Report

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. Los Angeles Mission College Prepared by DW 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 Los Angeles Mission College Prepared by DW Section 5.4 Conditional Probability and the General Multiplication Rule Objective A : Conditional Probability and the General Multiplication Rule A1. Multiplication Rule for Dependent Events If E and F are dependent events, then P( E and F ) P( E) P( F | E) . 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. Los Angeles Mission College Prepared by DW 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 P(1st redand2ndred) 5 5 25 P(1st red) P(2ndred) 7 7 49 (b) Without replacement → Dependent case P(1st redand2ndred) P(1st red) P(2ndred |1st red) Los Angeles Mission College 5 4 20 10 7 6 42 21 Prepared by DW Example 2: Three cards are drawn from a deck without replacement. Find the probability that all are jacks. Without replacement → Dependent case P(1st jack and 2nd jack and3rd jack) 4 3 2 1 1 1 1 0.00018 52 51 50 13 17 25 5525 (Almost zero percent of a chance) Los Angeles Mission College Prepared by DW A2. Conditional Probability P( E and F ) N ( E and F ) If E and F are any two events, then P( F | E ) . P( E ) N (E) 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. Los Angeles Mission College Prepared by DW 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 : P( B and S ) 0.65 P( B) 0.72 Find : P( S | B) 0.65 P( S and B) P( S | B) 0.903 P( B) 0.72 Los Angeles Mission College Prepared by DW 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 Los Angeles Mission College Prepared by DW 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. (Fa) (O) (N) Class Favor Oppose No opinion 50 (F) Freshman 15 27 8 (S) Sophomore 23 5 2 30 38 32 10 80 If a student is selected at random, find these probabilities. (a) The student is a freshman or favors the ban. P( F or Fa) P( F ) P( Fa) P( F and Fa) 50 38 15 80 80 80 73 80 Los Angeles Mission College Prepared by DW (b) Given that the student favors the ban, the student is a sophomore. P( S and Fa) P( S | Fa) P( Fa) 23 80 38 80 23 38 Los Angeles Mission College Prepared by DW 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? Without replacement → Dependent case P(1st Titleist and 2nd Titleist) 20 19 380 0.319 35 34 1190 (b) What is the probability that the first ball selected is a Titleist and the second is a Maxfli? Without replacement → Dependent case P(1st Titleist and 2nd Maxfli) 20 8 160 0.134 35 34 1190 Los Angeles Mission College Prepared by DW (c) What is the probability that the first ball selected is a Maxfli and the second is a Titleist? Without replacement → Dependent case P(1st Maxfli and 2nd Titleist) 160 8 20 0.134 35 34 1190 (d) What is the probability that one golf ball is a Titleist and the other is a Maxfli? Without replacement → Dependent case P(1st Titleist and 2nd Maxfli or1st Maxfli and 2nd Titleist) 8 20 20 8 320 0.269 35 34 35 34 1190 Los Angeles Mission College Prepared by DW 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 Los Angeles Mission College Prepared by DW Section 5.5 Counting Techniques Objective A : Multiplication Rule of Counting If Task 1 can be done in n ways and Task 2 can be done in m ways, Task 1 and Task 2 performed together can be done together in n m ways. Example 1 : Two dice are tossed. How many outcomes are in the sample space. 6 6 36 Example 2 : A password consists of two letters followed by one digit. How many different passwords can be created? (Note: Repetitions are allowed) 26 26 10 6760 Los Angeles Mission College Prepared by DW 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 Los Angeles Mission College Prepared by DW Section 5.5 Counting Techniques Objective B : Permutation and Combination Permutation The number of ways we can arrange n distinct objects, taking them r at one time, is n! n Pr (n r)! Order matters Combination The number of distinct combinations of n distinct objects that can be formed, taking them r at one time, is n! n Cr r!(n r)! Los Angeles Mission College Order doesn’t matters Prepared by DW Example 1 : Find (a) 5! (a) 5! (b) 25 P8 (c) 12 P4 (c) 5 4 3 2 1 12 (d) 25 C8 P4 11880 120 (b) P 25 8 43609104000 Los Angeles Mission College (d) 25 C8 1081575 Prepared by DW 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? n7 7 r 3 P3 210 Example 3 : If a person can select 3 presents from 10 presents, how many different combinations are there? n 10 10 r 3 C3 120 Los Angeles Mission College Prepared by DW 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 Los Angeles Mission College Prepared by DW Section 5.5 Counting Techniques Objective C : Using the Counting Techniques to Find Probabilities After using the multiplication rule, combination and permutation learned from this section to count the number of outcomes for a sample space, N (S ) and the number of outcomes for an event, N (E ) , we can calculate P (E ) by the formula P( E ) Los Angeles Mission College N (E) . N (S ) Prepared by DW 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? N (S ) 109 1,000,000,000 one billion (b) What is the probability of correctly guessing the Social Security number of the President of the United States? 1 N (E) 1 P( E ) 9 one out of a billion N ( S ) 10 1,000,000,000 Los Angeles Mission College Prepared by DW 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? N (S ) 100 C7 16, 007,560,800 N ( D) 55 C7 202,927,725 202,927, 725 N ( D) 0.0127 P ( D) N ( S ) 16, 007,560,800 (b) What is the probability that the committee is composed of all Republicans? N (S ) 100 C7 16, 007,560,800 C7 45,379, 620 45,379, 620 N ( R) 0.0028 P( D) N ( S ) 16, 007,560,800 N ( R) 45 Los Angeles Mission College Prepared by DW (c) What is the probability that the committee is composed of all three Democrats and four Republicans? N (S ) 100 C7 16, 007,560,800 N (3D and 4R) 55 C3 45 C4 3,908,883,825 P( D) N (3D and 4 R) 3,908,883,825 0.2442 N (S ) 16, 007,560,800 Los Angeles Mission College Prepared by DW