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**Overview Created by Tom Wegleitner, Centreville, Virginia**

Edited by Olga Pilipets, San Diego, California

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**Rare Event Rule for Inferential Statistics:**

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. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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Example Suppose that you developed a gender- selection method that increases the likelihood of a baby being a girl. Suppose that independent test results from 100 couples results in 98 girls in 100 births. Because the chance of such a result with no special treatment is very low we conclude that the gender-selection method is effective. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Fundamentals Created by Tom Wegleitner, Centreville, Virginia**

Edited by Olga Pilipets, San Diego, California

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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. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Definitions Event an outcome of an experiment or a procedure. Success**

a favorable outcome of an experiment or a procedure. Simple Event an outcome or an event that cannot be further broken down into simpler components Compound event any event combining 2 or more simple events Sample Space All possible outcomes Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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

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**Example Procedure Event Sample Space single birth (a simple event)**

A: baby girl {boy, girl} a series of 3 births (compound event) B: 2 girls and 1 boy {ggb, gbg, bgg, bbg, bgb, gbb, ggg, bbb} Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Computing Probability**

Basic Rules for Computing Probability Rule 1: Relative Frequency Approximation of Probability refers to the data being derived from observations, rather then theory It is the ratio of the total number of successes to the number of times an experiment is repeated. P(S) = number of successes number of trials Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Example: #10 p.147. Cell phones and Brain Cancer.**

In a study of 420,095 cell phone users in Denmark, it was found that 135 developed cancer of the brain or nervous system. a) Estimate the probability that a randomly selected cell phone user will develop such a cancer. b) Is the result very different from the probability of that was found for the general population? c) What does the result suggest about cell phones as a cause of such cancers, as has been claimed? worksheet Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Probability Limits The probability of an impossible event is 0.**

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

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**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.) Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**s = n Basic Rules for Computing Probability - cont P(S) =**

Rule 2: Classical Approach to Probability An experiment has n possible events (outcomes) and each of those outcomes has an equal chance of occurring. There are s of the outcomes that are considered to be a success, then the probability of success number of ways a success can occur s = P(S) = n total number of possible outcomes (events) Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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Example: guessing A quick quiz consists of a multiple choice question with five possible answers (a, b, c, d, e). If a question is answered with a random guess, find the probability that the answer is correct. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Example: #16 p.148. Pedestrian Walk buttons.**

New York City has 750 pedestrian walk buttons that work, and another 2500 that do not work (based on data from “For Exercise in New York Futility, Push Button, by Michael Luo, New York Times) a) If a pedestrian walk button is randomly selected in New York City, what is the probability that it works? b) Is the same probability likely to be a good estimate for a different city, such as Chicago? worksheet Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Computing Probability - cont**

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. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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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. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Finish problems from section 4.2**

Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Addition Rule Created by Tom Wegleitner, Centreville, Virginia**

Edited by Olga Pilipets, San Diego, California

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**Definition any event combining 2 or more simple events Compound Event**

Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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Example What is a probability of getting a Red Card or an Ace when selecting a single card from a deck? Event A: getting a red card Event B: getting an Ace What is a probability of getting a Red Card or an Ace Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Example cont-d Notation**

Probability of getting a Red Card or an Ace: P (A or B) Notation P(A or B) = P (in a single trial, event A occurs or event B occurs or they both occur) Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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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. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**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. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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Example What is a probability of getting a Red Card or an Ace of Hearts when selecting a single card from a deck? Event A: getting a red card Event B: getting an Ace P (A or B): probability of getting a Red Card or an Ace worksheet Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**P(A or B) = P(A) + P(B) – P(A and B)**

Compound Event 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. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Example: #18 p.157. Addition Rule**

Use the data in the table, which summarizes blood groups and Rh types for 100 typical people. These values may vary in different regions according to the ethnicity of the population. Group O A B AB Type Rh+ 39 35 8 4 Rh- 6 5 2 1 If one person is randomly selected, find P (group B or type Rh+) worksheet Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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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 Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Example:#6 p.156. Determine whether events are disjoint**

a) Randomly selecting a fruit fly with red eyes Randomly selecting a fruit fly with sepian (dark brown) eyes b) Receiving a phone call from a volunteer survey subject who opposes all cloning Receiving a phone call from a volunteer survey subject who approves of cloning of sheep c) Randomly selecting a nurse Randomly selecting a male Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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

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Example Suppose you were offered the following bet: You put up $10 and the other person puts up $5. You flip a coin two times. If you get two tails you loose. Event C : getting {TailTail} You want the complement of C to occur. C : {TailHead, HeadHead} Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**P(A) and P(A) are disjoint**

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

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**Venn Diagram for the Event A and its Complement**

Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Some background information for the Rule for Complimentary Events**

The probability of a sample space is always 1. Simply stated, when we are performing an experiment it is inevitable that we get a result that is somewhere in the sample space. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Rules of Complementary Events**

Since an event and it’s complement together cover entire sample space, hence the rule: P(A) + P(A) = 1 = 1 – P(A) P(A) = 1 – P(A) P(A) Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Example: #8 p.156. Finding complements**

A Reuters/Zogby poll showed that 61% of Americans say they believe that life exists elsewhere in galaxy. What is the probability of randomly selecting someone not having that belief? Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Pedestrian Intoxicated?**

Example: #10 p.157. Finding complements Use the data in the table, which summarizes results from 985 pedestrian deaths that were caused by accidents (based on data from the National Highway Traffic Safety Administration). Pedestrian Intoxicated? Yes No Driver Intoxicated? 59 79 266 581 If one of the pedestrian deaths is randomly selected, find the probability that the pedestrian was not intoxicated or the driver was not intoxicated worksheet Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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**Recap In this section we have discussed: Compound events.**

Formal addition rule. Intuitive addition rule. Disjoint events. Complementary events. Copyright © 2007 Pearson Education, Inc Publishing as Pearson Addison-Wesley.

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