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.

The Possibility of Throwing a Football 100 Yards

Continuing from last week's post we are exploring the possibility of throwing a football one hundred yards.

That in a moment. But first, I heard on a sports radio show that for the upcoming Super Bowl game the Patriots are favored by 3 points. They also said that the over/under for total points that both teams would score was 57 (as I recall. It was a few days ago.) That is, you can bet that both teams combined will score more than 57 points or you can bet they'll score less. Of course I thought, "Hey, nice system of equations problem."

If x is the points scored by the Atlanta Falcons and y is the number of points scored by the New England Patriots, then we have:

y = x+3 and x+y = 57

That leads to x+(x+3) = 57
So, 2x+3 = 57
So, x = 27 and y = 30
Thus the odds makers are figuring:  New England 30, Atlanta 27.

Now, back to: Is it possible to throw a ball 100 yards?

You can look back on the last post to fill in what you might have missed, but currently we are at this formula.

d = (v²/g)ᆞ(sin(2θ))

The distance we are hoping to cover is 100 yards, which is 300 feet. The optimal throwing angle is 45 degrees. The acceleration due to gravity is 32. So,

300 = (v²/32)ᆞ(sin(2(45)))

Solving this gives us v = 97.98 feet per second which is 66.8 miles per hour.

I thought this should be possible. Ater all, there are people that throw a baseball over 100 mph. Surely someone could throw at 66.8 mph. Yes, footballs are heavier and probably have more air resistance, but still...

Turns out that 60 mph is about the best anyone is going to be able to do. You can check it out here
http://ftw.usatoday.com/2014/03/how-fast-football-throw-nfl-combine-logan-thomas. They've tested it out and no one has done better than that in the times they've been keeping track. Maybe someday someone will come a long that throws a football a lot faster than anyone else. But until then the longest passes will be well under 100 yards.

Throwing the Length of a Football Field

I had heard once that there was a quarterback (Roman Gabriel of the L.A. Rams) that could throw a football the length of the field. I think we need to investigate this. I found the formula for to calculate the range of any projectile:



We will work this out using the English System of measurement. If the football field was 100 meters long instead of 100 yards (It would only be 9 yards longer), the metric system would probably catch on in the U.S. a lot faster. Alas, that won't be happening any time soon.

So we let d = 300 feet, g = 32 feet per second, the angle is the optimum throwing angle of 45 degrees, the initial height is 6 feet, which I figure is about where the ball would leave the quarterback's hand. The velocity is what it would take to throw it that far.

This is not the way to go with this. For one thing, solving it is really hard. Trust me. I did it and its rough. You could assign it for a massive amount of extra credit, but otherwise it isn't worth it. Secondly, I didn't take into account that we are measuring to where it hits the ground which is not the same elevation that it took off anyway.

Let's just take a starting height of zero. That will make things easier and it turns out demonstrates an nice application of a trig identity.

So here we go. With a starting height of zero, the above equation becomes:

d = (vcos(θ)/g)ᆞ( vsin(θ) + vsin(θ))

d = (vcos(θ)/g)ᆞ(2vsin(θ))

d = (v²/g)ᆞ(2sin(θ)cos(θ))

But, there is a double angle identity that states: sin(2θ) = 2sin(θ)cos(θ). So our formula becomes:

d = (v²/g)ᆞ(sin(2θ))

This then becomes the formula that is commonly found for range of a projectile with a starting height of zero.

Nice application of an identity.

I feel that is quite enough excitement for one day. Next time we'll answer the question of whether Roman likely threw the ball 100 yards.

Dropping Pennies from the Empire State Building

You probably heard about the idea of dropping a penny from the Empire State Building and its

probability of doing some damage. The word from various sources is - No, it won't do much of any damage. If it hit you, you might think, "Hey what was that?"

They say (they being those in charge of such things) that a big part of the reason is the flat shape of the penny and the air resistance it would meet. Granted, but what if there is no resistance? Let's try it out.

(Final velocity)² = (Initial velocity)² + 2gd is a formula for a free falling object. (Time out - I, for the first time, used html to show exponents. Wow!! Big day for me.)

Mass of penny = 3.1 grams. (I'm going with that, but I saw that it is now 2.5 grams. And I didn't even notice.) Height of Empire State Building is 86.42 meters, and initial velocity of zero. That gives us a final velocity is 86.42 meters per second or 193.32 miles per hour. But as they say, that is a moot point as air resistance is not going to allow it to go that fast. Even if it is going fairly fast, it doesn't weigh very much.

Then I thought, what if you drop a baseball and what would it's impact be? And what would be a measure of its impact. Would we be looking at momentum or kinetic energy? If I had gotten an A instead of a gift B in physics, I might have an answer for that.

Momentum = mass x velocity
Kinetic Energy = mass x (velocity)²

So take the penny's momentum/energy. What would the velocity of the baseball be to have the same impact as the penny?

To make a long story short, I figured if having the same momentum, the baseball would be going 1.488 m/s or 3.329 miles per hour.

If looking at kinetic energy, the baseball's velocity would be 28.275 miles per hour.

Neither speed would hurt a whole bunch. Hopefully no one drops something like an anvil.

10 Cool Things about Rene Descartes

This is a blog about high school applications of mathematics. However, sometimes I've thrown in some historical things or bios. I was thinking maybe once a month I might focus on a particular mathematician. It might be interesting when presenting, say the topic of analytic geometry, some info on the so-called "Father of analytic geometry" Rene Descartes. We'll see how it goe.

I thought, also, instead of a dry bio, make it into a top ten list. These won't necessarily be the 10 most vital things about the mathematician. Just some things I found interesting.

Without further ado: Ten Cool Things about Rene Descartes

1. His hometown, LaHaye, France is now known as Descartes, France in his honor.
2. He earned a law degree from the University of Poitiers.
3. he did not publicize his theories on the solar system being heliocentric after seeing the treatment Galilleo received.
4. In spite of this, Pope Alexander still put his works on a list of prohibited books.
5. He never married, although had a daughter who died at the age of five.
6. Feeling there were to many things being accepted as true, he started with only "Cogito, ergo sum" - "I think, therefore I am.
7. At the turn of the millennium, the A&E Network conducted a survey of historians on the 100 most important people of the millennium. Descartes came in #32.
8. The concept of exponents predated his time, but Descartes had the idea of writing them as superscripts.
9. He showed that a viewers angle from the center to the edge of a rainbow is 42 degrees.
10. He wrote proofs verifying the existence of God.
11. 

World Population Growth

Here is an interesting graph. (https://ourworldindata.org/world-population-growth/) It shows the world's population growth up to the present day and then someone's estimates as to what will happen in the next few decades. It probably takes a little looking at for it to make sense. I combines two graphs in one. The horizontal axis shows the passage time in years and the vertical shows growth rates. The graph also shows the total population although these numbers are just recorded on the graph rather than being recorded on the vertical axis. As line graphs go, it's a pretty busy graph.

It seems to me that math teachers could make use of the graph in pretty much any high school mathematics class.

This actually started for me with information I found in the 2017 World Almanac. It showed population estimates going much farther back in time than this graph shows. You could look at that information as a set of ordered pairs with years being represented as x-values and world population (in billions) as y-values. The almanac states that in the year one the population was an estimated 300 million. That gave me an ordered pair of (1,0.3). Proceeding in this manner gave me ordered pairs of (1,0.3), (1250,0.4), (1500,0.5), (1804,1), (1927,2), (1960, 3), (1974,4), (1987,5), (1999,6), (2011,7).

Just using the raw data, an Algebra I class might simply write ordered pairs, or without seeing the above graph, choosing appropriately labeled axes to graph the data.

Higher math classes could look at finding an equation to model the data. I had more trouble than I thought I would. I guess that is because, as the graph shows, the rate has varied over time just in the last couple centuries, let alone millennia. Leaving out the first few ordered pairs and adjusting the data such as changing (1804,1) to (0,1) and so on, I was able to find an equation that had a correlation of r = .9647. Students could maybe experiment with similar things to get a best fitting curve.

Calculus students would be able to examine the blue population growth curve and discuss how it ties into first and second derivatives. It is interesting that the person making the future projections seems to think our current point in time seems to correspond to an inflection point. Students could discuss what that really means and mathematically and socially.