Modular 9 Ch 5.4 to 5.5 Copyright of the definitions and examples is reserved to Pearson Education, Inc.. In order to use this PowerPoint presentation, the required textbook for the class is the Fundamentals of Statistics, Informed Decisions Using Data, Michael Sullivan, III, fourth edition. Prepared by DWLos Angeles Mission College

Section 5.4 Conditional Probability and the General Multiplication Rule Objective A : Conditional Probability and the General Multiplication Rule Objective B : Application Section 5.5 Counting Techniques Objective A : Multiplication Rule of Counting Objective B : Permutation or Combination Objective C : Using the Counting Techniques to Find Probabilities Prepared by DWLos Angeles Mission College

Section 5.4 Conditional Probability and the General Multiplication Rule If E and F are dependent events, then. The probability of E and F is the probability of event E occurring times the probability of event F occurring, given the occurrence of event E. Objective A : Conditional Probability and the General Multiplication Rule A1. Multiplication Rule for Dependent Events Prepared by DWLos Angeles Mission College

Example 1: A box has 5 red balls and 2 white balls. If two balls are randomly selected (one after the other), what is the probability that they both are red? (a) With replacement → Independent case (b) Without replacement → Dependent case Prepared by DWLos Angeles Mission College

Example 2: Three cards are drawn from a deck without replacement. Find the probability that all are jacks. Without replacement → Dependent case (Almost zero percent of a chance) Prepared by DWLos Angeles Mission College

A2. Conditional Probability The probability of event F occurring, given the occurrence of event E, is found by dividing the probability of E and F by the probability of E. If E and F are any two events, then. Prepared by DWLos Angeles Mission College

Example 1 : At a local Country Club, 65% of the members play bridge and swim, and 72% play bridge. If a member is selected at random, find the probability that the member swims, given that the member plays bridge. Let B be the event of playing bridge. Let S be the event of swimming. Given : Find : Prepared by DWLos Angeles Mission College

Section 5.4 Conditional Probability and the General Multiplication Rule Objective A : Conditional Probability and the General Multiplication Rule Objective B : Application Section 5.5 Counting Techniques Objective A : Multiplication Rule of Counting Objective B : Permutation or Combination Objective C : Using the Counting Techniques to Find Probabilities Prepared by DWLos Angeles Mission College

Example 1 : Eighty students in a school cafeteria were asked if they favored a ban on smoking in the cafeteria. The results of the survey are shown in the table. Class Favor Oppose No opinion Freshman Sophomore If a student is selected at random, find these probabilities. (a) The student is a freshman or favors the ban. (F) (Fa) (O) (N) (S) Prepared by DWLos Angeles Mission College

(b) Given that the student favors the ban, the student is a sophomore. Prepared by DWLos Angeles Mission College

Prepared by DWLos Angeles Mission College Example 2 : The local golf store sells an “onion bag” that contains 35 “experienced” golf balls. Suppose that the bag contains 20 Titleists, 8 Maxflis and 7 Top-Flites. (a) What is the probability that two randomly selected golf balls are both Titleists? (b) What is the probability that the first ball selected is a Titleist and the second is a Maxfli? Without replacement → Dependent case Without replacement → Dependent case

Prepared by DWLos Angeles Mission College (c) What is the probability that the first ball selected is a Maxfli and the second is a Titleist? (d) What is the probability that one golf ball is a Titleist and the other is a Maxfli? Without replacement → Dependent case Without replacement → Dependent case

Section 5.4 Conditional Probability and the General Multiplication Rule Objective A : Conditional Probability and the General Multiplication Rule Objective B : Application Section 5.5 Counting Techniques Objective A : Multiplication Rule of Counting Objective B : Permutation or Combination Objective C : Using the Counting Techniques to Find Probabilities Prepared by DWLos Angeles Mission College

Section 5.5 Counting Techniques Objective A : Multiplication Rule of Counting Example 1 : Two dice are tossed. How many outcomes are in the sample space. Example 2 : A password consists of two letters followed by one digit. How many different passwords can be created? (Note: Repetitions are allowed) If Task 1 can be done in ways and Task 2 can be done in ways, Task 1 and Task 2 performed together can be done together in ways. Prepared by DWLos Angeles Mission College

Section 5.4 Conditional Probability and the General Multiplication Rule Objective A : Conditional Probability and the General Multiplication Rule Objective B : Application Section 5.5 Counting Techniques Objective A : Multiplication Rule of Counting Objective B : Permutation and Combination Objective C : Using the Counting Techniques to Find Probabilities Prepared by DWLos Angeles Mission College

Objective B : Permutation and Combination Order matters Order doesn’t matters Permutation The number of ways we can arrange distinct objects, taking them at one time, is Combination The number of distinct combinations of distinct objects that can be formed, taking them at one time, is Section 5.5 Counting Techniques Prepared by DWLos Angeles Mission College

Example 1 : Find (a) (b) (c) (d) (a) (b) (c) (d) Prepared by DWLos Angeles Mission College

Example 2 : An inspector must select 3 tests to perform in a certain order on a manufactured part. He has a choice of 7 tests. How many ways can be performed 3 different tests? Example 3 : If a person can select 3 presents from 10 presents, how many different combinations are there? Prepared by DWLos Angeles Mission College

Section 5.4 Conditional Probability and the General Multiplication Rule Objective A : Conditional Probability and the General Multiplication Rule Objective B : Application Section 5.5 Counting Techniques Objective A : Multiplication Rule of Counting Objective B : Permutation or Combination Objective C : Using the Counting Techniques to Find Probabilities Prepared by DWLos Angeles Mission College

Objective C : Using the Counting Techniques to Find Probabilities, we can calculate by the formula. After using the multiplication rule, combination and permutation learned from this section to count the number of outcomes for a sample space, and the number of outcomes for an event, Section 5.5 Counting Techniques Prepared by DWLos Angeles Mission College

Example 1 : A Social Security number is used to identify each resident of the United States uniquely. The number is of the form xxx-xx-xxxx, where each x is a digit from 0 to 9. (a) How many Social Security numbers can be formed? (b) What is the probability of correctly guessing the Social Security number of the President of the United States? Prepared by DWLos Angeles Mission College

Prepared by DWLos Angeles Mission College Example 2 : Suppose that there are 55 Democrats and 45 Republicans in the U.S. Senate. A committee of seven senators is to be formed by selecting members of the Senate randomly. (a) What is the probability that the committee is composed of all Democrats? (b) What is the probability that the committee is composed of all Republicans?

Prepared by DWLos Angeles Mission College (c) What is the probability that the committee is composed of all three Democrats and four Republicans?