Section 4.3-1Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Warm Up

• Express the indicated degree of likelihood as a probability value:

• “There is a 40% chance of rain tomorrow.”

A bag contains 6 red marbles, 3 blue marbles, and 7 green marbles. If a marble is randomly selected from the bag, what is the probability that it is blue?

Assume that one student in a class of 27 students is randomly selected to win a prize. Would it be “unusual” for you to win? (Assume “unusual” is a probability less than or equal to 0.05)

Section 4.3-2Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Chapter 4Probability

4-1 Review and Preview

4-2 Basic Concepts of Probability

4-3 Addition Rule

4-4 Multiplication Rule: Basics

4-5 Multiplication Rule: Complements and Conditional Probability

4-6 Counting

4-7 Probabilities Through Simulations

4-8 Bayes’ Theorem

Section 4.3-3Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Objective

Students will use the addition rule for finding probabilities of the form P(A or B).

Section 4.3-4Copyright © 2014, 2012, 2010 Pearson Education, Inc.

P(A or B) is the probability that either event A occurs or event B occurs (or they both occur).

The key word in this section is “or.”

In mathematics we usually use the inclusive or, which means A ,B or both.

P(A or B)

Section 4.3-5Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Compound Event any event combining 2 or more simple events

Compound Event

Notation

A or B is a compound event because it combines the two simple events A and B.

We are moving from studying probabilities of single events to probabilities of compound events.

Section 4.3-6Copyright © 2014, 2012, 2010 Pearson Education, Inc.

1. Find the total number of ways A can occur.2. Find the total number of ways B can occur.

3. Find the total number of ways A and B occur together. (the overlap P(A and B))

4. Find the size of the sample space. 4. Add the probability of A to the probability of B

and subtract the overlap.Subtracting the overlap prevents double counting)

Algorithm for finding P(A or B)

Section 4.3-7Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Recall

Section 4.3-8Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Specifically

Formal Addition Rule

P(A or B) = P(A) + P(B) – P(A and B)

Section 4.3-9Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Positive test result

Negative test result

Subject uses drugs

44 6

Subject does not use drugs

90 860

Pre – Employment Drug Screening Results

Find the probability P(positive test result or subject uses drugs)

There are 44 results who use drugs and tested positive and out of 1000 total results total so P(positive test result and drug user) = 44/1000 = 22/500

There are 50 drug user results out of 1000 results total so P(positive test result) = 50/1000 = 25/500

There are 134 positive test results out of 1000 results total so P(positive test result) = 134/1000 = 67/500

Section 4.3-10Copyright © 2014, 2012, 2010 Pearson Education, Inc.

So P(positive test result or subject uses drugs)= 67/500 +25/500 -22/500 = 70/500 = 7/50

Section 4.3-11Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Caution

When finding P(A or B), be sure to add in such a way that every outcome is counted only once. Watch out for double counting.

Section 4.3-12Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Disjoint (Mutually Exclusive)

Two events A and B are disjoint (or mutually exclusive) if they cannot occur at the same time.

Disjoint events do not overlap.

Section 4.3-13Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Decide if the events are disjoint or not. A =The event of randomly selecting someone taking a statistics course.

B =The event of randomly selecting someone who is a female.

A =The event of randomly selecting someone who is a Registered Democrat.

B =The event of randomly selecting someone who is a registered Republican.

A and B are not disjoint

A and B are disjoint

Section 4.3-14Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Disjoint (A visual perspective)

Venn Diagram for Events That Are Not Disjoint

Venn Diagram for Disjoint Events

Section 4.3-15Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Alternate notation

Section 4.3-16Copyright © 2014, 2012, 2010 Pearson Education, Inc.

P(A or B) (A visual perspective)

Venn Diagram for Events That Are Not Disjoint

Venn Diagram for Disjoint Events

Section 4.3-17Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Complementary EventsRecall A and are complements.

Since it is impossible for both to occur together at the same time, they must be disjoint.A

( ) ( ) 1P A P A

( ) 1 ( )P A P A ( ) 1 ( )P A P A

Section 4.3-18Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Complements (A visual perspective)

Section 4.3-19Copyright © 2014, 2012, 2010 Pearson Education, Inc.

Example

Based on data from a Harris Interactive Poll, the probability of randomly selecting someone who believes in the devil is0.6. P(believes in the devil) = 0.6. Use this to find P(person does not believe in the devil)

Section 4.3-20Copyright © 2014, 2012, 2010 Pearson Education, Inc.