## Mathematics Illuminated

# Making Sense of Randomness

## How can we make sense out of the seemingly random results of throwing a pair of dice or even the haphazard flow of heavy traffic in the city? How can we talk meaningfully about any situation that is unpredictable or has an uncertain outcome? Well, welcome to the mathematics of probability.

Probability is the mathematical study of randomness, or events in which the outcome is uncertain. This unit examines probability, tracing its evolution from a way to improve chances at the gaming table to modern applications of understanding traffic flow and financial markets.

### Unit Goals

- Mathematical consideration and understanding of chance took a curiously long time to arise.
- The outcome of any particular event can be unpredictable, but the distribution of outcomes of many independent events that are trials of a single experiment can be predicted with great accuracy.
- The simple probability of a particular (“favorable”) outcome is the ratio of outcomes that are favorable to the total number of outcomes.
- The Galton board is a useful model for understanding many concepts in probability.
- The Law of Large Numbers says that theoretical and experimental probabilities agree with increasing precision as one examines the results of repeated independent events.
- The distribution that results from repeated binary events is known as a binomial distribution.
- A standard deviation is a measurement of the spread of the data points around the average (mean).
- A normal distribution is determined solely by its mean and standard deviation.
- The Central Limit Theorem says that the average of repeated independent events is, in the long run, normally distributed.
- Conditional probability and Markov chains provide a way to deal with events that are not independent of one another.
- The BML Traffic model represents the frontier of modern probabilistic understanding.

### Unit Glossary

### ADDITIONAL UNIT RESOURCES: BIBLIOGRAPHY

# Bibliography

## WEBSITES

http://www.dartmouth.edu/~chance/

Barth, Mike. “Industry Loss and Expense Ratio Comparisons Between HMOs and Life/Health Insurers,” *NAIC Research Quarterly*, vol. 3, no. 1 (January 1997).

Berlinghoff, William P and Fernando Q. Gouvea. *Math Through the Ages: A Gentle History for Teachers and Others*. Farmington, ME: Oxton House Publishers, 2002.

Berlinghoff, William P. and Kerry E. Grant. *A Mathematics Sampler: Topics for the Liberal Arts*, 3^{rd} ed. New York: Ardsley House Publishers, Inc., 1992.

Bernstein, Peter L. *Against the Gods: The Remarkable Story of Risk*. New York: John Wiley and Sons, 1996.

Bogart, Kenneth, Clifford Stein, and Robert L. Drysdale. *Discrete Mathematics for Computer Science*(Mathematics Across the Curriculum). Emeryville, CA: Key College Press, 2006.

Burton, David M. *History of Mathematics: An introduction*, 4^{th} ed. New York: WCB/McGraw-Hill, 1999.

Casti, John L. *Five More Golden Rules: Knots, Codes, Chaos, and Other Great Theories of 20 ^{th}-Century Mathematics*. New York: John Wiley and Sons, Inc., 2000.

David, F.N. Games, Gods and Gambling; *The Origins and History of Probability and Statistical Ideas from the Earliest Times to the Newtonian Era*. New York: Hafner Publishing Company, 1962.

De Fermat, Pierre. “Fermat and Pascal on Probability” from Oeuvres de Fermat, Tannery and Henry, eds. University of York. http://www.york.ac.uk/depts/ maths/histstat/pascal.pdf (accessed October 26, 2005).

Desrosieres, Alain. *The Politics of Large Numbers *(translated by Camille Naish). Cambridge, MA: Harvard University Press, 1998.

D’Souza, R. M. “Coexisting phases and lattice dependence of a cellular automata model for traffic flow,” *Physical Review* E, vol. 71 (2005).

Epstein, Richard A. *The Theory of Gambling and Statistical Logic*. New York: Academic Press, 1977.

Gigerenzer, Gerd et al.* The Empire of Chance: How Probability Changed Science and Everyday Life*. New York: Cambridge University Press, 1989.

Ghahramani, Saeed. *Fundamentals of Probability*, 2^{nd} ed. Upper Saddle River, NJ: Prentice Hall, Inc., 2000.

Gordon, Hugh. *Discrete Probability*. New York: Springer, 1997.

Grinstead, Charles M. and J. Laurie Snell. *Introduction to Probability*: 2^{nd} rev. ed. Providence, RI : American Mathematical Society, 1997.

Griffiths, T.L. and Tenebaum, J.B. “Probability, Algorithmic Complexity, and Subjective Randomness,” Proceedings of the Twenty-Fifth Annual Conference of the Cognitive Science Society, (2003).

Gross, Benedict and Joe Harris.* The Magic of Numbers.* Upper Saddle River, NJ: Pearson Education, Inc/ Prentice Hall, 2004.

Kelly, J.L. Jr. “A New Interpretation of Information Rate,” *Bell Systems Technical Journal*, vol. 35 (September 1956).

Larsen, Richard J and Morris L. Marx. *An Introduction to Probability and its Applications.* Englewood Cliffs, NJ: Prentice Hall, Inc., 1985.

Martin, Bruce. “Fate… or Blind Chance? What Seems Like Eerie Predestination Is Merely Coincidence,” *The Washington Post*, September 9, 1998, final edition.

Relf, Simon and Dennis Almeida. “Exploring the ‘Birthdays Problem’ and Some of its Variants Through Computer Simulation,” *International Journal of Mathematical Education in Science & Technology*, vol. 30, no. 1 (January/ February 1999).

Smith, David Eugene. *A Source Book in Mathematics.* New York: McGraw-Hill Book Company, Inc., 1929.

Tannenbaum, Peter. *Excursions in Modern Mathematics*, 5^{th} ed. Upper Saddle River, NJ: Pearson Education, Inc., 2004.

## LECTURES

D’Souza, R.M. “The Science of Complex Networks.” CSE Seminar at University of California – Davis, February 2006. HYPERLINK “http://mae.ucdavis.edu/dsouza/ talks.html” http://mae.ucdavis.edu/dsouza/talks.html (accessed 2007).