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THE RELATIVE FREQUENCY DEFINITION OF PROBABILITY:ADDITION LAW

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MTH001 ­ Elementary Mathematics
LECTURE # 31:
Relative Frequency Definition of Probability
·
Axiomatic Definition of Probability
·
Laws of Probability
·
·  Rule of Complementation
·  Addition Theorem
THE RELATIVE FREQUENCY DEFINITION OF PROBABILITY
(`A POSTERIORI' DEFINITION OF PROBABILITY):
If a random experiment is repeated a large number of times, say n times, under
identical conditions and if an event A is observed to occur m times, then the probability of
the event A is defined as the LIMIT of the relative frequency m/n as n tends to infinitely.
Symbolically, we write
m
P(  A) = Lim
n
n→∞
The definition assumes that as n increases indefinitely, the ratio m/n tends to become stable
at the numerical value P(A). The relationship between relative frequency and probability can
also be represented as follows:
Relative Frequency Probability
as n → ∞
As its name suggests, the relative frequency definition relates to the relative frequency with
which are event occurs in the long run. In situations where we can say that an experiment
has been repeated a very large number of times, the relative frequency definition can be
applied.
As such, this definition is very useful in those practical situations where we are
interested in computing a probability in numerical form but where the classical definition
cannot be applied.(Numerous real-life situations are such where various possible outcomes
of an experiment are NOT equally likely). This type of probability is also called empirical
probability as it is based on EMPIRICAL evidence i.e. on OBSERVATIONAL data.
It can also be called STATISTICAL PROBABILITY for it is this very probability that forms
the basis of mathematical statistics.
Let us try to understand this concept by means of two examples:
1) from a coin-tossing experiment and
2) from data on the numbers of boys and girls born.
EXAMPLE-1:
Coin-Tossing:
No one can tell which way a coin will fall but we expect the proportion of leads and tails after
a large no. of tosses to be nearly equal. An experiment to demonstrate this point was
performed by Kerrich in Denmark in 1946. He tossed a coin 10,000 times, and obtained
altogether 5067 heads and 4933 tails.
The behavior of the proportion of heads throughout the experiment is shown as in the
following figure:
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MTH001 ­ Elementary Mathematics
The proportion; of heads in a sequence of tosses of a coin (Kerrich, 1946):
1.0
.8
.6
.5
.2
0
10
30
100
300
1000
3000
10000
3
Number of tosses (logarithmic scale)
As you can see, the curve fluctuates widely at first, but begins to settle down to a more or
less stable value as the number of spins increases. It seems reasonable to suppose that the
fluctuations would continue to diminish if the experiment were continued indefinitely, and the
proportion of heads would cluster more and more closely about a limiting value which would
be very near, if not exactly, one-half.
This hypothetical limiting value is the (statistical) probability of heads.
Let us now take an example closely related to our daily lives --- that relating to the sex ratio:-
In this context, the first point to note is that it has been known since the eighteenth century
that in reliable birth statistics based on sufficiently large numbers (in at least some parts of
the world), there is always a slight excess of boys,
Laplace records that, among the 215,599 births in thirty districts of France in the years 1800
to 1802, there were 110,312 boys and 105,287 girls.
The proportions of boys and girls were thus 0.512 and 0.488 respectively (indicating a slight
excess of boys over girls).In a smaller number of births one would, however, expect
considerable deviations from these proportions.
This point can be illustrated with the help of the following example:
EXAMPLE-2:
The following table shows the proportions of male births that have been worked out for the
major regions of England as well as the rural districts of Dorset (for the year 1956):
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MTH001 ­ Elementary Mathematics
Proportions of Male Births in various Regions
and Rural Districts of England in 1956
(Source: Annual Statistical Review)
Propo rtio n
Propo rtio n
Region of
Rura l Districts of
of Ma le
of Ma le
Eng land
Do rset
Births
Births
No rthern
.514
Beaminste r
.38
E. & W. Riding
.513
Blandfo rd
.47
No rth Weste rn
.512
Bridpo rt
.53
No rth Midland
.517
Do rcheste r
.50
Midland
.514
Shaftesbury
.59
Eastern
.516
She rbo rne
.44
London and S.
.514
Sturminste r
.54
Eastern
Wa re ham and
Southe rn
.514
.53
Purbeck
Wimbo rne &
South Weste rn
.513
.54
Cranbo rne
All Rura l District's
Who le country
.514
.512
of Do rset
As you can see, the figures for the rural districts of Dorset, based on about 200 births each,
fluctuate between 0.38 and 0.59. While those for the major regions of England, which are
each based on about 100,000 births, do not fluctuate much, rather, they range between
0.512 and 0.517 only. The larger sample size is clearly the reason for the greater constancy
of the latter. We can imagine that if the sample were increased indefinitely, the proportion of
boys would tend to a limiting value which is unlikely to differ much from 0.514, the proportion
of male births for the whole country.
This hypothetical limiting value is the (statistical) probability of a male birth.
The overall discussion regarding the various ways in which probability can be defined is
presented in the following diagram:
Probability
Non-Quantifiable
Quantifiable
(Inductive,
Subjective or
Personalistic
Probability)
" A Priori "
Statistical
Probability
Probability
(Verifiable
(Empirical or
through
" A Posteriori "
Empirical
Probability)
Evidence)
(A statistician's
main concern)
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MTH001 ­ Elementary Mathematics
As far as quantifiable probability is concerned, in those situations where the various possible
outcomes of our experiment are equally likely, we can compute the probability prior to
actually conducting the experiment --- otherwise, as is generally the case, we can compute
a probability only after the experiment has been conducted (and this is why it is also called
`a posteriori' probability).
Non-quantifiable probability is the one that is called Inductive Probability.
It refers to the degree of belief which it is reasonable to place in a proposition on given
evidence.
An important point to be noted is that it is difficult to express inductive probabilities
numerically ­­ to construct a numerical scale of inductive probabilities, with 0 standing for
impossibility and for logical certainty. An important point to be noted is that it is difficult to
express inductive probabilities numerically ­­ to construct a numerical scale of inductive
probabilities, with 0 standing for impossibility and for logical certainty.
Most statisticians have arrived at the conclusion that inductive probability cannot, in general,
he measured and, therefore cannot be use in the mathematical theory of statistics.
This conclusion is not, perhaps, very surprising since there seems
no reason why rational degree of belief should be measurable any more than, say, degrees
of beauty. Some paintings are very beautiful, some are quite beautiful, and some are ugly,
but it would be observed to try to construct a numerical scale of beauty, on which Mona Lisa
had a beauty value of 0.96.Similarly some propositions are highly probable, some are quite
probable and some are improbable, but it does not seem possible to construct a numerical
scale of such (inductive) probabilities .Because of the fact that inductive probabilities are not
quantifiable and cannot be employed in a mathematical argument, this is the reason why the
usual methods of statistical inference such as tests of significance and confidence interval
are based entirely on the concept of statistical probability. Although we have discussed
three different ways of defining probability, the most formal definition is yet to come.
This is The Axiomatic Definition of Probability.
THE AXIOMATIC DEFINITION OF PROBABILITY:
This definition, introduced in 1933 by the Russian mathematician
Andrei N. Kolmogrov, is based on a set of AXIOMS.
Let S be a sample space with the sample points E1, E2, ... Ei, ...En. To each sample point,
we assign a real number, denoted by the symbol P(Ei), and called the probability of Ei, that
must satisfy the following basic axioms:
Axiom 1:
For any event Ei,
0 < P(Ei) < 1.
Axiom 2:
P(S) =1
for the sure event S.
Axiom 3:
If A and B are mutually exclusive events (subsets of S), then
P (A B) = P(A) + P(B).
It is to be emphasized that According to the axiomatic theory of probability:
SOME probability defined as a non-negative real number is to be ATTACHED to each
sample point Ei such that the sum of all such numbers must equal ONE.
The ASSIGNMENT of probabilities may be based on past evidence or on some other
underlying conditions.
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MTH001 ­ Elementary Mathematics
(If this assignment of probabilities is based on past evidence, we are talking about
EMPIRICAL probability, and if this assignment is based on underlying conditions that ensure
that the various possible outcomes of a random experiment are EQUALLY LIKELY, then we
are talking about the CLASSICAL definition of probability.
Let us consider another example:
EXAMPLE :
Table-1
below shows the numbers of births in England and Wales in 1956 classified by (a) sex
and (b) whether liveborn or stillborn.
Table-1
Number of births in England and Wales in 1956 by sex and whether live- or still born.
(Source Annual Statistical Review)
Livebo rn
Stillbo rn
Tota l
Ma le
359,881 (A)
8,609 (B )
368,490
F emale
340,454 (B )
7,796 (D)
348,250
Tota l
700,335
16,405
716,740
There are four possible events in this double classification:
·  Male livebirth (denoted by A),
·  Male stillbirth (denoted by B),
·  Female livebirth (denoted by C)
and
·  Female stillbirth (denoted by D),
The relative frequencies corresponding to the figures of Table-1 are given in Table-2:
Table-2
Proportion of births in England and Wales in 1956 by sex and whether live- or
stillborn.
(Source Annual Statistical Review)
Livebo rn
Stillbo rn
Tota l
Ma le
.5021
.0120
.5141
F emale
.4750
.0109
.4859
Tota l
.9771
.0229
1.0000
The total number of births is large enough for these relative frequencies to be treated for all
practical purposes as PROBABILITIES.
Let us denote the compound events `Male birth' and `Stillbirth' by the letters M
and S.Now a male birth occurs whenever either a male livebirth or a male stillbirth occurs,
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MTH001 ­ Elementary Mathematics
and so the proportion of male birth, regardless of whether they are live-or stillborn, is equal
to the sum of the proportions of these two types of birth; that is to say,
p(M)
= p(A or B) = p(A) + p(B)
= .5021 + .0120 = .5141
Similarly, a stillbirth occurs whenever either a male stillbirth or a female stillbirth occurs and
so the proportion of stillbirths, regardless of sex, is equal to the sum of the proportions of
these two events:
p(S)
= p(B or D) = p(B) + p(D)
= .0120 + .0109 = .0229.
Let us now consider some basic LAWS of probability.
These laws have important applications in solving probability problems.
LAW OF COMPLEMENTATION:
If A is the complement of an event A relative to the sample space S, then
P(A )  = 1 - P(  A).
Hence the probability of the complement of an event is equal to one minus the probability of
the event.
Complementary probabilities are very useful when we are wanting to solve
questions of the type `What is the probability that, in tossing two fair dice, at least one even
number will appear?'
EXAMPLE:
A coin is tossed 4 times in succession. What is the probability that at least one head occurs?
(1) The sample space S for this experiment consists of 24 = 16 sample points (as each toss
can result in 2 outcomes),and
(2) we assume that each outcome is equally likely.
If we let A represent the event that at least one head occurs, then A will consist
of MANY sample points, and the process of computing the probability of this event will
become somewhat cumbersome! So, instead of denoting this particular event by A, let us
denote its complement i.e. "No head" by A.
Thus the event A consists of the SINGLE sample point {TTTT}.
Therefore P(A ) = 1/16.
Hence by the law of complementation, we have
P(A )  = 1 - P(A ) = 1 -
1  15
=  .
16  16
The next law that we will consider is the Addition Law or the General Addition Theorem of
Probability:
ADDITION LAW:
If A and B are any two events defined in a sample space S, then
P(AB) = P(A) + P(B) ­ P(AB)
In words, this law may be stated as follows:
"If two events A and B are not mutually exclusive, then the probability that at
least one of them occurs, is given by the sum of the separate probabilities of events A and
B minus the probability of the joint event A B."
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Table of Contents:
  1. Recommended Books:Set of Integers, SYMBOLIC REPRESENTATION
  2. Truth Tables for:DE MORGAN’S LAWS, TAUTOLOGY
  3. APPLYING LAWS OF LOGIC:TRANSLATING ENGLISH SENTENCES TO SYMBOLS
  4. BICONDITIONAL:LOGICAL EQUIVALENCE INVOLVING BICONDITIONAL
  5. BICONDITIONAL:ARGUMENT, VALID AND INVALID ARGUMENT
  6. BICONDITIONAL:TABULAR FORM, SUBSET, EQUAL SETS
  7. BICONDITIONAL:UNION, VENN DIAGRAM FOR UNION
  8. ORDERED PAIR:BINARY RELATION, BINARY RELATION
  9. REFLEXIVE RELATION:SYMMETRIC RELATION, TRANSITIVE RELATION
  10. REFLEXIVE RELATION:IRREFLEXIVE RELATION, ANTISYMMETRIC RELATION
  11. RELATIONS AND FUNCTIONS:FUNCTIONS AND NONFUNCTIONS
  12. INJECTIVE FUNCTION or ONE-TO-ONE FUNCTION:FUNCTION NOT ONTO
  13. SEQUENCE:ARITHMETIC SEQUENCE, GEOMETRIC SEQUENCE:
  14. SERIES:SUMMATION NOTATION, COMPUTING SUMMATIONS:
  15. Applications of Basic Mathematics Part 1:BASIC ARITHMETIC OPERATIONS
  16. Applications of Basic Mathematics Part 4:PERCENTAGE CHANGE
  17. Applications of Basic Mathematics Part 5:DECREASE IN RATE
  18. Applications of Basic Mathematics:NOTATIONS, ACCUMULATED VALUE
  19. Matrix and its dimension Types of matrix:TYPICAL APPLICATIONS
  20. MATRICES:Matrix Representation, ADDITION AND SUBTRACTION OF MATRICES
  21. RATIO AND PROPORTION MERCHANDISING:Punch recipe, PROPORTION
  22. WHAT IS STATISTICS?:CHARACTERISTICS OF THE SCIENCE OF STATISTICS
  23. WHAT IS STATISTICS?:COMPONENT BAR CHAR, MULTIPLE BAR CHART
  24. WHAT IS STATISTICS?:DESIRABLE PROPERTIES OF THE MODE, THE ARITHMETIC MEAN
  25. Median in Case of a Frequency Distribution of a Continuous Variable
  26. GEOMETRIC MEAN:HARMONIC MEAN, MID-QUARTILE RANGE
  27. GEOMETRIC MEAN:Number of Pupils, QUARTILE DEVIATION:
  28. GEOMETRIC MEAN:MEAN DEVIATION FOR GROUPED DATA
  29. COUNTING RULES:RULE OF PERMUTATION, RULE OF COMBINATION
  30. Definitions of Probability:MUTUALLY EXCLUSIVE EVENTS, Venn Diagram
  31. THE RELATIVE FREQUENCY DEFINITION OF PROBABILITY:ADDITION LAW
  32. THE RELATIVE FREQUENCY DEFINITION OF PROBABILITY:INDEPENDENT EVENTS