# Pressing Toward the Prize

## Posts Tagged ‘probability’

### Of Songs, Puzzles, and Mathematical Analysis

Posted on: October 13, 2010

While reading the September 2010 issue of “Math Horizons” magazine, I ran across some fun and interesting articles. In the “fun” category was an article written by Donald Byrd, a parody, or mathematicizing (is that a word??), of the seemingly endless song “100 Bottles of Beer on the Wall.” For the infinite version, with infinite bottles of beer on the wall, one falls down, and you are left with infinite bottles of beer, forever. The larger infinity version starts with uncountable bottles, if countable bottles fall,  then you still have uncountable bottles of beer. The indeterminate version: start with infinite bottles of beer, if infinite bottles of beer should happen to fall, you are left with indeterminate bottles of beer on the wall, and this is the end of that song! Other versions include geometric progression, Euler’s identity, differentials, identity, topological dimension, and fractal dimension. All in all, pretty entertaining.

Another article, written by Bruce Torrence and still in the “fun” category,  described a convention of magicians, mathematicians, and puzzle masters held in honor of Martin Gardner, “the prolific and magnetic author whose interests spanned the seemingly disparate disciplines of mathematics, puzzles, magic, and the spirited debunking of pseudoscience.” A particularly obscure puzzle was presented by one of the speakers before he abruptly ended his brief talk and sat down: “I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?” …brief pause… “The first thing you think is ‘What has Tuesday got to do with it?’ Well, it has everything to do with it.” As it turns out, this riddle became the talk of the convention, and was finally solved by counting all possible outcomes, including gender, births, and days of the week, and using basic probability theory. By the way, the answer is 13/27.

Finally, of extreme interest to me, was an article about the terrors of Mathematical Analysis. This is the course I will be taking in the spring, and it is the only course I still need to fulfill my major. The bad part is that I am already stressing about taking it. I have heard frightening things about how impossible this course is, and how I should just resign myself to being happy if I pass it, let alone trying to get a good grade in it. So Tina Rapke’s article “Confronting Analysis” was speaking to me. She shares about how she struggled with mathematical analysis in the beginning, actually “drowning” is the word she used, and how she eventually found understanding and success by hard work, tenacity, and consulting multiple resources, some of which she discusses in the article. She is now a Ph.D. candidate pursuing an interdisciplinary degree in mathematics and education, having written her PH.D. candidacy exam in analysis, and passing. Her advice to those who struggle with analysis? a) You are not alone! b) Be open to various textbooks and resources, and  c) Don’t give up!

I think I will keep this article close at hand… spring semester is coming!

### The Top of the Pyramid: Calculus or Statistics?

Posted on: September 20, 2010

I was recently discussing with my math professor and a fellow student the relevance, or lack thereof, of current math courses to high school students. When I commented I wasn’t sure if it was the relevance of the courses, or the marketing of those courses, that was the problem, my professor directed me to a video of a talk given by another math professor, Arthur Benjamin. In it he explains why he feels that making calculus the pinnacle of the high school math curriculum is not the best, and that students (and society) would be better served if learning probability and statistics was the zenith. He does not minimize the value of calculus for certain students, but he feels it is not relevant to our everyday lives in the way that probability and statistics are.

After taking my first statistics course, I was so impressed by the relevance of the material that I began to believe, and still do, that everyone in America should know statistics. If followed, Professor Benjamin’s “formula for changing math education” would make that possible. We are so inundated in our culture with facts, figures, and persuasive techniques involving statistics (think sports, politics, lotteries, advertising), that one needs to be able to sift through the rhetoric in order to recognize the reality. If one more fully understands the data being presented, one can make more informed decisions. It would seem, then, that I have had to re-think my position on high school math in favor of changing the current course offerings and emphasis to bring more relevance to our students. And as Professor Benjamin points out, probability and statistics involve uncertainty and risk, which are clearly relevant to our everyday lives.

### Kings, Dukes, and Truelists

Posted on: September 15, 2010

On Tanya Khovanova’s Math Blog I encountered a couple of interesting probability puzzles involving a truel. In one, the three men have an infinite number of bullets and they shoot in order until only one man is left alive, and in the other, each truelist has only one bullet. Ms. Khovanova gives the solution for the second one, based on probabilities and basic survival instincts, but leaves the first one to the reader. These puzzles remind me of a problem we were given in Professor Edgar’s M317 Proofs class:

“The five Dukes of Earl are scheduled to arrive at the royal palace on each of the first five days of May. Duke One is scheduled to arrive on the first day of May, Duke Two on the second, etc. Each Duke, upon arrival, can either kill the king or support the king. If he kills the king, he takes the king’s place, becomes the new king, and awaits the next Duke’s arrival. If he supports the king, all subsequent Dukes cancel their visits. A Duke’s first priority is to remain alive, and his second priority is to become king. Who is king on May 6?”

We all tried to solve this, and there was some interesting logic involved in our attempts, but even with the hint that working backwards would help, only one young lady in our class was able to solve it. She also solved it for six Dukes, and discovered a pattern that followed for even and odd numbers of Dukes. I have given a few hints, but I am not going to give the solution in case others might like to try it.

I have noticed that in order to solve any of these three puzzles, not only must probabilities for each action be considered, but one must have a good understanding of human nature as well. These are not merely about numbers, but also involve a fair amount of psychology. Two things I take away from this: 1) sometimes it is necessary to work backwards to find a solution, and 2) math does not exist in a vacuum!