**Title of the course:** Measure and Paradoxes in Probability

**Instructor:** Dr. Tatiana González Grandón

**Institution:** Humboldt Universität Berlin

**Dates:** 7-13 August 2017

**Prerequisites:** Calculus. familiarity with undergraduate-level probability theory is advantageous.

**Level:** Undergraduate (all levels), graduate

**Abstract:** In this 6 days course we will introduce students to probability theory with the enticing philosophical

underpinning of human beings wanting to measure uncertainty. For that, we first need to define what measure is?

What does it mean for a subset E of R to be measurable? This will lead us to compare the differences between

measure and cardinality. We will look at some beautiful but non-pathological subsets, which cannot be measured

and perplex us amid the Banach-Tarski Paradox. We will construct the basic ideas and tools to work with the

Lebesgue Integral, while drawing its parallels to Expected values of uncertain results. Moreover, in the deterministic

world it is unambiguous what it means for a sequence to converge to a limit, if however one has a sequence of Random

Variables, how can it converge?

Daily curriculum:

1. The problem of measure: Cardinality, Banach-Tarski Paradox.

2. Monty Hall Problem. What is Probability?

Probability space and examples.

3. Birthday problem.

Random variables as measurable functions.

4. Bertrand’s box paradox.

Expected Values and Lebesgue Integral.

5. Playing with infinity

Types of Convergence and Examples.

6. Distributions of random variables.

**Literature:**

Probability and Random Processes, Grimmett Geoffrey.

An Introduction to Measure Theory, Terence Tao.

**Language: **EN