Slide Slide 1 Created by Tom Wegleitner, Centreville, Virginia Edited by Olga Pilipets, San Diego, California Overview.

SlideSlide 1

Created by Tom Wegleitner, Centreville, VirginiaEdited by Olga Pilipets, San Diego, California

Overview

SlideSlide 2Copyright © 2007 Pearson

Education, Inc Publishing as Pearson Addison-Wesley.

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.

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Fundamentals



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.



Event

an outcome of an experiment or a procedure.

Simple Eventan outcome or an event that cannot be further broken down into simpler components

Sample SpaceAll possible outcomes

Compound eventany event combining 2 or more simple events

Success

a favorable outcome of an experiment or a procedure.

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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.

Notation for Probabilities

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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}



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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.



Basic Rules for Computing Probability

P(S) = number of successes

number of trials

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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 0.000340 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

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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.



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.)

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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



Basic Rules for Computing Probability - cont

P(S) =

number of ways a success can occur

total number of possible outcomes (events)

sn

=

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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.



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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

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Rule 3: Subjective Probabilities



Basic Rules for Computing Probability - cont

P(A), the probability of event A, is estimated

by using knowledge of the relevant

circumstances.

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As a procedure is repeated again and again, the relative frequency probability (from Rule 1) of an event tends to approach the actual probability.



Law of Large Numbers

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Finish problems from section 4.2



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Addition Rule

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Compound Event any event combining 2 or more simple events



Definition

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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



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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)



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.



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.

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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



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.

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Group

O A B AB

TypeRh+ 39 35 8 4

Rh- 6 5 2 1



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.

If one person is randomly selected, find P (group B or type Rh+)

worksheet



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

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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 cloningReceiving a phone call from a volunteer survey subject who approves of cloning of sheep

c) Randomly selecting a nurseRandomly selecting a male





Definition

The complement of event A, denoted by

A, consists of all outcomes in which the

event A does not occur.

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P(A) and P(A)are disjoint



It is impossible for an event and its complement to occur at the same time.





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.



P(A) + P(A) = 1

= 1 – P(A)

P(A) = 1 – P(A)

P(A)

Since an event and it’s complement together cover entire sample space, hence

the rule:

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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?



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Pedestrian Intoxicated?

Yes No

Driver Intoxicated?

Yes 59 79

No 266 581



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).

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



In this section we have discussed:

Compound events.

Formal addition rule.

Intuitive addition rule.

Disjoint events.

Complementary events.

Slide Slide 1 Created by Tom Wegleitner, Centreville, Virginia Edited by Olga Pilipets, San Diego, California Overview.

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