NCAA bracket odds

We're down to the "Elite 8" in the 2014 NCAA tournament. I've been hearing some discussion of the odds of filling a "perfect bracket." It has been discussed at On Point Radio, which I listen to frequently. The program aired on March 26, 2014. USA Today has a youtube video up of a DePaul math professor estimating the odds. He comes up with an estimate that a savvy bracket filler has about 1 in 128 billion odds of filling in a perfect bracket. It's March Madness.

Then I thought to myself, how did the professor get there? USA today does not go into the details. I wonder what the odds are of finding real math in USA today on a given today? Probably slimmer than 1 in 1 million. I'll point out that precision does not matter when we're finding astronomically long odds. I'll call this a ballpark estimate. With probabilities this small, the upshot is that you'll never win. It's worse than playing lotto.

Let's do some math. Sixty-four teams compete in the NCAA tournament. There are six rounds, including the final game. So we have 32+16+8+4+2+1 = 63 games played. In order to fill out a perfect bracket, I have to choose 63 winners correctly. If we were flipping a coin, there are

possible sequences of 63 coin flips. A bracket filler would have odds of about 1 in 9 quintillion of filling out the bracket correctly. There is no hope of winning here.

But we are not flipping a coin. Suppose you're a savvy odds maker. You know which teams are ranked highest in the tournament. You study basketball statistics, etc. I will quantify your savvy thusly: you have a 90% probability of picking a winner of 16 games in the first round. You have a 90% probability of picking a winner of 16 games in the second round. Your probability of picking all the other winners is 50/50, the same as flipping a fair coin. What is your probability of filling in the bracket correctly? You have to correctly choose 63 mutually exclusive events with a probability of .9 for 32 of them, and .5 for the other 31. Since the events are mutually independent, the individual probabilities multiply. That gives us a final probability of

or odds of

.

This is roughly twice as much probability as the DePaul math professor estimated. So, my hypothetical savvy guesser had roughly twice as much savvy, but he is still hopeless when it comes to filling in the bracket correctly.

Here is some R code to calculate and print the numbers here. We're right on the edge of R's basic precision, where p could be rounded down to zero. If I change the numbers slightly, I would need to use the Rmpfr package (or logarithms) to get a non-zero printout.

> p=c(0.9,0.5)
> games=c(32,31)
> prob=prod(p^games)
> prob
[1] 1.598934e-11
> 1/prob
[1] 62541682939
> format(1/prob,dig=3,sci=T)
[1] "6.25e+10"

Hyperreal Numbers and Calculus without Limits

Once upon a time, I stumbled into a hyperreal number field on the internet when searching for something related to calculus. Maybe I was calculating the volume of a Steinmetz Solid? The so-called hyperreals can be used to develop the calculus without limits. The claim is made that, perhaps, the use of "infinitesimals" is less confusing than the concept of the limit for struggling calculus students. In fact, there is a calculus text using this approach that one can download for free from this page. The author, H. Jerome Keisler, is a professor emeritus of mathematics at UW-Madison.

What are the hyperreals? They are an extension R* of the field of real numbers, which is usually denoted by R. R is a subset of R*. Without going in to too much detail, R* is obtained from R by adding in infinitesimals and infinitely large numbers. An infinitesimal number is infinitely small. What does this mean? For every real number a, a number ε > 0 is infinitesimal if -a < ε < a. That's it. In contrast, 1/ε is infinitely large (a "hyperinteger"). Conceptually, hyperreals are similar to complex numbers. However, unlike the complex numbers, they are depicted in one linear dimension, not a two-dimensional plane. The nitty gritty is all in Keisler's book. With the concept of the infinitesimal available, we can have the hyperreal quantity x + ε that is infinitely near x. This hyperreal x + ε can then be used to formulate the derivative, which won't be surprising to anyone who has studied calculus.

Here's figure 1.4.3 from Keisler's book. An "infinitesimal microscope" has been used to zoom in on the hyperreal number line:

The infinitesimal microscope is a captivating pedagogical concept. I claim that it's easier to grasp than the concept of a limit. Children learn about microscopes in grade school. Maybe they learn about it nowadays by zooming in on their iPads. In any case, R* is a useful mathematical concept that can be used to formulate the calculus without that ungainly Σ thing. All you need is a little ε guy. :-)  Keisler's book provides more than 800 pages of evidence.

The history of infinitesimals is interesting. The epilogue [pdf] to the aforementioned calculus text lays out the story. Archimedes anticipated both infinitesimals and the ε, δ approach of limits in some of his proofs. In the 17th century, the modern calculus was developed independently by Newton and Leibniz. In his reasoning, Newton used both infinitesimals and the concept of a limit as well as the so-called velocity method. Leibniz used infinitesimals. So there were three competing methods for doing calculus: infinitesimals, limits, and the velocity method. Jumping ahead (see Keisler's epilogue for whole story), the first truly rigorous treatment of calculus was formulated by Karl Weierstrass in the 1870's. I believe it was Weierstrass who introduced the  ε, δ notation that we still use today.

There can be little doubt that the limit approach is firmly entrenched in the current pedagogy. Other than Keisler's book, I am unaware of any other textbook that uses this approach. I never heard it in Altgeld Hall or Van Vleck. Mathematics teachers should be aware of this alternative approach and should consider using it to introduce students to derivatives and integrals.

Raising the Minimum Wage

President Obama, in his recent State of the Union (SOTU) speech, called on congress to raise the federal minimum wage and to start reducing the enormous inequality that has arisen in this country in the last 30 years. The people at the top of the economic food chain in this country are doing fantastically well right now, in fact they are better off financially than any similar cohort has been in the history of the world. Congressman George Miller (D-CA) and Senator Tom Harkin (D-IA) have introduced the Fair Minimum Wage Act of 2013, which would raise the minimum wage to $10.10/hr.

As Robert Reich has pointed out, this is a no-brainer. Republicans are claiming that raising the minimum wage will cost jobs. This is untrue. As Reich points out in the 2:29 video embedded below, minimum wage jobs are service sector jobs such as flipping burgers or changing the linen as a hotel maid. Business owners can easily absorb the cost of an increased minimum wage for pennies on the dollar, which they would pass on to customers. Furthermore, workers on the bottom rungs of the economic ladder spend everything they earn. The dollars spent on increasing their wages have an immediate stimulus effect.

Economists agree that raising the minimum wage would reduce poverty. Arindrajit Dube, an economist at UMass Amherst, has calculated that raising the federal minimum wage to $10.10/hr could lift 4.6 million people out of poverty. Of course, republicans are currently howling that any raising of the minimum wage will wreck the economy. There is no evidence to support this doomsaying. What we aren't told is that the real value of the minimum wage has been falling now for decades. This is illustrated by the following graph:

The blue points are the federal minimum wage by year in 2009 dollars. The maximum value occurred more than 45 years ago in 1968. The value was $9.86. This was a time of great economic prosperity in the US. The dollar values were computed by dividing wage values by the consumer price index (CPI) for a given year. The CPI was normalized to 2009. The blue curve is a lowess smooth computed in R with a "smoother span" of 0.4. Note that it has leveled off. The red data points are federal minimum wage data before the CPI adjustment. The first data point in 1938 was $0.25 after the minimum wage was instituted by Roosevelt. The 1938 value is the minimum value on either curve.

Looking at the graph above, one notices that the smoothed fit from 1938 – 1968 is quite linear. It's a simple matter to fit a straight line to these data using R and then extrapolate. Here is the result in graphical form. R-squared was 0.85 for the linear fit with lm.

Minimum wage in real dollars with linear fit 1938 – 1968

Where does this take us? If we had stayed on that linear growth curve, the minimum wage in 2013 would have risen to $18.10!

Reich has been hammering on economic inequality for years. He has put his arguments into movie form, Inequality for All. Watch the official trailer or youtube if you dare. Raising the federal minimum wage to $10.10/hr would put it back to where it was back in the 60s before it started falling. It's the right thing to do for the poorest Americans.

For the first graph above, the data and the R code used to generate it.

[Apr 7, 2014. Edited to remove an html display bug]

Randomness and literacy

The generation of random numbers is too important to be left to chance. Robert Coveyou

I banged my shin the other day. The bruise on my left tibia just below the knee is a painful reminder of this event. A distraction was provided by the phone ringing. I stood up, thinking about answering, and I smacked my leg right into the coffee table. The ringing of a phone is a perfect example of a random event. If it had not rung, would there be a bruise on my shin? Probably not.

Random numbers, randomness, and the generation of random numbers are important topics. Randomness is particularly relevant to current events because of its essential use in modern cryptography, which has been in the news lately with articles about Edward Snowden and the NSA. Embedded within a larger frame of mathematics, science, and current events, these topics can provide plenty of impetus for interesting conversation and mathematical diversion.

More to the point, I intend to discuss literacy in the mathematics and science classroom. What can we do to motivate students to learn mathematics? One technique I have used and will continue to use is critical literacy. The teacher can display, e.g., the text of a newspaper or magazine article that gets the math wrong, that provides an example of innumeracy. This text can then serve as a jumping-off point for a discussion of a proper mathematical analysis.

Another technique I find intriguing is the discrepant event. A discrepant event is a demonstration or a question with a surprising or startling conclusion.  An attention-grabbing event can be used to initiate the process. The discrepancy creates a cognitive springboard and forces the students to think about the subject matter. An example of a discrepant event was provided to me recently. A question was posed about fish bladders, an organ possessed by ray-finned fish such as the largemouth bass, Micropterus salmoides. I intend to expand on this topic in a later post.

A closely related technique is the thought-provoking question. How many years is one billion seconds? How many cells do you have in your body? These questions can provide a nice stimulus for a lesson and get the gears turning in the student's heads. Each of which has hundreds of thousands of hair follicles, of course.

Literacy can be used in the classroom to motivate, to captivate, and to initiate discussions. Mathematics is a complex topic, and motivating young students to learn can be challenging. It behooves us as teachers to have many arrows in our educational quivers.