A few fun Perfect Bracket stats questions for students

ESPN had 18.8 million entries in their bracket challenge for the NCAA men’s basketball tournament. There were also several other bracket contests going, too. Below are a couple of fun bracket-related questions for students learning about statistics:

(1) Perfect Brackets after the first round:

The ESPN contest went from 18.8 million entries to 164 perfect brackets at the end of the first round of games:

So . . .

(i) If you were running a contest that had 100,000 entries instead of 18.8 million, how many perfect brackets would you expect to have in your contest?

(ii) What do you think the probability of having 0 in your contest would be?

(2) Perfect brackets after Michigan St. beat Miami

The number of perfect brackets in the ESPN contest fell from 952 to 513

Prior to the Michigan St win you had 5 perfect brackets left in your contest. Given what happened in the ESPN contest how many do you think you’ll have after the Michigan St. win?

What do you think the probability is that you will have 0?

(3) The USC vs SMU game was the 22nd game of the tournament

You had 241 perfect brackets going in and 22 after USC won.

In the ESPN contest 81.6% of the entries picked SMU to win and 18.4% picked USC.

(i) Suppose you have a coin that flips heads 18.4% of the time. If you flip it 241 times what is the probability that you will have 22 or fewer heads? (probably best to use a computer for this one . . . )

(ii) Do you think having 22 brackets left rather than 44 (which would roughly be 18.1%) was random chance or was there another factor in the reduction?

(4)  Expected upsets

I’ll make up the numbers for purposes of this problem, but you can get the actual numbers here if you want:

Some statistics for the ESPN bracket tournament

Suppose that the 18.8 million entries have selected the winners of the games this way:

Teams 1 – 5:  95% of the entrants guessed these teams would win their 1st round game

Teams 6 – 10: 90% of the entrants guessed these teams would win their 1st round game

Teams 11 – 15:  85% of the entrants guessed these teams would win their 1st round game

Teams 16 – 20:  80% of the entrants guessed these teams would win their 1st round game

Teams 21 – 25:  70% of the entrants guessed these teams would win their 1st round game

How many games out of these 25 do you expect the team that was not favored by the ESPN contestants to win?  Why?

A risk pricing formula for simple situations

I was reading Nassim Taleb’s most recent paper this morning:

and this short passage caught my attention:

It reminded me of a simple risk pricing idea I had (and used) 15 years ago that nearly got me laughed out of the room. The idea isn’t applicable to general situations – just pretty simple ones in which you have low-probability, insurance-like risk. So a situation something like:

(i) The event you trying to price has a low probability -> I’ll say 1/k, and

(ii) you have a known (and binary) payout if the event happens, and

(iii) you keep whatever premium you collect if the event doesn’t happen.

The original source of the conversations was providing insurance for Pepsi’s “Play for a Billion” contest in which there was a 1 in 1000 chance that someone would win $1 billion. Most of the traditional ways that people think about insurance risk fall flat here. Do you want a 10% more than the “loss cost”? So, an extra$1 million to take a 1 in 1,000 shot a losing $1 billion? Maybe 50% more? My idea was to imagine taking the bet over time and look at the probability that you would collect enough money to pay one loss before the first loss actually happened. The math behind this idea makes for a nice Calculus 1 example. Assume for simplicity that your payout will be$1.

Probability of loss -> $1/k$

Premium collected each time -> $x / k$, so you are getting “x” times the loss cost.

It will thus take you $k / x$ trials to collect enough money to pay a single loss. The probability of not paying out in the first $k / x$ trials is:

$(1 - 1/k)^{(k/x)}$ = $( (1 - 1/k)^k )^{(1/x)} \approx e^{(-1/x)}$ since k is large.

This math gives us a very simple formula which is independent of the probability (as long as the probability is low). If, say, we want a 75% chance of collecting a $1 before paying out the$1 we have to charge 3.5x the loss cost since:

$e^{(1/x)} = 0.75$, and $x = - \ln{ 0.75 } \approx 3.5$

For a 95% change you’d have to charge about 9.5x the loss cost.

Anyway, as I said this approach was laughed at by just about everyone on the other side of the table. I’m glad to see serious discussion about time averages rising to the top now!

An interest rate question extending Kate Nowak’s rate post

Earlier in the week Kate Nowak wrote a neat post about rates. The perspective in the post (in my words) is coming from writing curriculum materials for 6th grade math:

Here’s an alternate perspective on the same (or at least similar) issue that I encountered at work this week.

Suppose I ask you to play the following game:

(1) You pay me $2 today. (2) I’ll then select an integer from 1 to 10 at random (uniformly) (3) At the end of year 1 you pay me$1, and if my random number was 1 I’ll pay you $10 and the game stops. If my number wasn’t 1 we’ll meet again next year. (4) In general, at the end of year n, you’ll pay me$1 and if the random number I picked was n, the game stops.

The interest rate question relating to this games is this: What is your expected rate of return for playing my little game?

Here are two different ways to think about it:

(1) Internal rate of return

You’ll see an expected set of cash flows that look something like this:

The “internal rate of return” on those cash flows is about 12%, so you might say (and I think that many people would be quite comfortable saying) that your expected rate of return playing my game is about 12%.

(2) Accounting for the costs and the investment returns differently

One possible objection to the internal rate of return calculation is that your cash outflows are really part of your investment in the game and so are quite different than the investment return. In fact, to play the game all the way through, in addition to the $2, you need to be sure that you have access to$10 over time to play.

So, you might prefer to discount your cash outflows at a less risky rate – I’ve picked 4% just for example purposes – and discount the inflows (the investment returns) at a risky rate to measure your return. That calculation looks something like this:

Using this method the expected investment return you’ll get for paying $2 to play my games is more like 8% per annum. So, what is the correct way to think about the rate of return for playing my game? I think the rate of return question here is pretty interesting to think about and gives a real life example of the things that Nowak is thinking about writing 6th grade curriculum. A neat probability paper Christopher Long shared with me A longer than usual, and more fun than usual sequence started on Twitter yesterday with this tweet: I was excited about it because we have some non-transitive “Grime” dice and the kids still play around with them. Here’s a little project we did when we got them: Non-Transitive Grime Dice Because of the original twitter post, Christopher Long ended up sharing this neat (and fairly accessible) probability paper: Long wrote several more (maybe 10) tweets following this one which are all pretty great follow ups to the paper. What especially caught my eye in the paper was section 5 on duration calculations. I’ve spent lot of time in the last couple of years thinking about how quantities like duration and internal rate of return change when you have both positive and negative cash flows. Originally my interest in these ideas came from noticing that many people in the financial area like to talk about internal rate of return (“IRR”) even when IRR seemed to be to be a pretty dubious way to analyze the particular financial product. The interest in duration calculations followed a similar path when people were marketing some “short duration” products that didn’t seem to have short durations at all (to me). The connection with fractional coins and dice seems fascinating. Although I haven’t really settled some of the ideas in my mind, I’m happy to have stumbled into this interesting connection to a problem that was already on my mind. Pretty amazing power that Twitter has 🙂 The most important piece of math I learned in 2015 My introduction to finance came back in graduate school when a friend working at an option shop in NY asked me to explain Edgar Peters’s Fractal Market Analysis to him. This is probably a pretty non-standard first introduction to finance, but it was a lucky one for me – I found the presentation to be compelling. Two other books that helped me refine my thinking on finance were both by Mandelbrot – The (Mis) Behavior of Markets and Fractals and Scaling in Finance. My only bit of background in finance that would be considered “standard” by the business school crowd came from reading Zvi Bodie’s paper “On the Risk of Stocks in the Long Run”. This paper asks a simple, but quite educationally useful, question about option pricing via Black-Scholes – what is the price a European put maturing in the future with a strike price equal to the current price of the asset brought forward to the maturity date at the risk free rate? The answer was a surprise to me (though, of course, I’d not studied much option pricing, so anything would have been a surprise, I suppose) and understanding that result proved to be useful many years later when the friendly Wall St. folks showed up wanting to write 15 and 20 year puts with us. With that all as background, this ~30 minute talk by Nassim Taleb has the most important piece of math that I learned in 2015: These screen shots in particular show two parts that have stuck with me since the moment I saw them. The first shows roughly how much data you need to be able to talk sensibly about something governed by a power law distributions, and the second makes the point that if you try to estimate the mean of a power law distribution ( with 1 < $\alpha$ < 2) from a data set, you will likely underestimate it. If you are analyzing anything that lives in (to borrow Taleb’s word) extremistan, you have to understand the ideas in Taleb’s talk. If you don’t, the slides I highlighted show two easy traps that you’ll fall into – and there many are others, too. An incredible article about data science If you or your students are interested in understanding ways that math can be applied to problems outside of academic / school settings, this recent article from “I Quant NY” is an absolute must read. Hat tip to Patrick Honner for pointing it out to me: So much of what is important in mathematical problem solving is on full display in the piece – noticing, wondering, basic number sense, and tons and tons of persistence. oh, and no equations more complicated than calculating a 20% tip. I’d guess that students ranging in age from middle school to graduate school can get something – and probably quite a lot – out of this article. The analysis, methods, and conclusions shared in the article provide such valuable lessons that I honestly can’t think of a better starting point to understand what quantitative analysis can bring to the table in the mythical “real world.” If you want one little sound bite / takeaway, let it be this passage: When Doing Data Science, Look at Your Raw Data. If there is one thing I have learned doing data science, it is to always look closely at your raw data in addition to your aggregate statistics. It would not have been possible to figure this out without looking at a subset of individual rides.” Bravo I Quant NY!! An interesting math example coming from the world of finance Saw this article tonight on Bloomberg: The Best and Worst Investments of 2014 What caught my eye was the description of the best performing bond fund (about 2/3 of the way down the page): “Vanguard’s index fund invests in U.S. government bonds that don’t mature for 20 to 30 years. They did well in 2014, reflecting expectations that inflation will remain low for quite a while. If you’d invested$10,000 in VEDTX on Jan. 1, it would be worth $14,506 today.” Here’s the math that students interested finance (and, of course, anyone else) might find interesting: According to Bloomberg data, the 30 year US Government bond had a yield of 3.969% on December 31, 2013, and a yield of 2.752% on December 31, 2014. So, throughout 2014 the yields of long-dated US government bonds – the exact type of bonds that the Vanguard fund invests in – were quite low. In that sort of yield environment how could a fund that invests in US government bonds have had a return of 45%? Answering this question will help you gain a little bit of knowledge about how bond prices move with interest rates. Let’s start with a simple example. Say that on December 31, 2013 someone had offered to pay you a 3.969% per year return on your initial investment which would result in you receiving$1,000 in 30 years. Just to be clear, you’ll receive no interest payments on this investment – the only time cash changes hands is on day 1 when you hand over the initial investment, and at year 30 when you are handed back $1,000. How much would you have to give that person today to make this investment (ignoring taxes and all sorts of other potential complications – just to keep things easy)? One way to get to the answer to this question is to take the final$1,000 payment and divide it by the 1.03969% return that you are promised each year for 30 years. That math tells you the initial investment would be $1,000 / (1.03969)^30 =$311.09.

Now, one year later these long-dated yields have gone down to 2.752%. How much is the $1,000 you receive in 29 years worth at that rate now? We can do the same math:$1,000 / (1.02752)^29 = 455.07. Wow – what a difference!

In just one year the so-called “present value” of the future payment has gone from $311 to$455 – a gain of 46.3%! That gain is due to combination of interest rates falling by 1.25% and the $1,000 payment being far off in the future so that the change in interest rates compounds for many years. I’m cheating a little in the above calculation by using the wrong rates. For a single payment in the future I should be using so-called “zero coupon” rates, but I’m just trying to illustrate where these high returns that the Vanguard fund achieved can come from. If we look at the 30 year coupon bond we can see a similar, though not exactly the same, move in prices. Again according to Bloomberg, the price of the official 30 year US government bone on December 31, 2013 was about$96 and that bond paid an annual coupon of $3.75. Roughly speaking that means you could have bought this bond for$96 on 12/31/2013, then received $3.75 every year for the next 30 years and then received$100 at the end of 30 years.

One year later the price of the exact same bond was about $120. So, at the end of the first year you could have sold the bond for$120 and kept all of the $3.75 in interest that you received during the year. You would have paid$96 and received $123.75 in interest and sales proceeds. That’s a return of about 29% on your$96 investment! Again, that return comes from the combination of falling interest rates and the fact that the maturity of the bond is so far in the future.

So, understanding the math on bonds helps us see how it is at least plausible that a US government bond fund could have 46% returns in a year when interest rates were quite low by historical standards. It may seem surprising that a seemingly small movement in interest rates – that is rates falling from 4% to 2.75% – could produce a 46% return, but that more or less exactly how the math works out. It was a lucky year to be taking bets on the values of long-dated bonds!

3 problems from the last week that made me think about math in my work

I’ve seen three interesting problems in the last week that have stuck in my mind. I’m a little embarrassed to admit that I don’t remember where I saw one of them, but here they are with two sources:

(1)

The project here begins with this situation: “Your class needs to raise $100 to go on a field trip. You decide to sell cups of iced tea and lemonade after school. At the stand, iced tea costs$0.50 per cup and lemonade costs $0.80 per cup,” and continues with questions about graphing, lines, slopes, and things like that. (2) From the puzzle corner section of MIT’s Technology Review: Link to the November / December Puzzle Corner “Some men sat in a circle, so that each had two neighbors. Each had a certain number of coins. The first had one coin more than the second, who had one coin more than the third, and so on. The first gave one coin to the second, who gave two coins to the third, and so on, each giving one coin more than he received, for as long as possible. There were then two neighbors, one of whom had four times as much as the other. How many men were there in the circle, and how much money did each one have?” I worked through this problem with my kids here: Family Math 215 (3) From an unknown source, and perhaps not quite transcribed perfectly (sorry, again, for forgetting who showed me this problem): “10 couples attend a party. At the party each person shakes the hand of every person that he/she doesn’t already know. Assume, naturally, that each member of a couple knows each other! At some point one of the people at the party asks everyone else how many hands that he/she shook. The answers come back 0, 1, 2, 3, 4, 5, 6, 7, and 8. How many hands did the person who asked this question shake? Each of these three questions gave me something to think about, though I’ve spent the most time thinking about the first question for reasons that I’m hard pressed to explain. What’s interesting to me is that I’m starting to think that the process of working through (2) and (3) is more relevant to the math that I do on a day to day basis at work than the process of working through (1) is. That seems a little odd to me since (2) and (3) are pretty contrived, almost math contest-like, problems and (1) is more naturally geared to a business setting. But I think that it is the process of searching for patterns and ideas on the way to the solution of (2) and (3) that makes them appeal to me. Rarely are the problems that I have to think through at work as clear cut as (1) is. It isn’t that I think (1) is a poor problem or exercise. Quite the opposite, actually, and ways to play around with it have been on my mind for several days now. I think, though, that the mathematical thinking required to solve (2) and (3) are more valuable in business. Some examples of problems that I’ve had to think through at work: (A) You are approached by a well-known professional sports team who is playing in three different tournaments. The owners of the team have promised the players a total of$1,000,000 of bonuses if the team wins all three tournaments – but they have to win all three or the bonuses will not be paid. The friendly odds makers say that the chance of the team winning each tournament is 50%, 50%, and 10%. The owners would like to buy an insurance policy from you that will cover the cost of the bonuses should the team win all three tournaments. How much would you charge for this insurance?

(B) You are approached by a well-known professional sports team who has signed a famous athlete to a contract that will pay a salary of $10,000,000. The contract is guaranteed even if the player is injured and unable to play. The team would like to buy a policy from you that will pay the$10,000,000 in the event of an injury preventing the player from playing. You look at the historical data and find (and I’m making this number up) that 5% of the time a player similar to this player is injured and misses the season. You also see several stories in the press about performance enhancing drugs that could be either masking, or perhaps preventing, injuries in the historical data that you’ve reviewed. How much would you charge for this insurance?

(C) You are asked to insure a large prize at a poker tournament. Suppose for simplicity that the prize is $100,000 and everyone agrees that the mathematical chance of the prize being awarded is exactly 1 in 100. What other things do you need to consider before agreeing to insure this prize for$5,000?

So, none of these examples really match any of the three original problems exactly. But also none of the work problems really fall into the category of questions where the “right” answers just pop out of formulas. All three of the work questions require you to explore the situation a bit before heading down the path to an answer. I think that’s why questions (2) and (3) above appeal to me a bit more and (1) does even though I recognize that the lessons being taught in (1) are important to getting to the solution of both (2) and (3), and all of the problems I mentioned from work.

Expected value and Dice

Saw this post on twitter tonight:

The exercise for the students is a neat one involving biased and unbiased dice.  If you read the Twitter thread a little further you’ll see suggestions about running Chi squared stats, too.  All great stats examples.

In 2003 and 2004 I was involved in an television game show called “Play for a Billion.”  On the show 1,000 contestants tried to guess a 6 digit number from a number that I had selected.   If anyone did successfully guess the number I had selected they would have won one billion dollars.

I did an interview for the show, but it didn’t make it on air.  A first cut of that interview is below.  I thought it might be to use for a stats example because I picked my six digit number using dice.  To determine if it would be ok to use the dice I had to run through a bunch of stats that were pretty similar to what the exercise above is asking the students to do.

This project was one of the most fun math-related projects that I’ve worked on in my career.

Responding to David Coffey’s challenge

Yesterday Davd Coffey published a fascinating piece in which he asked the question – “Whose fault is it that you aren’t good at math?”:

http://deltascape.blogspot.com/2014/10/whose-fault-is-it-that-you-arent-good.html

At the end of his piece he challenges the reader to rewrite two paragraphs from another blog just as he did for his piece.  I wanted to take that challenge but found that Coffey’s single sentence “It’s because of your experiences” pretty much captured everything that I was going to say.  Damn those clear and concise writers! So, I’ll have to take an incomplete on the specific challenge but want to respond in a different way instead.

I think that most people in the US do not have a great connection with math.  There are probably millions of different reasons, but I do think that Coffey’s idea is right on – most people’s experience with math doesn’t leave them wanting to learn more about it or even study it for one minute more than they have to.  From where I sit, though, that seems like such a shame.

There is some good news, though. If you really do want to have a better experience with math there are all sorts of things right at your fingertips these days. In fact, if you are looking to have some positive experiences with math there’s probably never been a better time than right now to find them.  My response to Coffey’s challenge will be to point out a few that I’ve run across lately.

Actually I’ve picked 10 items from the last couple of years that I’ve either really enjoyed or found to be incredibly important to learn more about. There are three examples from academic math, three from work / industry, three from education, and I’ll end by talking about a sometimes politically hot idea where some school math can provide important information and insight.

(Academic 1) The 2014 Fields Medals

Every four years the Fields Medals are awarded to some of the world’s outstanding young mathematicians.  Maybe I was just paying more attention this time around, but a couple of the stories from the 2014 awards were totally captivating.  I found Quanta magazine’s profiles of Manjul Bhargava and Maryam Mirzakhani to be particularly fascinating. If you find these two articles to be interesting, I would definitely encourage you to read the profiles of the other two winners as well:

http://www.simonsfoundation.org/quanta/20140812-the-musical-magical-number-theorist/

http://www.simonsfoundation.org/quanta/20140812-a-tenacious-explorer-of-abstract-surfaces/

These two are typical mathematicians in the same way that Roger Federer and Serena Williams are typical tennis players. One difference probably is that Federer and Williams probably don’t get asked “what will I use tennis for?” very often 🙂  It is neat for me to see people doing groundbreaking work in math getting recognition. I believe that math, and academic math especially, hasn’t been so great at communicating with the public. These sorts of articles help make the world of academic math seem much less mysterious.

One of Bhargava’s early papers was made public after the awards were announced.  It was an amazing paper about factorials and it was (surprisingly) not super difficult to read (with means it only took a few days for me to get the hang of it).  Working through this paper inspired me to play around with Bhargava’s ideas a little more and even stumble on a fun tidbit hiding inside of his factorials analagous to Euler’s identity $e^{\pi i} = -1$.

https://mikesmathpage.wordpress.com/2014/08/27/a-fun-surprise-with-eulers-identity-coming-from-manjul-bhargavas-generalized-factorials/

It was exciting for me to play around with some of the early work of one of today’s top mathematicians. I’d guess that Bhargava’s paper has some great research possibilities for students interested in math.

There was quite a buzz last year when Yitang “Tom” Zhang of the University of New Hampshire proved a theorem about prime numbers that had been unsolved for 100’s of years.  He showed that there were infinitely many prime numbers that had a gap between them of no more than 70,000,000.   It is a hard theorem to understand, obviously, but it set the world of math buzzing.   Following the publication of Zhang’s result many mathematicians from all over the world worked together in an online “polymath” project and lowered Zhang’s 70,000,000 bound to around 600.  A far better description of the theorem and the back story is given in Jordan Ellenberg’s “How not to be Wrong” – itself a great read about math!  I tried to talk about the result a little with my kids, too, so they could get just a peek at the kinds of problems that academic mathematicians think about:

https://mikesmathpage.wordpress.com/2014/05/31/prime-gaps-and-a-james-tanton-problem/

Zhang, who was essentially unknown in the math world prior to proving this theorem, became a star following the publication of his theorem and has recently been awarded a MacArthur “Genius” grant. The video interview of him tells an incredible story:

http://www.macfound.org/fellows/927/%20

For me, Zhang’s story of following his dreams through some difficult times and eventually making an incredible breakthrough in mathematics is an incredibly inspiring story.

(Academic 3)  Laura Taalman’s 3D printing blog

I had the great fortune of meeting Taalman in person about a month ago.  She’s now serving as the mathematican in residence at the Museum of Mathematics in New York City – a super great win for the Museum.

I’d ran across her work online when we bought a 3D printer earlier in the year.  She was in the middle of a project in which she was making a new 3d printed object every day for a year.  It was (and is) an incredible undertaking:

http://www.makerhome.blogspot.com/2014/08/day-364-customizable-hinged-polyhedra.html

If you search back through the blog you’ll find many fascinating math projects and you’ll also see how to use some basic software to make your own 3D printing models.  This project inspired me to explore how 3D printing could help expose kids to ideas in math that they’d not seen before:

https://mikesmathpage.wordpress.com/2014/08/16/3d-printing-and-negative-space/

Taalman’s year long project on 3d printing is one of the most amazing things I’ve ever seen online. As 3d printers become more common in schools I hope more and more people will get to appreciate her work.  In the mean time, if you are in the NYC area and have the opportunity to attend one of her 3D printing projects at the Museum of Math – don’t pass it up!

Next up – Work and Industry

(Work 1) One of the most amazing applications of math that I’ve seen recently came from the Rosetta satellite project.    The Rosetta spacecraft was launched in 2004 and recently began orbiting a comet if you can believe that!  This project is truly one of the greatest scientific accomplishments of the last decade.   Recently (or at least I saw it recently) the Rosetta team published a simulation that allows you to see the path that the spacecraft took to intercept the comet:

http://sci.esa.int/where_is_rosetta/

The key to being able to map out the path of the spacecraft ahead of time is a field of math called differential equations.  The simulation from the Rosetta team is an absolutely beautiful example of differential equations in action.   I remember seeing a much simpler problem from the 1959 Putnam example as a double star (!!) problem in my differential equations book as a kid:

“A sparrow flowing horizontally in a straight line is 50 feet below an eagle and 100 feet above a hawk.  Both the hawk and the eagle fly directly towards the sparrow, reaching it at the same time.    How far does each bird fly?”

That problem seemed so cool to me and inspired me to want to learn more about differential equations.  Mapping out the required path of the spacecraft is a (much!) more complicated version of this problem. I think the Rosetta visual – as well as the pictures coming back from the comet – will inspire kids in the same way the eagle and hawk problem inspired me.  Such a beautiful illustration of math in action from the Rosetta team.

(Work 2)  In early 2014 Quicken announced that they were offering a $1 billion dollar prize if someone could correctly pick each winning team in every game of the NCAA men’s college basketball tournament. I was lucky to be involved in this promotion and it is one of the most interesting applications of probability that I’ve ever seen. One obvious question is this – what are the odds that someone could pick all of these games correctly in the first place? It is fun to see how the answers to this question change as your assumptions about the likelihood of picking games correctly change. There are other fun questions, too, and even a surprise application of the “birthday paradox” as we wondered about how many people would end up picking the games in exactly the same way. This project also led to a fun one to talk about with my kids: https://mikesmathpage.wordpress.com/2014/03/22/perfect-brackets/%20 It isn’t often that some really neat math shows up in my work, so this was a really special treat for me! (Work 3) The financial crisis – One of the important things to understand form the financial crisis is how the use and misuse of mathematical models ended up causing severe financial problems. Whether it was the statistical modelling used by the rating agencies to rate mortgage backed securities, the modeling used by regulators to measure what would happen in “stress” situations, or the modelling used by the buyers and sellers of securities, the modelling missed by a mile. It was hardly the first time math had gone horrible wrong in the financial markets. Roger Lowenstein’s excellent book “When Genius Failed” details some of the flaws in the models that were used in the financial markets in the late 1990s. This time wasn’t different! Next time won’t be different either and these lessons about not understanding correlation, gap risk, or simply that it is impossible to model human behavior are forgotten quickly. The more deal hungry people get, the faster the lessons are forgotten. This tweet from Nassim Taleb from October 8, 2014 sums up some of the modelling problems well as anything I can offer: https://twitter.com/nntaleb/status/519841458657198080 What should really concern people, though, is not just that the failures in the modeling are nearly forgotten, but these fun little complicated models are creeping away from the financial markets into other areas. Cathy O’Neil (a mathematician with experience in the financial markets) writes about these models occasionally on her blog. Here’s one of her pieces on the the use of statistical models to measure teacher performance: Why Chetty’s Value-Added Model studies leave me unconvinced If you want to understand how people are trying to either measure or mislead with models, it is important to understand the math behind those models. As these models continue to creep and creep into all aspects of life, understanding how they work will become even more important. Next up – Education (Education 1) I think there are plenty of examples from both theory and practice that are not part of a standard curriculum that kids will find interesting and engaging. I try as hard as I can to create fun discussions with my kids using as many of these examples as I can find. Two of my favorites from the last year involve Graham’s number and infinite series. I’d never heard of Graham’s number until I saw a tweet about it from Evelyn Lamb. After researching it a little bit I found a cool Numberphile video about Graham’s number. Trying to understand this number will change the way you think about numbers – would you have ever thought there could be a number that was so big that it was basically impossible to even talk about how big it was? A neat break from typical arithmetic for sure: https://mikesmathpage.wordpress.com/2014/04/12/an-attempt-to-explain-grahams-number-to-kids/ One other fun example that is really pure theory came from Jordan Ellenberg’s book “How Not to be Wrong.” In that book he talks through a concept he calls “algebraic intimidation.” The specific example he uses is the typical proof that 0.99999.. [repeating forever] = 1. I tried out some of his “intimidation” examples with my kids and they loved it: https://mikesmathpage.wordpress.com/2014/09/24/jordan-ellenbergs-algebraic-intimidation/ As I wrote above, people love to use models to justify their positions. A lot of that justification is just like Ellenberg’s “algebraic intimidation.” It is important to learn early on not to be intimidated by math. (Education 2) Social media turns out to be a great way to learn from math educators. I’m constantly surprised by how many ideas / lessons / problems / and other(!) get shared on blogs and on twitter, for example. When it comes to teaching my own kids Fawn Nguyen has been a constant source of ideas and inspiration. Here’s a fun problem she shared just a month ago: https://mikesmathpage.wordpress.com/2014/09/03/fawn-nguyen-shares-a-really-neat-math-forum-problem/ and if you want something for kids that is as far away from “do problems 1 – 50 in your algebra book” as you can get, check out Fawn’s Visual Patterns website: http://www.visualpatterns.org/%20 You might just find yourself exploring patters 1 through 50 just for fun!! (Education 3) Another teacher who’s online contributions leave you wondering “how do you find the time?” is Patrick Honner from New York City. His critical reviews of some of the NY state math exams are important reading for anyone interested in learning about the testing culture we all seem to live in: http://mrhonner.com/regents-recaps%20 Away from the exam reviews, when you see his thoughts and ideas about interesting math (and, in particular, math that will be interesting for students) you can’t help but get caught up in the excitement. Take his “proof without words” showing the sum of a neat infinite series, for example: https://mikesmathpage.wordpress.com/2013/12/21/numberphiles-pebbling-the-chessboard-game-and-mr-honners-square/ Going back to what inspired this blog post in the first place. If you are wondering about how to get excited about math, or how to find some great math experiences, it is hard to think of a better starting place on line than simply following both Fawn Nguyen and Patric Honner on twitter. (Last) The final example – Pensions An often controversial topic where math is essential to understanding the economic issues is pensions, and funding of pensions in particular. This tweet from yesterday got me thinking about using this topic as an example: One thing that makes understanding pensions particularly difficult is that seemingly small changes in assumptions produce large changes in the economics. From the math point of view, though, the math that explains that isn’t all that hard The example I’ll use below just involves the polynomial: $x^{25} + 25x^{24} + 300 x^{23} + 2300 x^{22} + . . . + 300 x^2 + 25 x + 1$ Don’t fall for my attempt at algebraic intimidation,though, a much better representation of this polynomial is the factored form $(1 + x)^{25}$ With a pension someone promises to make a payment, say, 25 years from now as part of your retirement. In order to fulfill that promise that someone (likely a corporation or a municipal entity) either needs to set aside some money today or put aside some money over time. But how much? Lots of entities think that they can earn 8% per year on their investments, so the question of how much money to set aside today boils down to finding out how much money$1 turns into in 25 years if you earn 8% per year.  It turns out that you can just evaluate the polynomial above at x = 8%, so $(1 + 8\%)^{25}$, which is approximately 6.85.   Alternately, if you’ve promised to pay $100 25 years from now and you think you can earn 8% per year on your investments, simply divide$100 by 6.85 to see how much you need to set aside on day 1.  This math tells you that you need to set aside about $14.60 on day 1 to “fully fund” that future$100 payment under the 8% growth assumption.

But let’s say that you think you can only earn 6% on your money instead of 8% – how much should you set aside to cover that $100 in 25 years now? Take a guess first, because the answer is surprising. We look at $(1 + 6\%)^{25}$ here and find a value of roughly 4.29. Taking$100 / 4.29 we get $23.31 this time – an increase of roughly 60% in the amount of money we need to set aside on day 1 to cover the promise. Amazing how much the values change with a seemingly small change in growth rate! How could the difference between earning 8% annually and 6% annually be so large? As I write this, 25 year “risk free” rates in the US are about 3.2%. You need to set aside about$45.45 if you think you can earn the risk free rate.

So, should you worry about all of these growth rate assumptions in pensions?  Well, people like to assume they can invest and earn lots more than the “risk free” rate.  They sort of doubly like to assume that when it means that they can put less money aside to cover those future obligations.  They triply like it when the consequences of being wrong occur when they are long gone. Sort of a financial and political free lunch.

But there are serious questions. For example, how much risk should you be taking when you are investing people’s retirement money?   Most people bristle at the suggestion of Social Security money being invested in the stock market, but few seem to be bothered by the fact that billions and probably trillions of dollars of pension money are invested both in the stock market and in other sorts of private investments like hedge funds.

If all of the other fun examples I’ve given above aren’t enough to get you interested in math, hopefully getting a better understanding of your own financial future will serve as good motivation.    The math – just a little bit of polynomials and maybe exponentials if you want to get slightly more tricky – is maybe not as interesting as some of the other math mentioned above, but trillions of dollars in retirement money hinges on that math.  Your own retirement might, too.  It certainly is a good enough example, I think, to refute the idea that there are no important applications of expanding and factoring polynomials 🙂

So, to wrap up – sorry to David Coffey for turning a two paragraph challenge into 2000 words. I hope, though, that someone asking themselves the question about their own math ability is able to find positive experiences with math. There are so many great examples out there these days, and so many different directions that those examples can take you. I do think that everyone will find a fun little positive experience if they are determined to look around long enough.