Boreal B[l]og

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.”  

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.

I’d look at the audience and find two rows where I saw 22 people.  “What do you think the probability in these 22 there is at least one pair with the same birthday?”   I’d ask that because most people would think it is quite unlikely.  Our brains tell us that, but our brains can deceive us, not only the brains of others.  Turns out, the probability is about 50%.  I might even start with 7 rows containing 70 people, where the probability is 99.9%.  “This does not make intuitive sense,” I would add, “but is easily proven by taking the approach of the probability that two people don’t have the same birthday.”  My statistics advisor did this in a class where I happened to be attending as a graduate student, and the first student’s birthday matched mine.  He said the look on the face of the students was priceless. “Why does this matter?  It matters because sometimes the way you think is wrong, flat out wrong!  Your brain lied to you.  Your brain said the likelihood of two people in the same room’s having the same birthday was small, very unlikely with 22 people, which is not true.  Our brains lie to us about speed, direction, and up and down. They worry more about improbable losses than probable gains, and how certain major events in our lives shape our thinking, even if they are very unlikely to ever happen agan.   The solution to the birthday problem is also a good life lesson:  figure out what you don’t want and whatever is left over is what you want.”

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.” 

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.”

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…

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.