# Week 12

## Topics

This is the first of three weeks devoted to elementary probability and statistics. This week, we will cover parts of chapters 7 and 8.

• Section 7-3 : Basic Counting Principles
• Section 7-4 : Permutations and Combinations
• Section 8-1 : Sample Spaces, Events, and Probability
• Section 8-2 : Union, Intersection, and Complement of Events

## What is Probability?

Probability deals with random occurrences. Probabilists talk about conducting experiments. An experiment could be rolling a die, drawing a hand of cards from a deck, or measuring someone's height. The set of all possible outcomes of an experiment is called the sample space. This could be a discrete set (the numbers 1 through 6 when rolling a die) or a continuous range (someone's height). An event is a subset of the sample space. For example, when you roll a die, the event "rolling an even number" is the subset {2,4,6} of the sample space {1,2,3,4,5,6}. When you measure someone's height, the event "taller than 6 feet" is the subset (6,infinity) of the sample space (0,infinity). (A smaller sample space would also do, of course).

You cannot predict whether a given event will occur or not, but you can say how likely it is. The underlying assumption is that if you conduct the experiment many, many times, the event will be observed a certain percentage of the time. That percentage is the probability of that event.

The book distinguishes between empirical probability, which is observed from actual experiments, and theoretical probability, which is assigned to the event based on a priori assumptions. For example, if you throw a die 600 times, and 105 times the number 1 comes up, you would assign an empirical probability of 105/600 to the event of throwing a 1. The theoretical probability is based on the assumption that all numbers are equally likely, and would be 1/6 = 100/600. Mostly, we work with theoretical probabilities.

## Overview of the Sections

The first two sections are a preparation for computing probabilities. If we assume that all outcomes of the experiment are equally likely, the probability that the outcome is of a certain type is simply

(number of outcomes of that type) / (number of all possible outcomes).

Both of the first two sections deal with counting how many possible outcomes there are, in various settings.

You should find Section 7-3 quite easy.

Section 7-4 is a little harder. It deals with permutations and combinations. Both of these have to do with selecting several items from a larger set. It is a permutation if the order of selection matters; it is a combination if the order does not matter. For example: in how many ways can we select a first place, second place, and third place winner out of a field of 20 athletes? That is a permutation. In how many ways can we select a committee of 3 people out of 20? That is a combination.

The calculations for permutations and combinations involve some rather large numbers. I recommend that you study your calculator manual to see if your calculator has these functions built in. At least, look for the factorial function.

You need to know some basic set theory for these sections, but we already reviewed that earlier this term. If necessary, look at section 7-2 again.

Section 8-1 introduces the basic terms of probability: sample spaces, events, elementary outcomes, probability.

In Section 8-2 you will learn about some rules for working with probability. More rules will come in later sections. You can skip the subsections about odds.

## Assignments

Read the textbook and do the homework assignment HW 9.

Last Updated: Wednesday, August 5, 2015