 # Slide Slide 1 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Lecture Slides Elementary Statistics Tenth Edition.

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Slide Slide 1 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Lecture Slides Elementary Statistics Tenth Edition

Slide Slide 2 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Chapter 4 Probability 4-1 Overview 4-2 Fundamentals 4-3 Addition Rule 4-4 Multiplication Rule: Basics

Slide Slide 3 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Created by Tom Wegleitner, Centreville, Virginia Section 4-1 Overview

Slide Slide 4 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Overview Rare Event Rule for Inferential Statistics: If, under a given assumption, the probability of a particular observed event is extremely small, we conclude that the assumption is probably not correct. Statisticians use the rare event rule for inferential statistics.

Slide Slide 5 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Created by Tom Wegleitner, Centreville, Virginia Section 4-2 Fundamentals

Slide Slide 6 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Key Concept This section introduces the basic concept of the probability of an event. Three different methods for finding probability values will be presented. The most important objective of this section is to learn how to interpret probability values.

Slide Slide 7 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Definitions  Event any collection of results or outcomes of a procedure  Simple Event an outcome or an event that cannot be further broken down into simpler components  Sample Space for a procedure consists of all possible simple events; that is, the sample space consists of all outcomes that cannot be broken down any further

Slide Slide 8 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example  Events  Simple Events  Sample Space  Tossing two quarters 2 heads – boys win 2 tails – girls win 1 head & 1 tail – Herman wins

Slide Slide 9 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Notation for Probabilities P - denotes a probability. A, B, and C - denote specific events. P (A) - denotes the probability of event A occurring.

Slide Slide 10 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Basic Rules for Computing Probability Rule 1: Relative Frequency Approximation of Probability Conduct (or observe) an experiment, and count the number of times event A actually occurs. Based on these actual results, P(A) is estimated as follows: P(A) =P(A) = number of times A occurred number of times trial was repeated

Slide Slide 11 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Basic Rules for Computing Probability - continued Rule 2: Classical Approach to Probability (Requires Equally Likely Outcomes) Assume that a given procedure has n different simple events and that each of those simple events has an equal chance of occurring. If event A can occur in s of these n ways, then P(A) = number of ways A can occur number of different simple events s n =

Slide Slide 12 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Basic Rules for Computing Probability - cont Rule 3: Subjective Probabilities P(A), the probability of event A, is estimated by using knowledge of the relevant circumstances.

Slide Slide 13 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Law of Large Numbers As a procedure is repeated again and again, the relative frequency probability (from Rule 1) of an event tends to approach the actual probability.

Slide Slide 14 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Probability Limits  The probability of an event that is certain to occur is 1.  The probability of an impossible event is 0.  For any event A, the probability of A is between 0 and 1 inclusive. That is, 0  P(A)  1.

Slide Slide 15 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Possible Values for Probabilities

Slide Slide 16 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Rounding Off Probabilities When expressing the value of a probability, either give the exact fraction or decimal or round off final decimal results to three significant digits. (Suggestion: When the probability is not a simple fraction such as 2/3 or 5/9, express it as a decimal so that the number can be better understood.)

Slide Slide 17 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Rounding off examples A probability of 0.021491 The probability of 1/3 The probability of ½ The probability of 432/7,842

Slide Slide 18 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example 1 Find the probability that NBA basketball player Reggie Miller makes a free throw after being fouled. At one point in his career, he made 5,915 free throws in 6,679 attempts.

Slide Slide 19 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example 2 As part of a study about cards, you take the following cards from a deck…ace, king, queen and jack. Then you shuffle the four cards and randomly select one of them. What is the probability that it is a king?

Slide Slide 20 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example 3 What is the probability that your car will be hit by a meteorite this year?

Slide Slide 21 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example 4 Adults are randomly selected for a Gallup poll, and they are asked if they think that cloning humans should or should not be allowed. 91 said that cloning humans should be allowed, 901 said it should not be allowed and 20 had no opinion. Estimate the probability that a person believes that cloning of humans should be allowed.

Slide Slide 22 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example 5 Find the probability that when a couple has three children, they will have exactly 2 boys. Assume that boys and girls are equally likely and that the gender of any child is not influenced by the gender of any other child.

Slide Slide 23 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example 6 If a year is selected at random, find the probability that Thanksgiving Day will be on a (a)Wednesday? (b)Thursday?

Slide Slide 24 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Definition The complement of event A, denoted by A, consists of all outcomes in which the event A does not occur.

Slide Slide 25 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example 7 In reality, more boys are born than girls. In a typical sample, there are 205 newborn babies, 105 of which are boys. If one baby is randomly selected from the group, what is the probability that the baby is not a boy?

Slide Slide 26 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Odds The actual odds against event A occurring are the ratio P(A) /P(A), usually expressed in the form of a:b (or “a to b”), where a and b are integers having no common factors. The actual odds in favor of event A occurring are the reciprocal of the actual odds against the event. If the odds against A are a:b, then the odds in favor of A are b:a. The payoff odds against event A represent the ratio of the net profit (if you win) to the amount bet. payoff odds against event A = (net profit) : (amount bet)

Slide Slide 27 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example: If you bet \$5 on the number 13 in roulette, your probability of winning is 1/38 and the payoff odds are given by the casino as 35:1. Find the actual odds against the outcome of 13. How much net profit would you make if you win by betting on 13?

Slide Slide 28 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Recap In this section we have discussed:  Rare event rule for inferential statistics.  Probability rules.  Law of large numbers.  Complementary events.  Rounding off probabilities.  Odds.

Slide Slide 29 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Created by Tom Wegleitner, Centreville, Virginia Section 4-3 Addition Rule

Slide Slide 30 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Key Concept The main objective of this section is to present the addition rule as a device for finding probabilities that can be expressed as P(A or B), the probability that either event A occurs or event B occurs (or they both occur) as the single outcome of the procedure.

Slide Slide 31 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Compound Event any event combining 2 or more simple events Definition Notation P(A or B) = P (in a single trial, event A occurs or event B occurs or they both occur)

Slide Slide 32 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. When finding the probability that event A occurs or event B occurs, find the total number of ways A can occur and the number of ways B can occur, but find the total in such a way that no outcome is counted more than once. General Rule for a Compound Event

Slide Slide 33 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Compound Event Intuitive Addition Rule To find P(A or B), find the sum of the number of ways event A can occur and the number of ways event B can occur, adding in such a way that every outcome is counted only once. P(A or B) is equal to that sum, divided by the total number of outcomes in the sample space. Formal Addition Rule P(A or B) = P(A) + P(B) – P(A and B) where P(A and B) denotes the probability that A and B both occur at the same time as an outcome in a trial or procedure.

Slide Slide 34 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Example Page 152; table 4 – 1 A person is randomly chosen from the 300 people that were tested. Find the probability of selecting a subject who had a positive test result or used marijuana. Using the Intuitive Addition Rule Using the Formal Addition Rule P(A or B) = P(A) + P(B) – P(A and B) Response  yesno Positive test 11924 Negative test 3154

Slide Slide 35 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Definition Events A and B are disjoint (or mutually exclusive) if they cannot occur at the same time. (That is, disjoint events do not overlap.) Venn Diagram for Events That Are Not Disjoint Venn Diagram for Disjoint Events

Slide Slide 36 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Mutually exclusive— think of this as the 2 events together (mutually) agreeing to exclude (not include) each others' elements. They have agreed to be different or mutually exclusive

Slide Slide 37 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Are each of the 2 events disjoint (mutually exclusive)? Randomly selecting a fruit fly with red eyes Randomly selecting a fruit fly with green eyes A person who opposes all cloning A person who approves of cloning sheep Randomly selecting a nurse Randomly selecting a male Randomly selecting a grey sock from a drawer Randomly selecting a cotton sock from a drawer

Slide Slide 38 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Complementary Events P(A) and P(A) are disjoint It is impossible for an event and its complement to occur at the same time.

Slide Slide 39 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Rules of Complementary Events P(A) + P(A) = 1 = 1 – P(A) P(A) = 1 – P(A) P(A)

Slide Slide 40 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Venn Diagram for the Complement of Event A

Slide Slide 41 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Recap In this section we have discussed:  Compound events.  Formal addition rule.  Intuitive addition rule.  Disjoint events.  Complementary events.

Slide Slide 42 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Created by Tom Wegleitner, Centreville, Virginia Section 4-4 Multiplication Rule: Basics

Slide Slide 43 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Key Concept If the outcome of the first event A somehow affects the probability of the second event B, it is important to adjust the probability of B to reflect the occurrence of event A. The rule for finding P(A and B) is called the multiplication rule.

Slide Slide 44 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Notation P(A and B) = P(event A occurs in a first trial and event B occurs in a second trial)

Slide Slide 45 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Key Point – Conditional Probability The probability for the second event B should take into account the fact that the first event A has already occurred.

Slide Slide 46 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Notation for Conditional Probability P(B A) represents the probability of event B occurring after it is assumed that event A has already occurred (read B A as “B given A.”)

Slide Slide 47 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Definitions Independent Events Two events A and B are independent if the occurrence of one does not affect the probability of the occurrence of the other. Several events are similarly independent if the occurrence of any does not affect the probabilities of occurrence of the others.) If A and B are not independent, they are said to be dependent.

Slide Slide 48 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Are the following events dependent or independent? Finding that your calculator is working Finding that your cell phone is not working Finding that your kitchen toaster is not working Finding that your refrigerator is not working Drinking until your driving ability is impaired Being involved in a car crash

Slide Slide 49 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Formal Multiplication Rule  P(A and B) = P(A) P(B A)  Note that if A and B are independent events, P(B A) is really the same as P(B).

Slide Slide 50 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Intuitive Multiplication Rule When finding the probability that event A occurs in one trial and event B occurs in the next trial, multiply the probability of event A by the probability of event B, but be sure that the probability of event B takes into account the previous occurrence of event A.

Slide Slide 51 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Applying the Multiplication Rule

Slide Slide 52 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Using the formal multiplication rule P(A and B) = P(A) P(BIA) A new computer owner creates a password consisting of 2 characters. She randomly selects a letter of the alphabet as the first character and a digit from 0 – 9 as the second character. Find P(R3)

Slide Slide 53 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Using the formal multiplication rule P(A and B) = P(A) P(BIA) A new computer owner creates a password consisting of 4 characters. She randomly selects 2 different letters of the alphabet as the first 2 characters and 2 different digits from 0 – 9 as the second character. Find P(TG34)

Slide Slide 54 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Using the formal multiplication rule P(A and B) = P(A) P(BIA) In your class of 13 boys and 10 girls, the math teacher randomly selects three different students to help her sort papers during study hall. What is the probability that they are all girls?

Slide Slide 55 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Summary of Fundamentals  In the addition rule, the word “or” in P(A or B) suggests addition. Add P(A) and P(B), being careful to add in such a way that every outcome is counted only once.  In the multiplication rule, the word “and” in P(A and B) suggests multiplication. Multiply P(A) and P(B), but be sure that the probability of event B takes into account the previous occurrence of event A.

Slide Slide 56 The 5% Guideline for Cumbersome Calculations If a sample size is no more than 5% of the size of the population, treat the selections as being independent (even if the selections are made without replacement, so they are technically dependent).

Slide Slide 57 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Recap In this section we have discussed:  Notation for P(A and B).  Notation for conditional probability.  Independent events.  Formal and intuitive multiplication rules.

Slide Slide 58 Section 4-5 Multiplication Rule: Complements and Conditional Probability

Slide Slide 59 Key Concepts Probability of “at least one”: Find the probability that among several trials, we get at least one of some specified event. Conditional probability: Find the probability of an event when we have additional information that some other event has already occurred.

Slide Slide 60 Complements: The Probability of “At Least One”  The complement of getting at least one item of a particular type is that you get no items of that type.  “At least one” is equivalent to “one or more.”

Slide Slide 61 Finding the Probability of “At Least One” To find the probability of at least one of something, calculate the probability of none, then subtract that result from 1. That is, P(at least one) = 1 – P(none).

Slide Slide 62 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Find the probability of a couple having at least one girl among three children. Assume that boys and girls are equally likely.

Slide Slide 63 Conditional Probability A conditional probability of an event is a probability obtained with the additional information that some other event has already occurred. P(B|A) denotes the conditional probability of event B occurring, given that event A has already occurred, and it can be found by dividing the probability of events A and B both occurring by the probability of event A: P(B A) = P(A and B) P(A)

Slide Slide 64 Intuitive Approach to Conditional Probability The conditional probability of B given A can be found by assuming that event A has occurred, and then calculating the probability that event B will occur.

Slide Slide 65 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. No (did not lie)Yes(lied) Positive test result (Polygraph test indicated that subject lied) 15 False positive 42 True positive Negative test result (Polygraph test indicated that subject did not lie) 32 True negative 9 False negative Example 1: If one of 98 subjects is randomly selected, find the probability that the subject had a positive test result, given that the subject actually lied.

Slide Slide 66 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. No (did not lie)Yes(lied) Positive test result (Polygraph test indicated that subject lied) 15 False positive 42 True positive Negative test result (Polygraph test indicated that subject did not lie) 32 True negative 9 False negative Example 2: If one of 98 subjects is randomly selected, find the probability that the subject actually lied, given that he or she had a positive test result.

Slide Slide 67 Confusion of the Inverse To incorrectly believe that P(A|B) and P(B|A) are the same, or to incorrectly use one value for the other, is often called confusion of the inverse.

Slide Slide 68 Recap In this section we have discussed:  Concept of “at least one.”  Conditional probability.  Intuitive approach to conditional probability.

Slide Slide 69 Section 4-7 Counting

Slide Slide 70 Key Concept In many probability problems, the big obstacle is finding the total number of outcomes, and this section presents several methods for finding such numbers without directly listing and counting the possibilities.

Slide Slide 71 Fundamental Counting Rule For a sequence of two events in which the first event can occur m ways and the second event can occur n ways, the events together can occur a total of m n ways.

Slide Slide 72 Notation The factorial symbol ! denotes the product of decreasing positive whole numbers. For example, By special definition, 0! = 1. Let’s find this button on your calculator now.

Slide Slide 73 A collection of n different items can be arranged in order n! different ways. (This factorial rule reflects the fact that the first item may be selected in n different ways, the second item may be selected in n – 1 ways, and so on.) Factorial Rule

Slide Slide 74 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. During the summer, you are planning to visit 6 national parks: Glacier, Yosemite, Yellowstone, Arches, Zion, and Grand Canyon. You would like to plan the most efficient route and you decide to list all the possible routes. How many different routes are possible?

Slide Slide 75 Permutations Rule (when items are all different) (n - r)! n r P = n!n! If the preceding requirements are satisfied, the number of permutations (or sequences) of r items selected from n available items (without replacement) is Requirements: 1.There are n different items available. (This rule does not apply if some of the items are identical to others.) 2.We select r of the n items (without replacement). 3.We consider rearrangements of the same items to be different sequences. (The permutation of ABC is different from CBA and is counted separately.)

Slide Slide 76 (n - r )! r! n!n! n C r = Combinations Rule If the preceding requirements are satisfied, the number of combinations of r items selected from n different items is Requirements: 1.There are n different items available. 2.We select r of the n items (without replacement). 3.We consider rearrangements of the same items to be the same. (The combination of ABC is the same as CBA.)

Slide Slide 77 When different orderings of the same items are to be counted separately, we have a permutation problem, but when different orderings are not to be counted separately, we have a combination problem. Permutations versus Combinations

Slide Slide 78 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley. Suppose you have 25 managers who you need to choose 5 from to form a group that will train new employees. How many different ways can you choose the 5 managers? What if you wanted to choose 5 managers to promote to the positions of president, vice-president, executive vice- president, secretary and treasurer? How many ways can you choose them now?

Slide Slide 79 Recap In this section we have discussed:  The fundamental counting rule.  The permutations rule (when items are all different).  The permutations rule (when some items are identical to others).  The combinations rule.  The factorial rule.