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RES 341RES 341RESEARCH AND RESEARCH AND

EVALUATIONEVALUATION

WORKSHOP 4

By Dr. Serhat Eren

University OF PHOENIXSpring 2002

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CHAPTER VII: PROBABILITY CHAPTER VII: PROBABILITY

7.1 CHAPTER OBJECTIVES7.1 CHAPTER OBJECTIVES

Basic Probability Rules

Random Variables and Probability Distributions

The Binomial Probability Distribution

The Normal Probability Distribution

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7.2 BASIC RULES OF PROBABILITY7.2 BASIC RULES OF PROBABILITY

Probability is a measure of how likely it is that something will occur.

An experiment is any action whose outcomes are recordable data.

The sample space, S, is the set of all possible outcomes of an experiment.

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7.2 BASIC RULES OF PROBABILITY7.2 BASIC RULES OF PROBABILITY

As a very simple experiment, consider rolling a single, six-sided die and recording the number of spots on the top facing side.

You know that there are six possible outcomes of the experiment, so the sample space is

S = {1, 2, 3, 4, 5, 6}

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7.2.1 Probability of an EventAn event, A, is an outcome or a set of

outcomes that are of interest to the experimenter.

The likelihood that the event A will occur is called the probability of A and is written P(A).

The probability of an event A, P(A), is a measure of the likelihood that an event A will occur.

7.2 BASIC RULES OF PROBABILITY7.2 BASIC RULES OF PROBABILITY

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When each of the outcomes in a sample space is equally likely, then the probability of A can be calculated using the following formula:

Where nA is the number of outcomes that correspond to the event, A, and N is the total number of outcomes in the sample space, S.

N

n

nTotal

NumberAP A

outcomes possible ofumber

occur can A that waysof )(

7.2 BASIC RULES OF PROBABILITY7.2 BASIC RULES OF PROBABILITY

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There are some facts about probabilities that must be true:– 0P(A)1. This says that the probability of

an event must be a number between 0 and 1 inclusive.

– P(S)=1. This says that the sum of the probabilities for the entire sample space must be equal to 1, or that essentially.

– If an event A MUST happen, then P(A)=1, and if the event cannot happen, then P(A)=0.

7.2 BASIC RULES OF PROBABILITY7.2 BASIC RULES OF PROBABILITY

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The complement of an event A, denoted A', is the set of all outcomes in the sample space, S, that do not correspond to the event A.

Since the event A is a set of outcomes in S and the complement of the event A' is the set of all outcomes in S that do NOT correspond to A, we can see that

P(A) + P(A') = 1

7.2 BASIC RULES OF PROBABILITY7.2 BASIC RULES OF PROBABILITY

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7.2.2 Combinations of Events-OR and ANDThe event AORB describes the event that

either A happens or B happens or they both happen.

The event AANDB is the event that A and B both occur.

Two events, A and B, are said to be mutually exclusive if they have no outcomes in common.

7.2 BASIC RULES OF PROBABILITY7.2 BASIC RULES OF PROBABILITY

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When two events are mutually exclusive, then the probability that A occurs or B occurs, P(AORB), is the sum of the individual probabilities.

P(AORB) = P(A) + P(B)

The simple addition rule easily extends to any number of mutually exclusive events. For example, if A, B, C, and D are four mutually exclusive events, then

P(AORBORCORD) = P(A) + P(B) + P(C) + P(D)

7.2 BASIC RULES OF PROBABILITY7.2 BASIC RULES OF PROBABILITY

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How can we adjust the simple addition rule of probability to work in situations when the events are NOT mutually exclusive?

The problem occurs when an outcome is included in both events in AANDB. The answer obtained by adding the individual probabilities was too large because the probability that both AANDB occur, P(AANDB), is included in both individual probabilities.

7.2 BASIC RULES OF PROBABILITY7.2 BASIC RULES OF PROBABILITY

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If we consider this we can come up with a general addition rule for probability that considers the probability of AORB when the events are not mutually exclusive:

P(AORB) = P(A) + P(B) - P(AANDB)

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When data are collected on two related variables, they are organized by using a cross-classification table or contingency table.

A contingency table is a table whose possible values of one variable and whose columns represent possible values for a second variable. The entries in the table are the number of times that each pair of values occurs.

7.2 BASIC RULES OF PROBABILITY7.2 BASIC RULES OF PROBABILITY

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7.2.3 Probabilities as Relative FrequenciesIf the data are a good representation of the

population, then we can also use the relative frequencies as estimates of the true probabilities for the population. Probabilities calculated in this way are often called empirical probabilities.

An empirical probability is one that is calculated from sample data and is an estimate for the true probability.

7.2 BASIC RULES OF PROBABILITY7.2 BASIC RULES OF PROBABILITY

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