Rating the Polls

I was going to call this week's post "March Mathness" and talk a little about the NCAA tournament. Let's do that next week. Let me go ahead, though, and apologize for the title now. I'm sure I'm not the only one to use this type of play on "March Madness". That still doesn't make it right.

There is a nice website by the name of fivethirtyeight.com. It presents information regarding polls and polling data (The 538 part comes from the fact that there are 538 electors in the electoral college.) One interesting part of the website is looking at various polls (there are a lot more than I would have imagined - they rate over three hundred polling firms).

The reason I got there is because I was trying to figure out how their site, can come up with in-game information like Arizona is ahead of St. Johns 55 to 46 with 3:38 left to play, thus Arizona has an 89% chance of winning. Wow. It's clear Arizona would probably win, but how do they come up with a percent like that? Anyway, we'll look at that next week.

I got side-tracked with a section that speaks to how they rate various polls. For example the Trump/Clinton election did not come out as most had predicted. Some polls are better than others. They rate them all. For example, one of the best seems to be the ABC News/Washington Post poll. On the other had, an organization called Research 2000 is not. An overview of their methodology is at:

https://fivethirtyeight.com/features/how-fivethirtyeight-calculates-pollster-ratings/

They don't really give enough information to show exactly how they do it. That would probably be beyond me anyway. Let me tell you something they have used in the past. It is an especially cool math application since it has a square root stuck in there.

Total Error = Square Root of (Sampling Error + Temporal Error + Pollster Induced Error)

Why don't polls come out exactly right:

1. Sampling Error:  Sampling not enough people or not getting a representative sample
2. Temporal Error:  The farther away it time a poll is from the event; the more error
3. Pollster Induced Error:  Seems to be kind of a catch-all category for other things that can go wrong, such as assuming a too high or too low voter turnout.

Something else interesting they talk about is the concept of "herding". The companies that do the polling want to look good. It does not look good if they've wandered too fall away from the rest of the herd. If most every other poll has candidate A having around 55% of the vote and you predict he'll have 73%, you might make an "adjustment" to your results. Or you simply chose to not publish those results in which your company seems to be way off. 

That and other factors make it pretty complicated. Polling itself is complicated and then ranking the pollster even more so. 

I hope I've done justice to what they do. If you read what they have to see on their website you can see the complexity involved.

Next week, March Madness. Don't worry it will still be going on. In fact, it is actually March and slopping over into a little bit if April Madness. 

Top Ten Applications of Pi

It is pi day!!! It snuck up on me. I was going to make a top ten of cool facts about pi - by stealing them from the internet, of course. But most of those are not math applications, just cool trivia with virtually no applications, e.g., in a Star Trek episode, Spoke once defeated a computer by commanding it to compute the last digit of pi.

Brainstorm. Combine the two into the top ten applications of pi. And also doing so by stealing them from the internet. All wrapped up in one tidy package. By the way, I understand some of these applications fully and some not very well at all. I will try to give a short internet-stolen explanation for these.

1. Circumference of a circle.
2. Area of a circle.
3. Volume of a sphere.
4. Surface area of a sphere.
5. Cosmological constant - Value of the energy density of the vacuum of space. Originally introduced by Albert Einstein in 1917.
6. Heisenberg's uncertainty principle - Precision with which certain pairs of physical properties of a particle (such as momentum and position) can be known. Joke break: Heisenberg is driving down the road and is pulled over by a police officer. Police: "Do you have any idea how fast you were going?" Heisenberg: "No, but I know exactly where I am."
7. Einstein's field equations of general relativity - Describes the fundamental interaction of gravitation as a result of space time being curved by mass and  energy.
8. Coulomb's Law - Force between two charged particles.
9. Period of a pendulum.
10. Buckling formula - Finds the stress an object can handle before "buckling" would occur.

Hidden Figures

This is kind of a review of the movie/book Hidden Figures. Before seeing either one, I was a little concerned that it was going to be preachy. We need to treat African American people / women with respect. Hollywood, you don't need to tell us that. The people that already know that don't need to be told. And the people that don't know that probably aren't going to pay money to see this movie anyway. But it wasn't like that - the movie or the book. It just told their story.

If you want the true story, books are usually a better bet than the movie. The movie seemed quite in line with the book, but there were a few things. The two hour movie obviously had to leave a lot of stuff out that was in the book. There was also a time one of the ladies was at a chalkboard. She was impressing the room with the math she was doing. Some of it was a little unrealistic. Most people can't rattle off sin(23) to ten-thousandths place from memory - stuff like that. But, I'm quibbling. The movie was really good.

I got the book. And to be honest, I did some skimming in parts. I may get around to reading every word at some point, but that will be a ways off. There were a couple of cool things dealing with math applications that I thought was interesting. First of all, the book is Hidden Figures and is written by Margo Lee Shetterly. There. I hope that covers me from violating any copy write issues. Regardless, here we go.

Where do systems of equations take place in real life? Well check this out: "Modeling flight at transonic speeds was a particularly knotty problem, because of the subsonic and supersonic winds that passed over the plane or model simultaneously. Aerodynamic equations describing transonic airflows might contain as many as thirty-five variables. Because each point in the airflow was dependent on the others, an error made in one part of the series would cause an error in all the others. Calculating the pressure distribution over a particular airfoil at a transonic speed could easily take a month to complete for the most experienced of mathematicians." (pages 137, 138)

I once had the father of a student tell me about an older child of his that was in the NASA astronaut program. He said that their astronauts had to have a very good mathematics background in case something happened to the on-board computers. Another part of this book spoke to that issue:  "An astronaut stranded hundreds of thousands of miles from Earth is like a mariner from a previous age, adrift in the most remote part of the ocean. So what do you do when the computers go out? This was precisely the question Katherine [Johnson] and her colleague Al Hamer had asked in the late 1960s, during the most intense preparations for the first Moon landing. And in 1967, Johnson and Hamer coauthored the first of a series of of reports describing a method for using visible stars to navigate a course without a guidance computer and ensure the space vehicle's safe return to earth. This was the method that was available to the stranded astronauts aboard Apollo 13." (page 248)


Speaking of outer space, I give both the book and the movie, four stars.

Weather Forecasting

Well, guess what happened? Gonzaga lost. You might remember my post from a couple weeks ago. The Gonzaga's men basketball team was undefeated with four games to go. We figured out they had a 91% chance of remaining undefeated. They won the first three, then lost the very last game of the season. So much for the 91% chance.

It was a pretty good mathematics application, I thought. If this was presented to a math class, how would the students react. My guess is that they would say the math failed. A 91% chance is not a sure thing, but there have been some studies that suggest people take it to be that. And on the other hand, people assume that a really low percentage is the same as no chance. This might lead to a good class discussion.

This made me think about other forecasts? Specifically, how do weather forecasters decide on the percent chance of rain. I found a post by a weather person at WESH TV, Amy Sweezey. Although we haven't met, and I don't even know what she looks like, she seems delightful. And smart. I learned some interesting things about how they make a percent estimation of rain for the day.

So what does a 40% chance of rain mean? It turns out that it depends.

Let's say that there is a 40% chance of rain over about half of the area in question.

Some weather forecasts will call this a 40% chance of rain. In a way it is. There is a 40% chance of rain somewhere in the area.

Some weather forecasts will call this situation a 20% chance of rain. And again in a way it is. There is a 20% chance that it will rain where you are currently standing.

As Amy says, "When it comes down to it, you cannot base your plans around a rain percentage." It's more important to know where, what time, and how heavy it will be.

It's because of this that many stations won't even do a percentage. Instead they might use descriptive words like "scattered showers", "isolated", "a few showers", and "likely".

10 Cool Things about Srinivasa Ramanujan

I new very little about this person until recently. One way math teachers know about him is the fact that he often get a half page bio in math books. He is probably included so the company can show that their book is multicultural and perhaps that will lead to more sales. I hate to be cynical like that, but I noticed that once when on a textbook adoption committee. We had a sheet in which potential adoptions were given points in various categories. One was how multicultural it was. I guess it makes financial sense for the companies to load up multicultural references.

While that is a good idea, students often catch on to some of the overreaching. One story problem example was (yes, an actual example) "One day Running Bear was ..." Its fine, but pretty transparent what they are trying to do.

Back to Ramanujan (accent on the "nu" syllable and pronounced "new"). He didn't really do a lot of applied mathematics. Most of his work was in number theory and analysis.

He is featured quite a bit in the book The Music of the Primes. There is a book and a movie about his life The Man Who Knew Infinity. I recently saw the movie and I thought it was very good. And unlike many "biographical" movies, it didn't take liberties with the person's life.

Here are a few, I think, interesting tidbits of the life of Ramanujan:

1. Born and died in India, but much of his mathematical work was done at Trinity College in England.
2. Lived from 1887 to 1920, only 33 years.
3. In the movie he is played by Dev Patel, who was also the main actor in Slumdog Millionaire. G.H. Hardy was played by Jeremy Irons, who for me will always be the voice of Scar in The Lion King.
4. He had almost no formal training in mathematics.
5. He came to the attention of famous mathematician G.H. Hardy at Trinity College after writing to him.
6. Despite an invitation, he was at first hesitant to go as crossing the seas, according to his religion, could make him an outcast.
7. Suffered poor health all his life, but it was aggravated by the food at Trinity not matching well with his vegetarianism.
8. As his heath and depression became worse he attempted suicide by jumping in front of a London train.
9. A famous quotation is, "An equation for me has no meaning unless it expresses a thought of God."
10. A major coup was finding one of his lost notebooks in 1976.

Gonzaga Probabilities

A lot of games on TV will put up statistics way too fast. They're up for a few seconds and its tough to take it all in. Maybe it is the Detroit Lions' total rushing and passing yardage for each of the past three seasons. You got about five seconds to see it and try to make some sense of it. They obviously have some point they are trying to make with all those numbers, but tough to take in. Sometimes you just have to stare at things for awhile.

I did see one a few days ago that didn't have too many numbers in it. Gonzaga is currently the number one college basketball team. In fact they're undefeated. What are the chances they stay undefeated? It's 91%, they claim. They also showed the chance of winning each of their remaining games. Those chances are:

• 98%
• 99%
• 98%
• 96%

How did they come up with those numbers? I'm not sure. They didn't explain that. I tried looking on the internet to find out. That didn't work so well. Different associations (Vegas, ESPN, etc.) have their own different methods. Most of them let you in only partially on how they figure things. Turns out they take a lot of things into account. Your win-loss record, opponents win-loss record, playing home or away, point differentials, injuries, recent win-loss records, etc.

Then how they take all that data and come up with a percent is a mystery. What isn't a mystery is the 91% chance of winning all four of those games to remain undefeated. It is an "and" probability problem, which mean we can just multiply those individual probabilities together.

       (0.98)(0.99)(0.98)(0.96) = 0.9128 or just over 91%

On a related note, the University of Connecticut women's team won their hundredth game in a row last night. Pretty impressive. Impressive especially considering the following:

• A team that usually wins 80% of its games has a (0.8)¹⁰⁰ = 0.00000002% chance of winning 100 in a row.
• The best NBA record ever is Golden State last year. They won 89% of their games. Thus, a 0.00087% chance of winning 100.
• How about a team that wins 99% of its games? They have a 36.6% chance of winning a hundred in a row.

Here is a little higher mathematical problem. What kind of team would it take to have a 50% chance of going on a 100 game winning streak?

• x¹⁰⁰ = 0.5 
• 100(log(x)) = log(0.5)
• x = 0.9931

So you need to usually have a 99.31% chance of winning any single game you would play to have a 50-50 shot at winning 100 in a row.

In your mind this is maybe not the most important math application ever, but many students are into this kind of thing. And it does beat flipping coins and pulling various colored socks from drawers.

The Importance of Proofs

This blog is devoted to showing applications of high school mathematics. At first I thought the following topic wouldn't fit here, but then I decided it does. That is, the idea of proof. It really is tough for students to see the necessity of proofs. Usually encountered mainly in a geometry, they can be a real chore both for students and teachers.

Proofs are hard. So combine that with the fact that what is being proved seems pretty obvious, you get the double whammy of why students hate proofs - they are both hard and pointless.

Do we really need to prove that the sum of the first n odd positive integers is n squared. It wouldn't take to much time to get students to accept that fact without doing a proof. 

The square root of two is irrational. "I found the square root of two on my calculator. The decimals don't repeat. Yes. I'll buy it."

It seems that statements that are true for the first ten or so examples are probably going to end up being true. In fact, it's a little difficult to find situations where that doesn't hold true. If I make a statement that a coin I have will always come up heads, you might have your doubts. If I flip it ten time and I get heads each time you probably think that statement is, in fact, true, because it probably is a two headed coin.

I had heard quite a while ago that there was some statement having to do with prime numbers what was true way down the line, but then turned out not to be false at some point. I never knew exactly what that was, though.  

Well here it is. 

Suppose we want to find out if P is a prime number. Find 2^P and then divide it by P. If the remainder is 2, then P is a prime number. If it is something other than 2, P isn't prime.

I was a doubter at first. I did the first several and they all worked out, though. I'll take one example. Let's try the number 17.

2¹⁷ = 131,072
131,072 divided by 2 is 7,710 remainder 2
Thus 17 is a prime number.

This really works - all the way to 340. Then it doesn't.

2³⁴¹ is so big I can't tell you what it is. I could get my calculator to go as high as to the 331 power, then it refused to do any more.

Anyway, pretty amazing. It works for 1, 2, 3, 4, 5, ..., 340, correctly determining if the number is prime. Then this method claims 341 is prime. But it isn't, because 11 x 31 is 341.

There you go. Proofs aren't pointless after all. Still hard, but not pointless.

            *****************************************************************

Now, about last week's post. We worked out a score for the yet to be played Super Bowl of New England 30, Atlanta 27.

And how did that turn out? Pretty darn close I must say. New England 34, Atlanta 28.