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Basic Concepts of Discrete Probability

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Title: Basic Concepts of Discrete Probability


1
Basic Concepts of Discrete Probability
  • (Theory of Sets Continuation)

2
Functions
  • If X is a set and Y is a set, and there is a
    sequence of well-specified operations for
    assigning a well-defined object to
    every element , and by applying these
    sequence of operations to every member of a set X
    we obtain a set Y, then this sequence of
    operations forms a rule, which is commonly known
    as function.

3
Functions
  • The set X is called the domain of the function,
    and the set Y is called the range.
  • Functions that have numerical values ere called
    numerical functions.
  • A numerical function establishing the
    correspondence between the elements of a set
    having a finite number of elements and a set of
    natural numbers will be of our particular
    interest

4
The number of elements in a set
  • The number of elements in a set may or may not be
    finite.
  • If the elements of the set can be placed in
    one-to-one correspondence with the set of natural
    numbers 1, 2, 3,, we say that the set has a
    denumerable or countable number of elements
    otherwise the set is nondenumerable (e.g., a set
    of points on any line or the set of points in
    0,1).

5
The number of elements in a set
  • If a set is denumerable and
    is a finite number, we say that the power of
    the set X (the cardinality of the set X) is X
    (the number of elements in the set X).
  • If a set is equivalent to the set of points in
    0,1 that is, it is denumerable, we say that the
    set has the power of continuum.

6
The number of elements in a set
  • The following properties hold for finite sets
  • If A and B are two disjoint sets, then
  • For any finite sets A, B, and C

7
Basic Concepts of Discrete Probability
  • Elements of the Probability Theory

8
Sample Space
  • When probability is applied to something, we
    usually mean an experiment with certain outcomes.
  • An outcome is any one of the possibilities that
    may be expected from the experiment.
  • The totality of all these outcomes forms a
    universal set which is called the sample space.

9
Sample Space
  • For example, if we checked occasionally the
    number of people in this classroom on Tuesday
    from 7pm to 9-45pm, we should consider this an
    experiment having 15 possible outcomes
    0,1,2,,13,14 that form a universal set.
  • 0 nobody is in the classroom, 14 all
    students taking the Information Theory Class and
    the instructor are in the classroom

10
Sample Space
  • A sample space containing at most a denumerable
    number of elements is called discrete.
  • A sample space containing a nondenumerable number
    of elements is called continuous.

11
Sample Space
  • A subset of a sample space containing any number
    of elements (outcomes) is called an event.
  • Null event is an empty subset. It represents an
    event that is impossible.
  • An event containing all sample points is an event
    that is certain to occur.

12
Probability Measure
  • The probability measure is a specific type of
    function which can be associated with sets.
  • Since an experiment is defined so that to each
    possible outcome of this experiment there
    corresponds a point in the sample space, the
    number of outcomes of the experiment is assumed
    to be at most denumerable.

13
Probability Measure
  • Let us consider an event of interest A as the set
    of outcomes ak.
  • Let a real function m(ak ) be the probability
    measure of the outcome ak.
  • The probability measure of an event is defined as
    the sum of the probability measures associated
    with all the outcomes ak of that event. It is
    also referred to as the additive probability
    measure

14
Probability Measure
  • The additive probability measure has the
    following important property the measure
    associated with the union of two disjoint sets is
    equal to the sum of their individual measures.

15
Probability Measure
  • Two events A and B are called disjoint if they
    contain no outcome in common two disjoint events
    cannot happen simultaneously. The probability
    measure has the following assumed properties

16
Probability Measure
  • If with an
    additive probability measure, then the following
    relations are valid

17
Probability Measure
  • For three disjoint sets
  • For three sets in general

18
Frequency of events
  • Let us consider a specific event Xk among all
    the possible events of the experiment under
    consideration. If the basic experiment is
    repeated N times among which the event Xk has
    appeared times, the ratio
  • is defined as the relative frequency of the
    occurrence of the event Xk.

19
Frequency of events. The Probability
  • If N is increased indefinitely, that is if
    then, intuitively speaking,
  • is the probability of the event Xk.

20
The Probability
  • The classical definition of Laplace says that the
    probability is the ratio of the number of
    favorable events to the total number of possible
    events.
  • All events in this definition are considered to
    be equally likely e.g., throwing of a true die
    by an honest person under prescribed
    circumstances
  • but not checking of the number of people in the
    classroom.

21
The Probability
  • The following properties are important

22
The Probability
  • To verify whether our empiric definition of
    probability satisfies the properties of the
    probability measure, we have to consider the
    probability of two events.
  • Let A and B be the events among the ones
    resulting from the experiment. Let the experiment
    be repeated n times.

23
The Probability
  • Each observation can belong to only one of the
    four following categories
  • 1) A has occurred, but not B ? AB
  • 2) B has occurred, but not A ? BA
  • 3) Both A and B have occurred ? AB
  • 4) Neither A nor B has occurred ? AB

24
The Probability
  • If the number of events of each category from the
    mentioned four ones is denoted by
  • then
  • relative frequency of A
    independent of B
  • relative frequency of
    B independent of A

25
The Probability
  • relative
    frequency of either A, B or both
  • relative frequency of A and B
    occurring together
  • relative frequency of A
    under condition that B has occurred
  • relative frequency of B
    under condition that A has occurred

26
The Probability
  • When the number of experiments tends to infinity,
    these relations lead to
  • If A and B are mutually exclusive events, then
    PAB0 PABPAPB

27
The Probability
  • Thus, we have proved that
  • Let us compare the latter with the properties of
    the probability measure

28
The Probability
  • We have just shown that our empirical definitions
    of the frequency and probability, respectively,
    satisfy all the properties of the probability
    measure.

29
Theorem of addition
  • For two events A and B of the sample space
  • The additive property of the probability measure
    suggests that
  • If A and B are mutually exclusive events, then

30
Theorem of addition
  • For two opposite events A and A AAU AA0,
    then

31
Theorem of addition
  • For the three events A, B, and C
  • In general, for a number of events

32
Theorem of addition
  • If the events are mutually exclusive, then
  • It is also clear now that

33
Conditional Probability
  • Let A and B be two events. The conditional
    probability of event A based on the hypothesis
    that event B has occurred is defined by the
    following relation

34
Conditional Probability
  • Returning to the example of two events
  • 1) A has occurred, but not B ? AB ?n1
  • 2) B has occurred, but not A ? BA ?n2
  • 3) Both A and B have occurred ? AB ?n3
  • 4) Neither A nor B has occurred ? AB ?n4

35
Conditional Probability
  • The two events A and B are called mutually
    independent if PABPA and PBAPB.
  • For mutually independent events PABPAPB.

36
Theorem of multiplication
  • The multiplication rule for the case of two
    events A and B can be obtained through the
    definition of the conditional probability
    PABPAPBAPBPAB.
  • For three events A,B, and C PABCPABPCABP
    APBAPCAB

37
Theorem of multiplication
  • For n events
  • For a countable infinite number of the mutually
    independent events
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