Probability for mathematicians

Some questions to ask

Q. Does probability theory belong to pure or applied mathematics?

A. You may joint the large community of `pure probabilists', or the large community of `applied probabilists' (or both, or their intersection....)
By the way: number theory has important applications in cryptography; so what? Is it pure or applied?

Q. Do probabilists get Fields medals?

A. Sometimes... Till now it happened twice: in 2006, to Wendelin Werner and in 2010 to Stanislav Smirnov.
Note also Wolf prize of 1980 to Kolmogorov, of 1987 to Ito, and Abel prize of 2007 to Varadhan.

Q. Is probability theory a part of measure theory?

A. Probability theory is embedded into measure theory. Likewise, measure theory (as well as algebra, geometry etc.) is embedded into set theory. So what?

Q. Is probability theory of any benefit to (say) geometry?

A. Sometimes... An example: martingales are of benefit to minimal surfaces, see R. Neel, 2008.

Q. Can a probabilistic result be highly nontrivial and elegant?

A. Sometimes... An example: a planar Brownian loop cuts out the area Pi/5 in the average, see C. Garban and J. Ferreras, 2005.

Q. The example above is far beyond our course. Is there something interesting within the reach of the course?

A1. For every branching process, the size of n-th generation (be it the number of animals, neutrons, or men of a given family), divided by its expectation, converges to a random limit (as n tends to infinity). See Williams, Chapter 0.

A2. There exists a real number such that its binary digits are 50 percents "0", 50 percents "1", while its decimal digits are 10 percents "0", 10 percents "1", ..., 10 percents "9". See normal numbers.

Q. Do we really need uncountable probability spaces (and measure theory)? There is an easier way: discrete probability in combination with a limiting procedure.

A1. Do you really need analysis? There is an easier way: finite differences in combination with a limiting procedure. Really?

A2. Think about a random direction in the three-dimensional Euclidean space. Try to translate it into the discrete language. Is it easy?

A3. Think about a normal number. Try to translate it into the discrete language. Is it easy?