Lecture 4

Events and the Sample Space

Published

September 7, 2023

Introduction

  • Basic concepts of probability so that we can draw conclusions about a sampled population

  • Example: Toss a single coin, probability of a heads (or tails) is 50%

  • If you toss 10 times and see 10 heads. If you have a fair coin, the probability that you’d see 10 heads is very low. It’s likely the coin is biased

  • Two ways

    • probability of event when the population is known,
    • probability of inference given you observe a sample from the population

4.1 Events and the sample space

  • An experiment is the process by which an observation or measurement is obtained.

Examples:

  • Recording a test grade
  • Measuring daily rainfall
  • Recording a persons’s opinion
  • Any others??

When an experiment is performed we observe an outcome called a simple event - Often written as \(E_i\)

Example -

What are the simple events that can happen when a 6 sided die is rolled?

  • \(E_1\) = 1
  • \(E_2\) = 2
  • …
  • \(E_6\) = 6

An event is a collection of simple events. for example:

  • A: Observe an odd number: \(A = \{E_1, E_3, E_5\}\)
  • B: Observe a number less than 4: \(B = \{E_1, E_2, E_3\}\)

Two events are mutually exclusive if when one event occurs the other cannot - and vise versa.

Note that \(A\) and \(B\) are not mutually exclusive - why?

  • What are some example of mutually exclusive events

  • The set of all simple events i called the sample space , S

Venn Diagram: (see page 129 of book) - Venn Diagram of die tossing, including all simple event and A/B

Some examples:

Ex 4.2

Toss a simple coin - what are the simple events that define the sample space?

Ex. 4.3 & 4.4

Blood Types (4 Simple Events)

These are mutually exclusive

  • what about adding Rh factor - (positive/negative)
  • Use a tree diagram to show how to come up with the simple events

Homework

[1] "4.1.1-6, 4.1.28"

Answers: Chapter 4 - Section 1