Non-Deterministic Mathematics

Non-Deterministic Mathematics: Patterns of Uncertainty Ideas with Probability

Washington University Middle School Mathematics Teachers’ Circle Richard Lodholz, February 6, 2008

Mathematician, Amir D. Aczel, in his book, Chance: A Guide to Gambling, Love, the Stock Market, & Just About Everything Else, describes probability as humanity’s attempt to understand the uncertainty of the universe.

In his book, Against The Gods: The Remarkable Story of Risk, economist, Peter L. Bernstein states that, “The revolutionary idea that defines the boundary between modern times and the past is the mastery of risk…” The modern conception of risk is rooted in the Hindu-Arabic numbering system that reached the West seven to eight hundred years ago, with the serious study springing during the Renaissance.

These references are given to convince anyone that the domain of probability is a perfect place in which to develop appreciation for and interest in mathematics. If you have never had a course in probability, consider one. If you want a quick and easy dose of probability, I suggest the Aczel book mentioned above.

We can learn concepts and develop mathematical skill in determining probabilities of events, but this could be called non-deterministic mathematics. The calculated measures provide patterns and predictions, but do not determine the event.

Investigation 1. In 1654, a French nobleman, the Chevalier de Mere, challenged the famous mathematician, Blaise Pascal to solve a puzzle. This restatement of the puzzle comes from John Allen Paulos, in his book, Innumeracy.

Two men bet on a series of coin flips. They agree that the first to win six such flips will be awarded $100. The game, however, is interrupted after only eight flips, with the first man leading 5 to 3. The question is: How should the pot be divided? One might say the first man should be awarded the full $100, because the bet was all or nothing and he was leading. Or, one could reason instead the first man should receive 5/8 of the pot and other one the remaining 3/8 because the score was 5 to 3. Pascal’s solution, one of the first problems in probability theory, was argued as follows. It might be argued that because the probability of the first man’s going on to win can be computed to be 7/8 (the only way the second man could have won is by winning three flips in a row, a feat with probability of 1/8 = 1/2 x 1/2 x 1/2), the first man should receive 7/8 of the pot and 1/8 for the second man.

The point of this storyline in history is that the criteria for deciding on any one of these divisions are nonmathematical. Mathematics can help determine our positions and what we might value, but we create the mathematics.

Lodholz page 1 Investigation 2. The Weird Dice

QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture.

The Storyline: These dice were discovered by Colonel George Sicherman, of Buffalo, New York and were originally reported by Martin Gardner in a 1978 article in Scientific American.

Check the site, (http://blog.plover.com/math/dice-and-polyhedra.html) for the notes below and additional interesting information on “The Sicherman Dice”.

Crazy dice refers to a standard mathematical problem or puzzle in elementary combinatorics, involving a re-labeling the faces of a pair of six-sided dice to reproduce the same frequency of sums as the standard labeling.

It is a standard exercise in elementary combinatorics to calculate the number of ways of rolling any given value with 2 fair six-sided dice (by taking the sum of the two rolls).

A question arises whether there are other ways of re-labeling the faces of the dice with positive integers that generate these sums with the same frequencies. The surprising answer to this question is that there does indeed exist such a way. These are the Sicherman dice.

Another interesting question is: which set, of dice is better for backgammon. Both sets have advantages: the standard set rolls doubles 1/6 of the time, whereas the Sicherman dice only roll doubles 1/9 of the time. (In backgammon, doubles count double, so that whereas a player who rolls (a,b) can move the pieces a total of (a+b) points, a player who rolls (a,a) can move pieces a total of 4a points.) The standard dice permit movement of 296/36 points per roll, and the Sicherman dice only 274/36 points per roll.

Investigation 3. The Non-transitive Dice

Lodholz page 2 QuickTime™ and a TIFF (LZW) decompressor are needed to see this picture.

The game is a two-person game in which each person chooses a die to toss. The highest number on one toss from each person determines the winner. What patterns in the pairings of dice will provide the best chance of winning.

Investigation 4. If a segment is broken at random in two places to form three pieces, what is the probability that these three pieces can form a triangle?

A nice connection to some geometry will produce an answer. Consider the illustration below with perpendiculars from a point of one side of an equilateral triangle to points on A R B the other two sides of the triangle.

A marvelous theorem in geometry states: If perpendiculars are constructed from a point on one side of an equilateral triangle to points P on the other two sides, then the sum of the lengths of these perpendicular segments S equals the altitude of the equilateral triangle.

C You can show this by drawing the two altitudes that are parallel respectively to PR and PS and then use similar triangles to the relationship of PR + PS = an altitude.

PS is to PC as the altitude from A is to PC + AP. Also PR/AP = (altitude from C)/(PC+AP). Solve each of these ratios for PS and PR respectively, and then add PS and PR to show the sum is the altitude.

Lodholz page 3 From this we can conclude that if perpendiculars are drawn from any point inside an equilateral triangle to the sides of the triangle then the sum of lengths of these perpendicular segments equals the altitude of the triangle. PS+PR+PQ=altitude

Two of these segments are greater than the third if and only if the point P is inside the equilateral triangle determined by the three midpoints of the sides of the original.

If a point determining the perpendiculars lies in the un-shaded region, it is clear that two of the perpendicular segments will have a total length shorter than the longest segment. So there will be no triangle possibility.

The area of the shaded triangle is 1/4 of the total area, so the probability of forming a triangle with the three segments is 1/4.

Investigation 5. The unexpected becomes the expected. (The following is taken from, The Golden Ratio, Mario Livio, pp.232-235)

Examining the first-digit for numbers taken from all sorts of data presents some interesting results. Astronomer/mathematician, Simon Newcomb (1835-1909) first discovered the “first-digit phenomenon” in 1881. He noticed that books of logarithms in the library, which were used for calculations, were considerably dirtier at the beginning (where numbers staring with 1 and 2 were printed) and progressively cleaner throughout. He went further than merely observing the pattern; he developed a formula that presented the appearance of 1 with a probability of 30%, P(2) = 17.6%, P(3) about 12.5%, P(4) about 9.7%, P(5) about 8%, P(6) about 6.7%: for 7 about 5.8%, for 8 about 5%, and for 9 about 4.6%.

Newcomb’s article appeared in the American Journal of Mathematics and the “law” he discovered went unnoticed, until 57 years later, when physicist, Frank Benford of General Electric rediscovered it (apparently independently). Benford tested it with extensive data on

Lodholz page 4 river basin areas, baseball statistics, and even numbers appearing in Reader’s Digest articles.

Not all numbers follow the law, but if you collect all the numbers appearing on the front page of several newspapers for a week, you will obtain a pretty good fit. Try it.

Attempts to put the law on a firm mathematical basis have shown little, but in 1995-96 mathematician, Ted Hill gave an explanation. Hill formulated the law statistically, in a new way: “If distributions are selected at random (in any unbiased way) and random samples are taken from each of these distributions, the significant-digit frequencies of the combined sample will converge to Benford’s distribution, even if some of the individual distributions selected do not follow the law.”

So, Hill showed that even if some distributions do not obey Benford’s Law by themselves, when you collect ever more of such numbers, the digits yield frequencies which conform ever closer to the law’s predictions.

Investigation 6. Phrasing the questions (from John Allen Paulos, Innumeracy) Imagine you are a general surrounded by an overwhelming force that will wipe out your 600-man army unless you take 1 of 2 escape routes.

Situation 1: If you take route A your intelligence officers explain that you will save 200 soldiers. If you take route B the same intelligence says there is a probability of 1/3 that all 600 will make it, and 2/3 probability that none will. Which route do you take?

Situation 2: If you take route A, 400 of your soldiers will die. If you choose route 2, the probability is 1/3 that none of the soldiers will die, and 2/3 that all 600 will die. Which route do you take?

The two situations are identical with only the phrasing changed. The first is positively stated with saving 200 lives (3 out of 4 people choose route A). The second situation is a negative statement about 400 soldiers killed (4 out 5 choose route B).

This is a great example to show the importance of analyzing survey questions before jumping to conclusions about results. It also points to the importance of probability as a necessary domain in mathematics education.

Investigation 7. Expected Improbabilities (from Innumeracy) Consider two baseball players, say Babe Ruth and Lou Gehrig. Ruth has a higher batting average the first half of the season. In the second half Ruth also has the higher average. The batting champion for the year is, of course, Lou Gehrig. Players must have at least 300 plate appearances to qualify for the title. Illustrate how this might be possible.

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