Download Part 1 - Angelfire

RES 341
RESEARCH AND EVALUATION
WORKSHOP 4
By
Dr. Serhat Eren
University OF PHOENIX
Spring 2002
1
CHAPTER VII: PROBABILITY
7.1 CHAPTER OBJECTIVES

Basic Probability Rules

Random Variables and Probability
Distributions

The Binomial Probability Distribution

The Normal Probability Distribution
2
7.2 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.
3
7.2 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}
4
7.2 BASIC RULES OF PROBABILITY
7.2.1 Probability of an Event
 An 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.
5
7.2 BASIC RULES OF PROBABILITY

When each of the outcomes in a sample
space is equally likely, then the
probability of A can be calculated using
the following formula:
nA
Number of ways that A can occur
P( A) 

Total number of possible outcomes
N

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.
6
7.2 BASIC RULES OF PROBABILITY

There are some facts about probabilities
that must be true:
– 0P(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
8
9
7.2 BASIC RULES OF PROBABILITY

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
10
11
7.2 BASIC RULES OF PROBABILITY
7.2.2 Combinations of Events-OR and AND
 The 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.
12
7.2 BASIC RULES OF PROBABILITY

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)
13
7.2 BASIC RULES OF PROBABILITY

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.
14
15
7.2 BASIC RULES OF PROBABILITY

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)
16
17
7.2 BASIC RULES OF PROBABILITY

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.
18
19
20
7.2 BASIC RULES OF PROBABILITY
7.2.3 Probabilities as Relative Frequencies
 If 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.
21
22
23