body Mass index is a function (depends upon) the square of the height, or the height multiplied by itself.

This concept of squaring something is important in many areas, such as the energy of a moving body, which is proportional to or depends upon the square of the velocity.  With hurricanes, velocity of winds increasing from 100 mph to 120 mph, 20%, is a 44% increase in energy, (120/100)(120/100).   A car moving at 60 mph has four times the kinetic energy it had at 30 mph.

Cubing something is to the third power or multiplying it by itself 3 times.  While 1 yard is 3 feet, 1 cubic yard is 3*3*3 or 27 cubic feet.  A meter is almost 10% longer than a yard (9.4%), and a cubic meter is 31% more than a cubic yard.  Gravitational attraction is inversely proportional to the square of the distance between objects; tides are inversely proportional to the cube of the distance between objects, which is why the Moon, so much less massive than the Sun, is responsible for 45% of the tidal pull on the Earth.

Fourth power?  Yes.  The radiation from a star can be considered to be equal to the fourth power of the temperature, useful for determining the temperature of distant stars.  And closer to home, the damage large vehicles cause to roads is roughly equivalent to the fourth power of the load equivalent factor, having to do with axle number and weight.

Other relationships?  Yes, too.  the acidity of a liquid is the negative log of its hydrogen ion concentration (pH), which is a nice way to call 0.0000001 moles/liter of hydrogen ion a pH of 7.  Therefore, what seems like a minor fall in the ocean pH from 8.2 to 8.1 represents a 26% increase in acidity.

It’s not always the magnitude of a number that matters—99.99% is not always good, and a pH’s falling from 8.2 to 8.1 will see the end of most coral reefs on Earth.

QUITTING BRIDGE…AGAIN

March 5, 2017

I like bridge, but I have found many who play it often less than charitable to those of us not skilled.  I started reading the bridge column fifteen years ago, read a few books about the game, liked it, and on a cruise ship to the 2005 eclipse, played a little. During the last few months of my father’s life, I played with him and his group.

I played “party bridge,” often disparaged by those who play duplicate, members of the American Contract Bridge League (ACBL) who get Master’s Points from tournaments, hoping some day to become a Life Master. I would be careful to disparage party bridge.  Good players can be anywhere, and dissing one style of play is like my saying that somebody who just learned basic algebra doesn’t know real math, because I know much more.  Good for the learner.  They learned something.

I wasn’t good at bridge.  Occasionally, I would do the right thing, because bridge is a game of probabilities, and sometimes the stars align.  With time, I did a few more good things, meaning I was learning, but too many people with whom I played were neither helpful nor nice.  “Four points?  I’ve never seen a response with four points.” I had four trumps, a side ace, and distribution, for those who know the game.  Or “Why didn’t you bid xxxx?”  There are a lot of mistakes one can make in bridge, and there is no shortage of critics, many of whom are dead wrong.  They have to be, because I was criticized on the same hand and being told contradictory things.  That happened a lot when I played basketball in city league, too, and I found it annoying.

At the beginning of the session, the head of the club said that comments about play were not to be offered unless asked for.  It was a nice thought, but it failed in practice.  One man was particularly nasty.  I didn’t understand his bidding, and he always had to have his style followed.  One day, he finessed me correctly for the king of hearts, knowing that eventually he would capture it.  I had 4 cards in hearts, including the king, and rather than not playing the king last, played it on the third round.  He took it and continued his play, rather surprised when I turned up with the missing heart at the end, sinking his contract.  I had learned through reading the technique of playing a “dead” honor sooner than expected.  I remained silent.  He hadn’t counted trump.

The last day I played at the club, my partner made a bid that I misinterpreted.  Had she passed, which she should have (she preempted over a preempt, for those who know the game, and one doesn’t do that), I would have defeated the contract four tricks.  Instead, my misinterpretation cost us being set two.  I was loudly criticized by the other three people at the table and never returned.

I read the bridge column every day; my wife and I occasionally deal out hands and play them.  These allow me time to safely think and process.

After a seven year hiatus, on a cruise to the 2016 eclipse in Indonesia, I decided to play on board, not surprisingly finding myself the worst player in the room.  Bridge is a sedentary game, and while I try not to be too judgmental, many there needed to do more physical activity.  I played duplicate bridge three afternoons, calling it quits after the third.  I was paired with a different person each day, and with a partner one doesn’t know, bridge is even more difficult. I wasn’t the only one who made mistakes, and the tone of voice may not have sounded critical to the owner, but it did to me, whether I was being criticized or somebody else.  There is a way to correct people that works, and good teachers know it.  Unfortunately, there are not many good teachers.

I ran into my last partner later in the cruise.  He had played for years and explained bridge players clearly, so clearly I wondered why I never figured it out.  You see, the irony is that I am good at numbers, probability, and have a decent memory, which should make me a great bridge player.  But I have a big deficit: I process slowly, bridge is a timed game, and most play it even faster.  I can’t keep track of cards when they are played quickly.  My partner simplified matters: “The best bridge players are options traders: they have to be quick with numbers and risk averse.”  That doesn’t describe me at all.

There are those who teach bridge, but I am reluctant to seek them out, because frankly, not many are good at teaching.  I am. I understand different styles of learning, I understand that not everybody knows something as well as I, and I try to be patient.  I do this when I tutor math, show people the night sky, or explain medical conditions.  I’m enthusiastic, not critical.

What I need is a bridge hand where everything is played slowly.  I need a chance to figure out who has what and decide what to play next, being gently guided with tips how to keep track.  The bridge I have played isn’t this way.  I know it exists somewhere, but not where I’ve been.  In a sense, bridge reminds me of learning German.  I was always in a group of better speakers, but I couldn’t find one who would work with me to make me better.  It is why after 3 years I eventually gave up trying to be fluent, yet can understand it well enough to teach beginners how to go about learning it, because the teachers and online methods I know are insufficient.

I will return to reading books on bridge and watching German videos alone. I enjoy both.  I will continue to devote efforts to volunteering as a math tutor both at the community college and online, where the comments about my teaching are “You are awesome,” or “Thank you for explaining everything so clearly <3.”

I understand math.  I understand that people have different learning styles, so I teach to the person.  Perhaps most importantly, I realize that many don’t “get” math the way I do and never will.  I am neither a language person nor a bridge person.  I can improve, but I am no longer going to hit my head against a wall trying to be something I cannot be.

Better I break down math walls and save some heads.  I’ll avoid options trading, too.

STATISTICS AT THE PINNACLE-PART 2

February 15, 2017

I’d talk about the lottery and expected values. People play the lottery, because eventually somebody wins.  We can predict quite accurately the probability that somebody will win.  “You see,” I’d say, “low probability events happen; they just happen with low probability.  Take the lottery with a 1 in 110 million chance of winning.  If 330 million tickets have been bought, the expected value of jackpot winners is 3.  That doesn’t mean that 3 will win, but it is expected.  We can easily, and I mean easily calculate the probability of 0,1,2,3, and 4 with a calculator and a few key strokes.  Three people nationwide might win.  Three people, in the entire country.  Yes, it has to be somebody.  But do you think it is going to be you?”

If you have a disease, you have certain symptoms.  Medicine is the study of people who have certain symptoms and tries to figure out the probability of their having a disease.  Physicians and others would do well to understand the idea that not all who test positive for a disease have the disease.  “Suppose a disease has a 0.1% prevalence in the population, or 1 in 1000 people has it.  We would do well to teach percentages early in math and often, too.  Suppose if you have the disease, you test positive for it 98% of the time.  If you don’t have the disease, you test negative for it 99% of the time.  You test positive.  What is the likelihood you have the disease?

What is important here is the background frequency of the disease.  The fact the disease occurs in only 1 of 1000 means that it is unlikely somebody who tests positive will have the disease: only 9%.

Anybody remember W. Edwards Deming?  He was ignored here but found the Japanese receptive to his ideas about data analysis and optimizing systems.  The Japanese cleaned our clocks in the automotive industry before the Big Three caught on, not because Japanese cars were fancy, but because they worked.  There is an apocryphal story about how a Japanese company was told by an American buyer that no more than 4% of ball bearings should be faulty.  In the next shipment, 4 at the top of every box were faulty.  When asked why they were there, the company spokesman said, ‘you didn’t want more than 4% faulty.  Here they are, on top.  The rest are perfect.’

“Deming taught that variability could be classified as “common cause” (noise) and “special cause” (signal, important).  It was he who said that considering every variation as significant was not only wasteful, such “tinkering” made the process worse.  How often do we hear comparisons of say a murder number in a city being more than last year’s and hearing somebody pontificate an explanation?  Have you ever heard that this is common cause variability, and that if you want to lower the murder number, you need to address the entire system?

Samples have to be random, which is a way of saying everybody in the population, the group of people one is studying, has a definable non-zero chance of being chosen.  That doesn’t mean, ‘They didn’t ask me, so the sample is no good.’  It’s no good if the sample is done in the Deep South and the sampler wants to extrapolate it to the whole country.  One living in New York or Ohio never had a chance of being sampled.  Most people think large samples mean more useful results, but bias in a sample of 200 continues to be bias in a sample of 200,000 if the methodology doesn’t change.  The mathematics of sampling are not difficult to understand, and if one wishes to be a little less confident, 90% rather than 95%, and the margin of error for a dichotomous (yes-no) question allowed to rise to 8 or 9%, rather than 1-2%, the sample size needed decreases dramatically.

“I’ve worn out my welcome, but let me finish by mentioning the concept of 2-3 standard deviations from the mean, which most people take as being a significant outlier.  That all depends whether the curve is Bell-shaped.  If it is, then the probability of something more than 2 standard deviations from the mean is 5%.  But it is possible, depending upon the distribution of the data, to have up to 25% of items more than 2 standard deviations from the mean, hardly a significant outlier.  For 3 standard deviations, it is 3 in 1000 chance with a bell-shaped curve, but with some distributions, up to 11% of the observations.  I wonder how many who have been 2-3 standard deviations on the wrong side of the curve have been punished unjustly.  Five standard deviations?  4%.  It is 1/25, which is 5 squared.  This is known as Chebyshev’s Inequality.

“Finally, I would like to see students learn how to make good graphs instead of the ones I see today.  I would make Edward Tufte’s books required reading.  I would like to see more line graphs, dot plots, and box-and-whisker graphs with fewer multi-color pie charts.  I said I could go on for three more pages about statistics.  I have. Statistics has many day-to-day encounters, it is often used poorly, both by those who don’t know it and worse, by those who want to fool you.  It’s not lies, damned lies, and statistics but rather lies and damned people who lie using statistics.”

SAY MORE, ARTHUR BENJAMIN!–STATISTICS AT THE PINNACLE–PART I

February 7, 2017

One of my good hiking friends posted a TED talk by Arthur Benjamin on why we should teach statistics at the pinnacle of math education, rather than calculus.  I had only two complaints with his talk: first, it was too short, fewer than 3 minutes.  He should have gone on for an hour with that audience.  They would have learned a lot from him. Second, I’d add that statistics can teach us a lot about life lessons.

I commented briefly, saying that I could easily write for 3 pages.  Then I thought, “Why not?” Few will read it, because it’s math, and well….

Anyway, I’d start off with a deprecatory statement about my field:  “We statisticians are almost never right.  That’s remarkable. Never right.  BUT, we know how wrong we are likely to be, because our estimates have a margin of error.  Any estimate that does not have a margin of error is, to us, worthless.  If that fact went to the Halls of Congress, if somebody said that “Social Security will be bankrupt by 2028,” I’d like someone to ask, “What is the margin of error?”  Why?  Because somebody made a prediction about the future with data.  If somebody made a different prediction with slightly different assumptions, they would have gotten a different answer.  How different?  That is what margins of error are all about.  We’re talking about the future, and we can’t predict the future with utter confidence.

“What is confidence?” I would ask. “Let me first define probability: Probability is the likelihood an event will occur in the future, not the past.  It can be 0, no possibility at all; 1, certain it will occur, or any number in between those two.  Those who gamble know something about probability.  Good bridge players know probabilities of various distributions of cards; six missing cards in a suit are more likely to divide 4-2 than 3-3.  What we need to learn in this society is the idea that probability is not always equal if there are only two options.  Heads-tails is 50-50; boy-girl is close enough, although not exactly 50%. Millions of illegals voted in the last election or did not, or vaccines cause autism vs. they don’t, and you still have two possibilities, but now they aren’t equal.  I wish the media would learn that and not assume all sides deserve equal billing.  As a corollary, I wish the media would remember that strong statements require strong evidence.

“Roll a die, and there is 1/6 chance a 3 will come up; all 6 possibilities have equal probability.  But when you roll two dice, there are 11 possible sums, from 2-12 inclusive, and their probabilities are not all 1/11.  If you disagree, please see me with your wallet in hand and we will play, because the expected value of my winnings, which is the likelihood of my profit or loss over a period of time, will be in my favor.  If I can bet on the fewest sums that will in the long run pay me money, I will choose 7, which has a 1/6 probability, 6 and 8, which each have (5/36) probability, and either 5 or 9, each of which has 1/9 probability.  In the long run, the probability will be 20/36 in my favor.  We need to teach that competing ideas do not necessarily have the same probability.  That means we shouldn’t give equal time to people who think alien abduction occurs, because it either does or doesn’t, and they feel they should have equal say.  When we get to more significant probabilistic questions, such as smoking significantly increases the likelihood of lung cancer or heart disease, or that polio vaccination dramatically decreases the likelihood of contracting polio, we can and should make appropriate public policy.  Liberal theories?  Nope, just laws of mathematics that can be proven and which may be applied to everyday life.

“Furthermore, probability can be independent or dependent, and failure to remember that was in part was behind the Challenger shuttle disaster. Independence means that the results of one trial don’t affect the next.  Dice don’t have a memory.  Dependence means that they do.  When one O-ring fails, the likelihood of another’s failing increased.  Pull three aces out of a deck of cards, and the probability I will draw an ace from the remaining cards is now 1/49.  That is a conditional probability.

“When we make an estimate of something, we need a margin of error, a wonderful concept which teaches us to be humble and say, “I could be wrong,” four words every man ought to learn before getting married, and a breath of fresh air again in the Hallowed Halls of Power.  A caution, however, in that a margin of error doesn’t mean anything goes, that “anything is possible.”  Anything is possible if one’s idea of possibility is a one in a trillion event matters.  Statistics discusses things like million, billion, and trillion, so let me describe likelihoods for various scenarios:

• 1 in 1000: about the likelihood of getting a straight flush in poker or correctly picking a second at random that I have chosen which occurred in the last 17 minutes.
• 1 in 10000: about the likelihood of guessing right a kilometer I am thinking of between Chicago and Tokyo, or picking a minute correctly that I am thinking of that occurred in the past week.
• 1 in 100,000: correctly picking a millimeter at random that I am thinking about on a football field from the back of end zone to the back of the opposite end zone.  Correctly pick an hour chosen at random in the past 12 years.
• 1 in a million: Correctly pick a person chosen at random in a large city; a second chosen at random in the last 12 days; an acre I am thinking of in a large wilderness area 50 x 30 miles size.
• 1 in two billion:  Correctly pick a second, chosen at random, from the 1 January 1955 to now.  A single second. Correctly pick a randomly chosen acre in the US.
• 1 in a trillion: Pick a day at random since the Earth was formed.

I think that every legislator be compelled to know the differences among million, billion and trillion before they are allowed to run for office, so we don’t get silly statements of “billions and billions, and billions of acres are locked up by the federal government.”  The whole country has fewer than 2 billion acres.  If you don’t have the sense of what a billionaire is, you can’t appreciate how much money that is.  A billionaire could spend two thousand dollars a minute for a full year, day and night, before they would run out of money.  Ten million dollar house bought Monday morning?  Paid off Thursday evening.

“We use something called a confidence interval.  That is a range around an estimate where we state how confident we are that the true value lies in the interval.  It isn’t probability, it’s confidence.  You see, there exists a true value, but it is unknown and unknowable.  The range we have will either contain that true value or it won’t.  That is a 100%-0% question and not helpful.  We have 95% confidence intervals to explain that if we were to take 100 different samples, obtain 100 different estimates and confidence intervals, 95% of them would contain the true value, but we wouldn’t know which 95.  See?  We don’t know the answer.  But we are highly confident we can construct an interval wherein it lies.

Knowing confidence intervals would have been useful for journalists who reported on the once famous 44,000-98,000 deaths annually due to medical errors.  They rounded the latter figure up to 100,000 and used it, but the point estimate of 71,000 was the single best number.  Zero was not possible, nor 10,000, nor a million, not possible if we are going to remain sensible about the world.

“Global climate change likelihood is prediction, which lends itself to statistics and to confidence intervals, and the IPCC was more than 95% confident years ago, a strong statement of science.  It means that the interval they calculated was highly likely not to contain 0, no temperature rise.  It is incumbent upon those who disagree to come up with a confidence interval so that we can look at their data and see what assumptions and calculations their models have.  This would prevent a lot of unnecessary arguing, and the arguments we have would be more appropriate.

“Means and medians are basic concepts people should understand, because a mean, the average, is affected greatly by outliers, whereas the median is not nearly as sensitive.  Housing prices and salaries are much better described by the median.

People talk about a non-existent term called the Law of Averages.  I’d not teach it, and maybe it would go away. There is The Law of Large Numbers, which says frequencies of events with the same likelihood of occurrence even out, given enough trials or instances.”

“I can see that a lot of you are yawning and looking fried.  I’m giving you a year’s curriculum in a few minutes.  Imagine, however, how useful all this stuff might be if I had a year to teach it to students.  I actually tried to do that in Tucson in 2011, for free, as a trial course, my swan song before leaving town 3 years later.  But I didn’t have an education degree, and the school had other priorities.  Such a shame, really.   OK, let’s take a break, and come back and I’ll finish the summary.”

A TIDE IN MY AFFAIRS

January 6, 2017

Warning: This post will contain some mathematical formulae and terms, which may scare or otherwise turn off some.  I hope such formulae do not detract from the beauty of what will be seen, because indeed, mathematics is beautiful.  It answers questions.  Is that not beauty?  In a week, pictures of the result will be shown.

I’m going over to Newport, Oregon next week to see the King Tides, something I had once never heard of.  I am almost a true Oregonian, but when I led a trip to the coast the last week, I forgot to look up the tides. That’s inexcusable.  Always know the tides when you are at the ocean.

Tides matter.  A lot.  In nature, many species thrive at border zones between one ecosystem and another.  They allow for organisms to live in varying degrees of wetness, rather than always wet or always dry.  They allow for tidal pools to become cut off from the ocean, where periodically they get refilled or organisms shuffled.  Without tides, the Earth would be a very different, far less diverse place.

What are tides, anyway?  They are common throughout the universe.  If one object tugs on another, it can deform the latter due to gravitational attraction, which may cause buckling or movement of the surface of the attracted object.  Jupiter’s moon Io gets tugged by massive Jupiter, causing volcanic eruptions on its surface.  The first was spotted by a woman, Linda Morabito, who saw a plume on Io, which had been once thought once to be dead, then had volcanism predicted.  Io is the most volcanically active place known in the solar system.

Both the Sun and Moon tug on the Earth.  While the Moon is much smaller, a mass 1/27,000,000 that of the Sun (mass is the amount of “stuff” something has; weight is the effect of gravity.  Diet removes mass; being in zero gravity does not, but it makes you weightless), the Moon exerts a majority (55%) of the tidal activity on the Earth.

For a long time, that 55% bothered me, because gravitation is proportional to the product of the masses but inversely proportional to the square of the distance, the distance between the two centers, or d, and the numbers didn’t work.

F=G m1 m2/d^2.

where G is the gravitational constant, m1 the mass of one body, m2 the mass to the second, and  d^2=d*d, the distance between them multiplied by itself.  The Moon is smaller, less massive, but it is much closer than the Sun.  Still, if one compares the large mass of the Sun with its admittedly larger distance from us (400 times further from the Moon, and the distance varies, which is important), the Sun ought have an effect 170 times greater than the Moon upon us.  It doesn’t, and that bothered me.  I show this below.  Gravity is the reason we circle the Sun and not the Moon; the Moon circles both of us.  I did not consider tidal forces, those which work differentially on a body, more on the near side than the far side.  These Ah-hah moments are one of the joys of life, when one understands a concept that has been murky for years.

The Moon tugs on the Earth, the oceans are pulled towards the Moon. Tides are maximal in general when the Moon is either overhead or at the opposite side, although that can vary considerably due to other factors and local conditions, which give rise to enormous tides at the Bay of Fundy or tidal bores on Turnagain Arm in Alaska.  The tide is greater (spring tides, nothing to do with the season) when the Moon is lined up with the Sun and the Earth, occurring about every 15 days, and lesser (neap tides) when the Moon is not aligned.  The square of the distance means that anything decreasing distance increases the tide, so when the Moon is close to us, which happens every 27.5 days, even not well aligned with the Earth and Sun, the tides are significantly affected. The Earth is 3 million miles closer to the Sun in early January compared to early July, and this increases tides as well, because while the Sun’s force is slightly less than the Moon’s, its distance from us is the least for the year. That’s why we’re going to Newport.

In Newport, king tides occur at full Moon in January, near perihelion.  The full Moon is opposite the Sun, meaning that it is in the northern part of the celestial sphere, over the northern hemisphere, and therefore is closer to the coastal cities there.

I also didn’t know why the Moon had a greater pull, given the gravity equation.  The numbers didn’t work. I thought—incorrectly— it was all gravity.

The tidal force looks at slight changes in the distance between the two bodies; the force is proportional to the cube of the distance between the bodies, d^3, or d*d*d, and a simplified proof is shown below.  Cubes are volumes, and the three factors are length, width, and depth.  When we compare the gravitational equation using the cube of the distance and twice the mass product, the Sun is responsible for about 45% of the tidal force; the Moon the rest.

Additionally, the lowest tide is not in January, as one would think, but is in the late spring early summer and at New Moon.  Why?  In May, the Earth is further from the Sun, so the Sun’s pull is less.  But at New Moon, which aligns with the Sun, the Moon is over the northern hemisphere. There are issues with the lunar nodes and the tilt of the Earth’s axis at different times of the year.  Tides are more complex than I thought, not due to simple gravitational pull but to a differential force that must be accounted for. When I go to Newport, I will be watching a 3 meter high tide and the -0.5 meter low tide, both a full meter higher than normal.

F(S-E)=Gm (S)*m(E)/d(S-E)^2. The Sun-Earth gravitational force is proportional to the product of the masses and inversely proportional to the distance between their centers. The same holds for the Moon-Earth.  It also holds between you and your computer, too.

F(M-E)=Gm (M)*m(E)/d(M-E)^2

Let’s take the ratio of the Sun-Moon forces which is dividing the top by the bottom.  Stay with me, because G and m(E) will disappear when we divide, because they are part of both.

Ratio=m(S)/d(S-E)^2 divided by m(M)/d(M-E)^2

When we divide, we invert the divisor, which is the value that is “going into” something.

If we divide 1 by 1/3, we invert the 1/3 and have 1 *3/1 or 3.  One-third goes into 1 three times.

If we do this math, we invert the denominators and have

Ratio=m(S)*d(M-E)^2 divided by m(E)*d(S-E)^2

We know these ratios.  The mass of the Sun is 27,000,000 that of the Moon.  The distance to the Moon is about 1/389 the distance to the Sun.  Let’s call it 1/400.  By the way, in the sky, the Moon is about the same angular size as the Sun, which is why we can just have total solar eclipses. The Sun is about 400 times the diameter of the Moon and is about 400 times further away, so they have about the same size when viewed from the Earth, one of the greatest cosmic coincidences there is.

The ratio of forces is about 27000000/400^2, or 169.  But the Sun is actually less powerful as the Moon in producing tides.  Tidal forces are differential and work differently on one side of the body versus the other.  Tidal forces are not the same as gravitational forces. They work as the inverse cube, not as the inverse square.  A cube here is d*d*d or d^3.  We measure volume when we know three factors—length, height and depth.

The ratio can be done by subtracting the force of the two objects from the front by the force from  the back.  Or, and this is why calculus was invented, we can take the derivative of the gravitational force with respect to the distance, because only the distance is changing, not the masses, and derivatives of constants are zero, making life a lot easier.  Here, we deal with the change of distance.

The derivative of Gm1m2/d^2 with respect to d is -2Gm1m2/d^3.  The bottom line, literally, is a cube, and the differential force for tides is a function of the cube of the distance, not the square.  If we look at the above ratio, we get 27,000,000/400^3 and it is 0.42.  If we use the average figure of 389 times further away, we get 0.46.  Tides are much more complex, but the idea of the inverse cube ratio is why the Moon exerts a greater tidal force on us than the Sun.

A second proof for tidal forces being proportional to the inverse cube of the distance is abbreviated, but goes something like this:

Force of Sun (Fs)= G(SE)/d^2, where G is the gravitational constant and SE is the Sun Earth distance.  We could make it the lunar distance if we wanted to.

The distance is slightly different on the other side of the Earth, so we will call that p.

F(SE-near or s1)-F(SE far or s2)=G (SE)/d^2-G(SE)/(d+p)^2

=G(SE){(1/(d+p)^2)-(1/(d^2)}, d is much greater than p or d>>p.  We have factored out G(SE), which is common to both.

Look at the parentheses, and using common denominator subtraction,

(d^2+2dp+p^2-d^2)/(d+p)^2d^2

=2dp+p^2/(d^4+2d^3p+d^2p^2)

=2dp/d^4,  skipping some steps, since as d gets very large, the denominator approaches d^4,

=2p/d^3

From earthsky.org, which is nowhere near scale but shows where tides come from.

GREAT EXPECTATIONS

March 23, 2016

It all started with a packaging error on some soy burgers I bought.  At home, I discovered a slit in the bottom of two that I didn’t see when I bought them.  Who knows how long the contents had been exposed to air?  How often do you check the packaging when you buy something?  For me, not often enough.

I returned to the store and told the manager, who was shocked.  I said I was going to look at the other packages they had to see if they, too, were defective.  When I checked the box that had the identical soy, sure enough, the two remaining packages in it had rips.  I took them back to the manager’s counter.

I had to wait, however, as I watched a woman with bleached blonde hair, maybe mid 40s, and a young man, who might well have been her son, discuss lottery tickets.

“We can’t afford too many,” she said.  “We need \$20 for xxxx.”  I couldn’t hear what “xxxx” was.

I was stunned, as I watched them purchase six lottery tickets, some sort of scratch type, for \$60, which the woman counted out using twenties, fives and ones.  It might have been a Raffle. You can’t win if you don’t play, right?  That was the catch phrase in Arizona.  You can’t lose if you don’t play would have been more accurate, and I can prove it.

Let us look at the expected value, which anybody playing the lottery should understand.  We ought to teach this in schools. If the expected value of something done is positive, over the long run it will be successful; if negative, over the long run, it will not be.

Here’s an example.  At a roulette table, you plunk down \$1 on a roll of two dice.  If they come up double 3s, you get a 30-fold return—\$30.  Well, it isn’t 30 fold, because you paid \$1 to begin with.  It is 29-fold.  Don’t laugh, casinos get rich on these “minor points.”  You have a 1 in 36 probability that the dice will both be 3: 1 in 6 for each, and we multiply the probabilities when one doesn’t affect the other, a term called  independence. The probability of winning \$30 is (1/36), and the probability of losing \$1 is (35/36).  If you multiply these and then add, you get +\$(30/36)-\$(35/36)=\$(-5/36).  That fraction comes out to about MINUS 14 cents per dollar bet.  Are there winners?  Yes.  The consistent winner is the casino.  Not only is it consistent, they can predict very accurately how much they will make, because in the long run (not a player’s time horizon), the casino will make 14 cents on each dollar wagered for that bet.

What I saw for a \$10 Raffle ticket:

1 in 250 chance of winning \$100

1 in 25,000 chance of winning \$20,000

1 in 250,000 chance of winning \$1,000,000

The exact calculations are more complicated, but the simple way works just fine.

You spend \$10, and you have a 1/250 chance to win \$100, an expected value of \$100/250 or \$0.40, 40 cents.  You also have a 1/25,000 probability of winning \$20,000, and that is \$0.80, so the expected return of both is \$1.20. Add to that the 1 in 250,000 chance of winning \$1 million, \$1,000,000/250,000=\$4.  The total expected return on \$10 spent is \$5.20, give or take.  That’s the plus.  The minus is \$10(249/250)=\$9.96.  The expected value is \$5.20-\$9.96=  MINUS \$4.76.  The pair spent \$60. If we looked at a lot of people buying, all of them, they would on average lose about \$28.50.  One of those \$20s that one comes in to the store with, and nearly a ten spot as well, won’t be seen again.  Do that weekly for a year, one will lose nearly \$1500 on average.

The lottery is very coy about posting long lists of numbers that won \$100.  Sounds like a great deal, except that first, they really won \$90, because they paid \$10 for the ticket.  Secondly, the list of those who lost is about 120 times longer.  We don’t see that one.

Here in Oregon, 93% of the money spent on the lottery goes to payouts.  Sounds good, except that the big payouts go to very few people.  BUT A LOT OF PEOPLE WIN A LITTLE, LIKE \$100, AND THAT’S A HUGE PROBLEM.  This is called variable ratio reinforcement.  Winning occasionally keeps one playing.  Never winning at all causes one to quit sooner.

There is a very strong correlation between low income and high use of the lottery. Why do people play?  Answer: they see this as the only way out of poverty.  The probability is exceedingly high, however, that they will only go deeper into poverty. The lottery is a regressive tax levied on those who can least afford it.  The lottery steals from those who don’t understand math, probability, or how our brain can lie to us.  When the money goes to education, a noble cause, it is being paid for by those who have the least money: the very poor spend 9% of their income on the lottery.  If one makes \$13,000 a year, that is nearly \$1100 they spend on the lottery.  If one has difficulty making ends meet, this is going to push them over the edge—with high probability—and I can define that probability exactly.

I never forgot what my statistics advisor said about expected values:  “If it is positive, I will beg, borrow, steal every dollar I have to play.”

It becomes absolutely certain at some point that somebody will win Powerball.  We can predict that as well.  If there is a Powerball with a probability of winning equal to 1 in 110 million (roughly equivalent to your guessing correctly a minute I choose between the signing of the Declaration of Independence in 1776 and now), and 330 million play the Powerball, the expected value of winners is 3.  Three people are expected to win; the probability of exactly 3 is about 22%; the probability that between 1 and 5 will win is nearly 7 in 8, and the probability nobody will win is about 5%, quite small.  In other words, one can make remarkably accurate probabilistic statements what is going to happen.  Did you guess my minute? (It was 3:32-3:33 a.m. 15 August 1846).

I’m lucky; I live comfortably.  Still, I bend down to pick up a dime or a penny if I see one.  If PetsMart gives me a \$3 coupon for doing a survey, I will do it.  If REI gives me a dividend, and I am planning to buy something, I will buy it with the dividend and furthermore try to get it when the item is on sale.  I use coupons when I shop, I comparison shop, I don’t drive 30 miles for cheaper gas, because it’s more expensive to do so, I pay my credit cards off every month, I try not to get a tax refund, because it means I loaned the government money, and I DON’T PLAY THE LOTTERY.

But occasionally, I do silly things with money.  The store agreed that the ripped packages I returned, and the two others in that I found that were also ripped, needed to be removed.  I found two good packages, but they were not on sale, so I actually had to pay to replace them.  I paid \$4.40 for doing the store and any customer who bought those a favor.  Small price for what I learned about the lottery.

The expected value for doing a good deed was negative.  In the long run…

ONLINE, ON COURSE

November 29, 2015

I received the following letters the past few weeks.  They made my day.

Thank you so much, I was struggling and your answer made it simple and understandable. UR GR8.

You are Amazing!

You have helped me in the past and always have accurate answers, I am so grateful you took your time out to help me today, thank you, I appreciate it so much!

Thank you, from the bottom of my heart, you have written this in a way that I totally understand. I will differently (sic) praise you to all my friends and family. God Bless you,

What did I do to deserve these?

Reason for the first:

The area of a triangle is 30 sq in. The base of the triangle measures 2 in more than twice the height of the triangle. Find the measures of the base and height.

The area of a triangle is (1/2)*base*height.  Remember that?  Therefore, the base*height must equal 60 for the area to be 30.  Let the height= x inches, then the base is 2x+2, two more than twice the height.  Then,

x*(2x+2)=60, 2x^2+2x=60, and dividing by 2, x^2+x=30.  We can write that as x^2+x-30=0 and factor it as (x+6)(x-5)=0.  That is 0 if x=-6 (not possible for a length) or if x=5.  So, the base is 5 inches and the height is 2(5)+2, or 12 inches.

For the last comment,  I answered the following, taking about a minute in my head, writing it down as I thought.

Write the slope-intercept equation for the line that passes through (-12, 10) and is perpendicular to 4x + 6y = 3.

One gets the slope first by rewriting the equation as 6y=-4x+3 and dividing by 6 to get y=-(2/3)x+1/2.  The slope is -2/3. The perpendicular line has a negative reciprocal slope.  Turn the fraction over (-3/2) and change the sign (3/2).  That is the slope of the perpendicular line.   Using the point slope formula where we know the slope and a point, x=-12, y=10, y-y1= (3/2) (x-x1).  That is y-10=(3/2)(x+12).  This becomes y-10=(3/2)x+18, and finally y=(3/2)x+28.  Also, y=mx+b, so 10=(3/2)((-12)+b.  That is 10=-18+b, so b=28.  Both methods work; the more ways one knows, the more ways to explain it to students.  One of the ways is likely to stick.

For the one who called me amazing?

Find the accumulated value on an investment of \$15,000.00 for 9 years at an interest rate of 11% if the money is compounded

a) Semi- annually b) Quarterly c) Monthly d) Continuously

Here, one uses the formula

Principal=Starting Principal{1+ rate/compounding per year} raised to (the number of years*compounding per year). P=Po{1+r/t}^nt.  Semi-annual is P=Po{1+(0.11/2)}^18, because it compounds twice a year and there are nine years.  This is \$39,322.  For continuously compounding, it is easier, P=Po*e^rt.  e^rt= e^(0.99), because 9*11%=0.99.  Po*e^0.99=\$40,368.52.  Continuously compounding gives you more money, although the difference between it and monthly is only \$200 less than continuously.  The last formula allows one to prove that the doubling time of money in years is 70 (or 72, which is easier to work with) divided by the interest rate in per cent.  I grew up in the age before calculators, and we had to do this by logs.  On a calculator, it takes about 15 seconds. Dividing 72 by 11 gives a doubling time of about 6 1/2 years, so \$15,000 should double once and be well on its way to doing it again.  The answer makes sense.

This is an online math help site.  More than 2000 tutors take part, some of whom have solved one problem, one nearly 70,000.  I’ve solved 2000.  About one in four thanks me.  That’s nice.

Several tutors offer their services for pay, \$1 per answer, \$2-\$5 to show the work.  I do it to relax.  Yes, relax.  This stuff is fun for me, and I have learned the easier the problem for me, the more grateful people tend to be.  I don’t need to hear anything, unless my answer is wrong or not understandable.  I’m there to help.  I don’t know names; I do know I have helped parents help their children.

I’ve learned much.  It has been a great review of my statistics, I now deal with ellipses better, and I understand geometric series better than I ever have before.

I usually want a challenge, so I choose what I want to solve.  I have a big advantage:  I grew up in the era of no Internet, Chemical Rubber Company tables of integrals, no calculators, only log tables to do complex calculations.  In other words, I learned math from first principles, from the ground up.  Yes, it helps to have a genetic ability to do this stuff.  I can’t play the violin, but I can find the vertex of a parabola mentally and write it in three different forms.  Kids need someone to help them understand how to do it, not in their head, but to allow them to understand these and similar problems.

The current list has perhaps 50 problems, and I often work down it until I find a problem I feel like doing.  If interested, I go to the list of unsolved problems.  Last I checked, statistics had about 40,000.  A lot of those are tough, and if I don’t have pen or paper around, I don’t do them.

When I tutor at the community college, I answer algebra questions online while waiting for non-virtual students to ask for help.  I guess I am volunteering, but I am having a lot of fun.  It’s nice to lay out quickly an answer in simple form for a person who is struggling.

The other day at the CC, I was asked to go into the higher level math room to help out.  That was a compliment, because I was felt to be good enough to help out there. I’m the go-to guy for statistics.  The other tutors are really smart, yet all of us at one time or another have trouble with something.  I may struggle at the high levels, but I often find myself pulling stuff out of the air from the past and making sense out of it.  Or better yet, I ask a student where he got a specific term in an equation.  The student looks puzzled then suddenly says, “Oh, wow, I didn’t see that before.  OK, I understand.  Thanks a lot.”  And he leaves.

I hadn’t a clue how to solve the problem, but I think I helped him.

Math is mentally taxing.  After doing about a dozen problems, I take a break.  It helps me later solve troublesome problems.  In the math lab, I have concentrated so deeply that one day when I walked out of the room, I forgot whether it was Tuesday or Friday.

I think the absent-minded professor was probably working overtime on a difficult problem.