{ "cells": [ { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# Week 1: Introduction, probability, combinatorics\n", "\n", " #### [Back to main page](https://petrosyan.page/fall2020math3215)\n" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "skip" } }, "source": [ "## Outcomes and events" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Suppose we are observing a phenomenon with a certain outcome (an experiment, the fluctuations of the stock market, etc). Let $S$ denote the space of all possible outcomes of this phenomenon. One such example is the toss of a coin. Each toss results in an outcome of either H (head) or T (tail) so in this case, the outcome space will be $S=\\{H, T\\}$. Or if we are pulling a card from a deck then the space of all outcomes is the set of all individual cards. \n", "\n", "Often times, we are not interested in a specific outcome but rather if the outcome has certain qualities. In the case of the pulled card, we may be interested if it is spades rather than being a specific card from spades. In other words, we are often interested in sets (collections) of outcomes. Such sets we conventionally call events in probability theory. \n", "\n", "* If we want the outcome to have quality $A$ and quality $B$ simultaneously, that will correspond to the event $A\\cap B$.\n", "* If we want the outcome to have quality $A$ and not have the quality $B$, it corresponds to the event $A\\setminus B$.\n", "* If we want the outcome to either have quality $A$ or $B$ then it corresponds to the event $A\\cup B$.\n", "* If we want the outcome to not have quality $A$ then it corresponds to the event $A^\\prime = S\\setminus A$.\n", "\n", "\n", "**Definition:** \n", "\n", "\n", "