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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.

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Dice Game

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.





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