**Instructors:**Janko Gravner, Bradley Ballinger**Prerequisites:**Precalculus**Typical Field Trips:**Exploratorium, California Academy of Science- This is a
option only.**FIRST CHOICE**

### Introduction

This cluster is designed to introduce students with a strong interest in mathematics to several advanced topics. These topics would ordinarily be studied at the advanced undergraduate level, but all lend themselves to an introductory course at the high school level. No prior experience in any of these topics is expected, but enthusiasm for and interest in mathematics is essential.

### Core Course - Probability

We will learn methods from the mathematical theory of probability that can solve many practical problems, such as this one: suppose you and three friends are each dealt 13 cards from a shuffled deck, and you observe that you were dealt a single ace; what is then the probably that your three friends also each have an ace? On the way, we will also learn many useful counting methods.

Probability is one one the most applicable areas of mathematics, and is used in statistics, finance, gambling, physics, artificial intelligence/machine learning, computer science, game theory, and philosophy to draw inferences about the expected frequency of events.

### Supplementary Course** - Introduction to Graph Theory **

A graph is a collection of points, called vertices, connected by edges. Using graphs to describe information has applications from scheduling tournaments to cryptography to understanding the structure of the internet. We’ll study the basics of graph theory, including graph coloring problems, Euler’s formula for graphs on surfaces, and graph planarity. We will also explore related ideas in the study of finite state machines and in topology.