STAT 230 Lecture Chapter 2, Fall 2011

Copyright © 2010 Pearson Addison-Wesley. All rights reserved. Chapter 2 Probability Copyright © 2010 Pearson Addison-Wesley. All rights reserved. Section 2.1 Sample Space Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 3 Definition 2.1 Example 2.2 An experiment consists of • Flipping a coin, then • Flip it again if the first one is heads, or • Toss a die if the first toss is a tail List the sample space Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 4 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 5 Figure 2.1 Tree diagram for Example 2.2 Example 2.3 Three items are selected from a manufacturing process Each item is tested and then classified as D for defective or N for nondefective List the elements of the sample space Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 6 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 7 Figure 2.2 Tree Diagram for Example 2.3 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. Section 2.2 Events Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 9 Definition 2.2 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 10 Definition 2.3 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 11 Definition 2.4 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 12 Definition 2.5 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 13 Definition 2.6 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 14 Figure 2.3 Events represented by various regions Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 15 Figure 2.4 Events of the sample space S Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 16 Figure 2.5 Venn diagram for Exercises 2.19 and 2.20 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. Section 2.3 Counting Sample Points Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 18 Rule 2.1 Example 2.14 • A developer of a a new subdivision offers prospective home buyers a choice of Tudor, rustic, colonial, and traditional exterior styling in ranch, two-story, and split level floor plans. • How many different choices does a buyer have? Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 19 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 20 Figure 2.6 Tree diagram for Example 2.14 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 21 Rule 2.2 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 22 Definition 2.7 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 23 Definition 2.8 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 24 Theorem 2.1 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 25 Theorem 2.2 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 26 Theorem 2.3 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 27 Theorem 2.4 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 28 Theorem 2.5 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 29 Theorem 2.6 Exercise 2.36 a) How many digit numbers can be formed from the digits 0, 1, 2, 3, 4, 5, and 6 if each digit can be used only once? b) How many of these are odd numbers? c) How many are greater than 330? Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 30 Exercise 2.43 • In how many ways can 5 different trees be planted in a circle? Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 31 Exercise 2.45 How many distinct permutations can be made from the letters of the word: • OPEN • BOOK? • INFINITY? Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 32 Exercise 2.46 • In how many ways can 3 oaks, 4 pines, and 2 maples be arranged along a property line if one does not distinguish among trees of the same kind? Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 33 Exercise 2.47 • How many ways are there to select 3 candidates from 8 equally qualified recent graduates for openings in an accounting firm? Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 34 Exercise 2.48 • How many ways are there hat no two students will have the same birth date in a class of size 60? Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 35 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. Section 2.4 Probability of an Event Interpretations of Probability Subjective probability: assigning a value based on one’s belief of how likely a certain event (that can not be repeated) • What is the chance that Joe will marry Zoe? • What is the chance that the current government will stay until the end of its full term? Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 37 Interpretations of Probability Frequentist’s probability: • Repeat an experiment a large number N of times • Let n(A) be the number of times that event A was observed • the chance of A is approximately n(A)/N The law of large numbers says: lim n(A)/N = P[A] Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 38 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 39 Definition 2.9 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 40 Rule 2.3 Example 2.28 • In a poker hand (i.e. five cards taken out of a deck of 52 cards), find the probability of holding 2 aces and 3 jacks? Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 41 Example 2.28 • The number of ways we can choose 2 aces is 4C2 • The number of ways we can choose 3 jacks is 4C3 • The number of ways we can choose 3 jacks is 52C5 • Hence the required answer is 4C2*4C3 /52C5 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 42 Exercise • In a poker hand, find the probability of holding at least 2 aces? Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 43 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. Section 2.5 Additive Rules Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 45 Theorem 2.7 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 46 Figure 2.7 Additive rule of probability Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 47 Corollary 2.1 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 48 Corollary 2.2 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 49 Corollary 2.3 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 50 Theorem 2.8 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 51 Theorem 2.9 Exercise 2.52 • In a senior class of 500 students, 210 smoke, 258 drink alcoholic beverages, 216 eat between meals, 122 smoke and drink, 83 eat between meals and drink, 97 smoke and eat between meals. 52engage in all of these bad health practices. Find the probability that a randomly chosen senior student: Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 52 Exercise 2.52 • In a senior class of 500 students, 210 smoke, 258 drink alcoholic beverages, 216 eat between meals, 122 smoke and drink, 83 eat between meals and drink, 97 smoke and eat between meals. 52engage in all of these bad health practices. Find the probability that a randomly chosen senior student: a) Smokes but does not drink b) Eats between meals and drinks but does not smoke c) Neither smokes nor eats between meals Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 53 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. Section 2.6 Conditional Probability, Independence, and the Product Rule Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 3 Definition 2.10 Explanation of Def. 2.10 • Knowing that A is given, we know that the outcome that was observed must be in A • It follows that the new “reduced” sample space is A (any outcome outside A becomes impossible) • Then for B to occur, the outcome must be in both A and B • The division by P(A) must be done so that P(A|A) = 1 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 4 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 5 Table 2.1 Categorization of the Adults in a Small Town Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 6 Definition 2.11 Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 7 Theorem 2.10 Example 2.37 • One bag contains 4 white balls and 3 black balls, and a second bag contains 3 white balls and 5 black balls. One ball is drawn from the first ball and placed unseen in the second bag. What is the probability that a ball now drawn from the second bag is black? Copyright © 2010 Pearson Addison-Wesley. All rights reserved. 2 - 8