BASIC CONCEPTS IN PROBABILITY

B

BASIC CONCEPTS IN PROBABILITY

We see that the theory of probability is at bottom only common sense reduced to calculation; it makes us appreciate with exactitude what reasonable minds feel by a sort of instinct, often without being able to account for it. It is remarkable that this science, which originated in the consideration of games of chance, should become the most important object of human knowledge.

Laplace

This chapter covers fundamental topics on probabilities of events.

Main Topics q Basics of Set Theory q Fundamental Concepts in Probability q Conditional Probability q Independent Events q Total Probability Theorem and Bayes' Rule q Combined Experiments and Bernoulli Trials

The materials covered in this chapter are essential for the study of the remaining chapters. Emphasis should be put on the understanding of concepts and how they can be applied.

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2.1 Basics of Set Theory

2.1 Basics of Set Theory

2.1.1 Basic Definitions

q set = a collection of objects, denoted by an upper case Latin letter Example: $ I J ' IG G GJ.

q element = an object in a set, denoted by a lower case Latin letter We say "D is an element of $," "D is in $," or "D belongs to $," denoted as D $.

q empty set = null set = a set with no elements, denoted by q space = the set with all the elements for the problem under consideration

(sometimes called universal set), denoted by 6

Convention:

Upper case Latin letter set Lower case Latin letter element If every element of set $ is also an element of set %, then $ is said to be a subset of %, denoted as $ | % or % } $. Set $ is said to be equal to set % if $ | % and $ } %, denoted as $ %. In this case, $ and % have exactly the same elements. Two sets are said to be disjoint if they do not have any element in common. Example 2.1: Consider the set of all positive integers smaller than 7: rule method $ I[ [ [ an integerJ tabular method $ I J

the space (universal set) of a 6-face die

Tabular form is not universally applicable. Example 2.2: Consider the set of all positive numbers smaller than 6:

rule method % I[ [ [ a real numberJ

There is no tabular form for this set because it is uncountable. Example 2.3: Consider the set of all positive integers:

& I[ [ ! [ an integerJ I J Example 2.4: The set of human genders * = Ifemale, maleJ

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2.1 Basics of Set Theory

2.1.2 Basic Set Operations

Definitions:

q The set of all elements of $ or % is called the union (or sum) of $ and %, denoted as $ > % or $ %. Union of disjoint sets $ and % may be denoted as $ @ %. Convention: "$ or %" = "either $ or % or both."

q The set of all elements common to $ and % is called the intersection (or product) of $ and %, denoted as $ ? % or $%.

q The set of all elements of $ that are not in % is called the difference of $ and %, denoted as $ b %.

q The set of all elements in the space 6 but not in $ is called the complement of $, denoted as $. It is equal to 6 b $.

A simple and instructive way of illustrating the relationships among sets is the so-called Venn diagram, as illustrated below.

6

$$>%%

$$?%%

6

$$b%%

6

6

$

$

Figure 2.1: Basic set operations.

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2.1 Basics of Set Theory

Example 2.5: Set Operations

For $, %, and & considered in Examples 2.1, 2.2, and 2.3:

$|& $>% $>& %>&

$?% $?& %?& $b% $b& %b$ %b& & b$ & b%

I[ [ [ a real numberJ & I[ [ a positive integer or a real number satisfying [ J This set has a mixed type. I J $ I J IJ I[ [ [ a noninteger real numberJ I[ [ [ a noninteger real numberJ I[ [ w [ an integerJ I J I[ [ w [ an integerJ I J

Space 6 depends on what we are considering. If we are considering only positive real numbers, then 6 I[ [ ! [ realJ. Thus,

$ I[ [ a positive real number other than J % I[ [ w [ a real numberJ & I[ [ a noninteger positive real numberJ

If, however, we are considering all real numbers, then 6 I[ [ realJ. Thus

$ I[ [ a real number other than J % I[ [ or [ w [ a real numberJ & I[ [ or [ a noninteger positive real numberJ

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2.1 Basics of Set Theory

2.1.3 Basic Algebra of Sets

Algebra of sets Algebra of numbers

Union > sum ""

Intersection ? product "c"

1

$>% %>$ DE ED

2

$?% %?$ DcE EcD

3

$ > % > & $ > % > & D E F D E F

4

$ ? % ? & $ ? % ? & D c E c F D c E c F

5 $ ? % > & $ ? % > $ ? & D c E F D c E D c F

6 $ > % ? & $ > % ? $ > & see below

Since $ ? $ $, $ ? % | $, $ ? & | $, and

D ED F D c D D c E D c F E c F

Line 6 in the table above follows from

$ > % ? $ > & $ ? $ > $ ? % > $ ? & >% ? &

_

^]

`

$

This illustrates that set algebra has its own rules.

De Morgan's laws:

$ > % ? &

$>% $?%

(2.1)

$?% $>%

(2.2)

Similarly,

$ > % > & $ > ' $ ? ' $ ? % > & $ ? % ? & $ ? % ? &

$ > c c c > $Q $ ? c c c ? $Q $ > $ ? $ > $

$ ? c c c ? $Q $ > c c c > $Q $ ? $ > $ ? $

Rules: (1) interchange > and ?; (2) interchange e and e. However, care should be taken when dealing with multiple nests, as demonstrated below.

Example 2.6:

$_ ?^] %` > & ' > & ' ? & $ ? % ? & $ > % ? &

(2.3)

'

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