1. Introducing the Course on Basic Electrical
2. Generation, Transmission and Distribution of Elect
3. sample
4. Introduction of Electric Circuit
5. Superposition Theorem in the context of dc voltage
6. Wye (Y) - Delta (Δ) OR Delta (Δ)-Wye (Y) Transform..
7. Node-voltage analysis of resistive circuit in the 
8. Loop Analysis of resistive circuit in the context
9. Generation of Sinusoidal Voltage Waveform (AC) and
10. Study of DC transients in R-L-C Circuits
11. Analysis of dc resistive network in presence of on
12. Thevenin's and Norton's theorems in the context
13. Superposition Theorem in the context of dc voltage
14. Three-phase Delta-Connected Balanced Load
15. Three-phase Balanced Supply
16. Resonance in Series and Parallel Circuits
17. Solution of Current in AC Parallel and Series-para
18. Solution of Current in AC Series and Parallel Circ
19. Solution of Current in R-L-C Series Circuits
20. Representation of Sinusoidal Signal by a Phasor
21. Equivalent Circuit and Power Flow Diagram of IM
22. Construction and Principle of Operation of IM
23. Rotating Magnetic Field in Three-phase Induction
24. Problem solving on Transformer
25. Auto-Transformer
26. Three Phase Transformer
27. Testing, Efficiency & Regulation
28. Practical Transformer
29. Ideal Transformer
30. Eddy Current & Hysteresis Loss
31. Magnetic Circuits
32. Measurement of Power in a Three-phase Circuit
33. Thevenin's and Norton's theorems in the context
34. Study of Single Phase Induction Type Energy Meter
35. Study of Electro-Dynamic Type Instruments
36. Study of DC-AC Measuring Instruments
37. Problem Solving on D.C Machines
38. Losses, Efficiency and Testing of D.C. Machines
39. D.C Shunt Motor
40. D.C Generators
41. EMF & Torque Equation
42. Principle of Operation of D.C Machines 
43. Constructional Features of D.C Machines
44. Starting Methods for Single-phase Induction Motor
45. Different Types of Starters for Induction Motor
46. Torque-Slip (speed) Characteristics of Induction M.

Posted in Labels: Topics 0 comments Posted by waytofeed at 9:03 AM  

This definition was given in 1930’s by Van Mises and A Kolmogorov. The limitation of both the definitions of probability given earlier are overcome by this approach to probability. The classical and relative frequency approach are special cases of this definition. It is based on the concepts of set theory.
Hence, to understand this approach, the concepts of set theory is essential.

Let  be a sample space and let p be a real-valued function defined on the subsets of . Then P is called probability function and the probability of the occurrence of event A is given by P(A) if the following axioms hold good.
Definition: Let (, S) be a sample space. A set function P defined on S is called a probability measure (or simply probability) if it satisfies the following conditions

Axiom 1: P(A) is a real number, i.e.,
P(A)  0 for every A  S or 0 P(A) 1.

Axiom 2: P() = 1.
Axiom 3: Let {Ai}, Ai  S i=1, 2, 3, …, be a disjoint sequence of sets, such that Ai  Aj =  for ij, then
We call P(A) the probability of an event. The axiom 3 is called countable additivity.
In other words, if the likelihood of the occurrence of an event A resulting from a statistical experiment is evaluated by means of a set of real numbers called weights or probabilities ranging from 0 to 1, then the probability of A is the sum of the weights of all the sample points in A.
Thus, if  is a sample space consisting of n events and if each event is equally likely to happen, then its probability is 1/n. Now if A is an event of  containing m elements of , i.e., n(A) = m, then
P(A) = 1/n + 1/n + 1/n +…+ up to m
terms
= m/n
P(A) = n(A)/n()
Where,
n(A) = No. of distinct elements in A
n() = No. of distinct elements in 

Example: If a coin is tossed thrice. What is the probability that at least one head occurs?
Solution:
 = {TTT, TTH, THT, HTT, THH, HTH, HHT, HHH}
If the coin is unbiased, each of these outcomes would be equally likely to occur. A probability or weight () can be assigned to each sample point. Then 8 = 1 or =1/8. If A denotes the event of at least one head occurring then, A = {TTH, THT, HTT, THH, HTH, HHT, HHH}
and
P(A) = 1/8+1/8+1/8+1/8+1/8+1/8+1/8
= 7/8

Example 2: If a coin is tossed thrice. Find the probability of getting (i) exactly one head (ii) exactly two heads, (iii) exactly one head or two heads, (iv) none of them are head
Theorem 1:
The probability of an impossible event is zero, i.e., P() = 0
Proof:
We know that, an impossible event contains no sample point, thus, the certain event  and the impossible event  are mutually exclusive.
It follows that
   = 
P(  ) = P()
P() + P() = P() from axiom 3
P() = 1-1 = 0 from axiom 1
P() = 0

Theorem 2: If Ac is the complementary event of A, the P(Ac) = 1 – P(A)

Proof:
Since A and Ac are complementary events, they are disjoint events. Also,
A  Ac = 
P(A  Ac) = P(A) + P(Ac) = P() =1
 P(Ac) = 1 – P(A)
Theorem 3:
For any two events A and B
P(Ac B) = P(B) - P(A  B)

Proof:
It is obvious that Ac A and A  B are disjoint events and
(A  B)  (Ac B) = B
Thus using axiom 3 we get,
P(B) = P(A  B) + P(Ac B)
 P(Ac B) = P(B) - P(A  B).

Similarly, it can be proved that
P(A  Bc) = P(A) - P(A  B).
Proof:
(i) Observe that A  Bc and B are
mutually exclusive events and
B  (A  Bc) = A, since B  A

 P(A) = P[B + P(A Bc)]
Using axiom 3, we get
= P(B) + P(A  Bc)
 P(A  Bc) = P(A) – P(B)

Proof:
We have B=(Ac B)  (A  B)
Now,
P(Ac B) = P(B) –P(AB) (1)
and
P(A  Bc) = P(A) – P(A  B) (2)
Adding (1) and (2), we get
P(ABc)+P(AcB)=P(A)+P(B)-2P(AB)
or
P(ABc)P(AcB)=P(A)+P(B)-2P(AB)

Laws of probability

Theorem 6:
If A and B are any two events, which are not mutually exclusive, then
P(A B) = P(A) + P(B) – P(A  B)

Proof:
We know that
A  B = A  (Ac  B)

Posted in 0 comments Posted by waytofeed at 8:54 AM  

This definition was given by Van Mises.

If a trial is repeated a number of times under essentially homogenous and identical conditions, then the limiting value of the ratio of the number of times the event happens to the number of trials, as the number of trials becomes indefinitely large is called the probability of happening of the event.
It is assumed here that the limit is finite and unique.

Symbolically, if in n trials, an event A happens m times, then the probability ‘p’ of the happening of A is
This definition is an effort to overcome the limitations of the mathematical definition and is a result of the law of statistical regularity. The law of statistical regularity states that if random experiments are repeated a large number of times, even though there is an unpredictable behaviour of
the individual results, the average results of long sequences of random experiments show a visible regularity.
Apparently, the two definitions of probability are different. The first one is the relative frequency of favourable cases to the total number of cases while in the later it is the limit of the relative frequency of the happening of the event.

Limitations of relative frequency approach

1. The condition of an experiment may
not remain the same in long series
of trials

2. The relative frequency may not
attain a unique value inspite of a
large number of trials
-field:
Let S be non-empty class of subsets of  then S is called a -field on  if
(a)   S
(b) For any A  S  Ac  S
(c) S is closed under the formation of
countable unions, i.e.,

Posted in 0 comments Posted by waytofeed at 8:51 AM  

In a random experiment, out of ‘n’ exhaustive, mutually exclusive, equally likely, independent events if ‘m’ of them are favorable to the occurrence of an event, say, ‘A’, then the probability of an event ‘A’, denoted by P(A) is

According to this definition, P(A) is found a priori without actual experimentation. For instance, if we keep using examples like fair coins, unbiased dice and standard deck of cards, we can state the answer in advance (a priori) without tossing a coin, or rolling a die or drawing a card.
Sometimes P(A) can be expressed by saying that “the odds in favour of A are m : (n-m) or the odds against A are (n-m) : m
P(A) +P(Ac) = 1 or P(Ac) = 1-P(A)

Limitations of classical definition probability

The classical definition of probability has certain drawbacks and fails at times in different situations as described below.
1.This definition emphasizes that the
events must be equally likely. Thus,
it fails when various outcomes of a
trail are not equally likely.
For example if a die is biased that gives numbers greater than 3 more often than the numbers less than 3, then the occurrence of numbers on the die is not equally probable.
Similarly, the definition also fails when we have to find the probability of an operation to be successful as the event of success or failure are not equally probable.
2. This definition is useful only when
we deal with card games, dice
games, coin tossing and the like. It
fails in situations when we try to
apply it to less orderly decision
problem we encounter in
management.
3. It does not consider those
situations that are unlikely but that
could conceivably happen. Like the
occurrence of a coin landing on its
edge or our room burning down
while watching TV etc., which are
extremely unlikely but not
impossible.
4. In case the exhaustive number of
cases in a trial is infinite, the
definition fails to give the required
probability.

5. In some situation there may be a
difference of opinion with respect to
the ways of forming the possible
and favourable outcomes.

Examples

1. Three light bulbs are selected at random from 15 bulbs of which 5 are defectives. Find the probability that
(i) None is defective
(ii) Exactly one is defective
(iii) At least one is defective
(iv) At most one is defective
Solution:
(i) - Exhaustive number of cases
15c3 = 455 ways
- Favourable cases
10c3 = 120 ways
Let A1 be an event that none of the bulb chosen is defective, then
P(A1)=120/455=0.26
(ii) - Exhaustive number of cases
15c3 = 455 ways
- Favourable cases
5c1 x 10c2 = 5 x 45 = 225 ways
Let A2 be an event that exactly one bulb chosen is defective, then
P(A2)=225/455=0.49
(iii) Let A3 be an event that at least one
bulb chosen is defective, then
P(A3)= 1-P(A1) = 1-0.26=0.74

(iv) Let A4 be an event that at most
one bulb chosen is defective, then
P(A4) = P(A1) +P(A2) = 0.26 + 0.49 =0.75

Posted in 0 comments Posted by waytofeed at 8:49 AM  

There are three different ways of defining probability. These three represent distinct conceptual approaches to the study of probability theory.

1. Classical approach
2. Relative frequency approach
3. Axiomatic approach

Posted in 0 comments Posted by waytofeed at 8:47 AM  

♣ Random experiment
☺An experiment in which results
are unpredictable
♣ Outcome
☺Results of a random experiment
♣ Exhaustive outcome
☺All possible occurrence of
outcomes in a random experiment
♣ Mutually exclusive event
☺Occurrence of one event
precludes or prevents the
occurrence of other event in a
random experiment.
♣ Equally likely event
☺Every outcome of a random
experiment which occur will have
an equal chance
♣ Independent event
☺Every outcome of a random
experiment will occur
independently of each other

♣ Favourable outcome
☺An event in which one is
interested in a random experiment
Sample space
The set of all possible outcomes of a random experiment is called sample space denoted by .

Example:
 = {NNN, NND, NDN, DNN, NDD, DND, DDN, DDD}
Event
A subset of the sample space  is known as an event. That is an event is a set consisting of possible outcomes of the random experiment. If the outcome of the experiment is contained in A, then we say that A has occurred.
Example: A = {x: x one defective chip is
found}
Algebra of events
Union: Suppose A and B are any two events then A  B = {x: x  A or x  B}

Example: If
A = {a card drawn at random from a
deck of cards is a jack}
B = {a card drawn at random from a
deck of cards is a diamond}
then A  B = {x: x is either a card
drawn randomly is a jack
or a diamond}.

Example 2: An unbiased die is thrown.
Let A = {an event that the outcome is
an odd number}
B = {an event that the outcome is
number  3}
then A  B = {x: x=1, 3, 4, 5, 6}.

Intersection: Suppose A and B are any two events then
A  B = {x: x  A and x  B}
Example 1: If
A = {a card drawn at random from a deck of
cards is a jack}
B = {a card drawn at random from a deck of
cards is a diamond}
then A  B = {x: x is a card drawn at
random from a deck of
cards is a jack}.

Example 2: An unbiased die is thrown.
Let A = {an event that an outcome is
an odd number}
B = {an event that an outcome is  3}
A  B = {x: x= 3, 5}.
Complement of events: Suppose  is a sample space and A be any event then
Ac = {x: x   but x  A}
Example 1: An unbiased die is thrown.
Let ={x: 1x6 }
A={an event that an outcome is  3}
Ac = {x: 1x2 }.

Posted in 0 comments Posted by waytofeed at 8:46 AM  

♣ Chance
♣ Un-predictable
♣ Non-deterministic
♣ Uncertain
♣ Random
♣ Stochastic

Posted in 0 comments Posted by waytofeed at 8:44 AM  

Posted in 0 comments Posted by waytofeed at 8:43 AM  

Sl No	Questions
1.	What do you mean by a set?  Give three examples to substantiate your answer. A set is a collection of objects called elements, which share some common properties. Examples   (i) The set of Natural numbers  (ii) The set of Whole numbers (iii) The set of integers (iv) The set of real numbers etc. A set is a collection of objects called elements, which share some common properties. Any object  belonging to a set is called an element of that set. The elements in a set are like: People, building, car, letters etc.
2.	Define the following with an illustrations to each
	(a) Universal set: A set is called a universal set if it includes every set under discussion.  A universal set is denoted by U
	(b) Null set: A set which does not contain any element is called an empty set or a null set denoted by j
	(c) Equavalent sets:
3.	Define the union and intersection of two sets. Give an example
	Let A and B be any two sets.  The union between these two sets, denoted by A  B is the set of all elements which are members of the set A or the set B or both.  Symbolically, it is written as A È B = {x/xεA v xεB v both} A È A = A
4.	Define
	(a)    Relative complement of the set A with respect to the set B Let A and B be any two sets.  The relative complement of B in A (or of B with respect to A) written A – B is the set consisting of all elements of A which are not elements of B, i.e.,          A – B = {x/x ε A L x Ï B }. The relative complement of B in A is also called the difference of A and B
	(b)   Complement of a set A (c)    Let U be the universal set.  For any set A,  the relative complement of A with respect to U, i.e., U – A is called the (absolute) complement of A and is denoted by Ac, i.e., (d)            Ac= U – A = {x/x ε U L x Ï A }.
5.	If Â={-¥, ¥}, A={x/x is a negative integers}, B={x/x is a nonpositive integer}
	Find (i )AÈÂ (ii) AÇÂ and (iii) AÇB, (iv) Ac (v) A-B
6.	Define the symmetric difference between the two sets Let A and B be any two sets.  The symmetric difference of A and B is the set consisting of all the elements that belong to A or B but not to both A and B.  It is dented by A Δ B i.e., A Δ B = {x/(xεA L xÏB) or (xεB L xÏA}.           = (A – B) U (B – A)
7.	If A={1, 2, 3, 4, 5, 8} and B={1, 2, 3, 5, 6, 7} find the symmetric difference the set A and the set B.
8.	If A, B, and C are any three sets prove that (i) (AÈB)ÈC = AÈ(BÈC) and (ii) (AÇB)ÇC = AÇ(BÇC)
9.	State De Morgan’s law for a swequence of n sets A1, A2, A3, …, An
10.	Mention the properties of Union, Intersection, Complement, and Symmetric difference between the sets. Let A, B, and C are sets and U be the universal set. Then   (i) Ac = U - A  (ii) A – B = A ∩ Bc  (iii) A – A = j (iv) A – j = A  (v) A – B = B – A which implies A = B (vi) A – B = A which implies A ∩ B= j (vii) A – B = j which implies AÍB Symmetric difference Let A and B any sets, then   (i) AΔA = j  (ii) A Δj = A  (iii) A Δ B = B Δ A (iv) A Δ B = (AÈB) –(A ∩ B)
11.	Represent the following using the Venn diagram
	(i) AÈB, (ii) AÇB, (iii) AÇBc, (iv) (AÈB)c, (v) (AÇB)c
12.	Define the Cartesian product of n-tuples. the Cartesian product of n sets A1, A2, …, An is defined as A1x A2x …x An = {(a1, a2, …, an)/ai ε Ai, for i = 1, 2, …, n}. The expression {(a1, a2, …, an) is called an ordered n-tuple.
13.	Let A, B, And C are finite sets.  Prove that
	(i) A X (BÈC) = (A X B) È (A X C)
	(ii) A X (BÇC) = (A X B) Ç (A X C)
14.	If A={1, 2, 4}, B={2, 5, 7}, and C={1, 3, 5} are any three sets verify
	(i) A X (BÈC) = (A X B) È (A X C)
	(ii) A X (BÇC) = (A X B) Ç (A X C)
15.	Show by means of an example
	(i) A X B ¹ B X A
	(ii) (A X B) X C = A X( B X C)
16.	Define (i) binary relation, (ii) Domain of a relation, and (iii) Range of a relation
17.	Mention the properties of relations.
18.	If R is the set of all real numbers and ° is the binary operation on R defined by a ° b=2a +3b, (a, b) Î R, show that ° is neither associative nor commutative.
19.	Examine whether the operatins ‘Å’ and ‘Ä’ defined by a Å b = a+2ab,  aÄ b =  an, for positive integers, obey the commutative, associative and distributive laws on not.
20.	Let Q be the set of rationals and an operation ° is defined over Q - {1} as a ° b = a + b – a.b, ‘+’ and ‘.’ being ordinary addition and multiplication.  Verify which of the algebraic laws satisfy the operatin ° over the set Q – {1}.
21.	For the relations R and S define (i) RÈS, (ii) RÇS, (iii) R-S, and (v) Rc
22.	For the relations R and S define the composite of (i) R and S and (ii) S and R.  Are they commutative?
	Let A = {1, 2, 3, 4} and R = {(1, 2), (2, 2), (3, 4), (4, 1)}.  Is R (i) Reflexive, (ii) Symmetric,
	(iii) Antisymmetric, (iv) Transitive?
23.	If A={1, 2, 3}, B={1, 2, 3, 4} and C={0, 1 , 2}, the relations R and S are defined as
	R={(1, 1), (1, 4), (2, 3), (3, 1), (3, 4)}
	S={(1, 0), (2, 0), (3, 1), (3, 2), (4, 1)}
	Find SoR. Can we find RoR? If so find RoR.
24.	Let X={1, 2, 3, 4, 5, 6, 7} and R={(x, y)/x-y is divisible by 3} in X.  Show that R is an equivalence relation.
25.	Let Z be the set of integers and let R be the relation called “congruence modulo 3” defined by
	R={(x, y)/x e Z Ù y e Z Ù (x-y) is divisible by 3}.  Is R an equivalence relation?
26.	Let A={v1, v2, v3, v4, v5}. Let R be a relation defined by R={(v1, v2), (v1, v3), (v3, v5), (v4, v1), (v4, v4), (v5, v2), (v5, v3), (v5, v1)}.  Write the adjacency matrix of R and draw a digraph for R.
27.	Define the following terms
	(i) Into function
	(ii) One-to-one function
	(iii) Bijective function
	(iv) Many-one mapping
28.	Which of the following relations on the set A = {1, 2, 3} is (i) reflexive, (ii) sy;mmetric, (iii) transitive, (iv) antisymmetric?
	R1 = {(1,1), (1, 2), (1, 3), (3, 3)}
	R2 = {(1,1), (1, 2), (2, 1), (2, 2), (3, 3)}
	R3 = {(1,1), (1, 2),  (2, 2), (3, 3)}
29.	Let A = {1, 2, 3, 4} and R = {(1,1), (1, 2), (2, 1), (2, 2), (3, 4), (3, 3) (4, 3), (4, 4) } be a relation on A.  Verify that R is an equivalence relation.
30.	On the set ;of all integers Z, consider the relation R = {(a, b)ç a º r (mod 2) and b º r (mod 2) } where r is a fixed integer.  Prove that R is an equivalence relation.
31.	Consider the sets A = {a, b, c} and B = {1, 2, 3} and relations R = {(a, 1), (b, 1), (c, 2), (c, 3)} and S = {(a, 1), (a, 2), (b, 1), (b, 2)} from A to B.  Determine
	(i) Rc, (ii) Sc, (iii) RÈS, (iv) RÇS, (v) R-1, (vi) S-1
32.	Let R be a relation from A to B.  Prove that
	(i) dom (R-1) = ran (R) (ii) ran(R-1) = dom (R)
33.	Let R and S be relations from set A to a set B, then the follwoing is true:
	(i) If R Í S, the R-1 Í S-1
	(ii) If R Í S, the Sc Í Rc
	(iii) (R Ç S)-1 = R-1 Ç S-1
	(iv) (R È S)-1 =  R-1 ÈS-1
	(v) (R ÇS )c = Rc ÈSc
	(vi) (R ÈS)c = Rc ÇSc
34.	Let R be a relation on a set A.  Then the following are true:
	(i) If R is reflexive, so is R-1
	(ii) R is reflexive if and only if Rc is reflexive
	(iii) R is sym;metric if and only if R = R-1
	(iv) If R is symmetric, so are R-1 and Rc
	(v) R is antisymmetric if and only if RÇR-1 = j
35.	Let R and S be relations from set A to a set B, then the follwoing is true:
	(i) If R and S are reflexive, so are R È S and R Ç S
	(ii) If R and S are symmetric, so are R È S and R Ç S
	(iii) If R and S are transitive, so is R Ç S
37.	Let R and S be relations from set A to a set B, then the follwoing is true:
	(i) If R is an equivalence relation, so is R-1
	(ii) If R and S are equivalence relations, so is R Ç S
38.	Let A, B, and C are three sets, R be a relation from A to B and S be a relation from B to C.  Prove that (SoR)-1 = R-1 o S-1
39.	Show that the composition of relations is associative.
40.	Let R = {(1, 2), (2, 2), (3, 4)} and S = {(1, 3), (2, 5), (3, 1), (4, 2)} be relations on the set A = {1, 2, 3, 4, 5}. Find (i) SoR, (ii) RoS, (iii) RoSoR, (iv) SoRoS
41.	Let X = { xçx is real and x ³ -1} and Y = { xçx is real and x ³ 0}. Consider the function f: A®B defined by f(x) = Ö(a+1), for all xÎX.  Show that f is invertible and determine f-1
42.	Let X = Y= Z = R, be the set of all real numbers and f:X®Y and g: Y®Z be defined by
	f(x) = 2x + 1 and g(y) = 1/3y, "x Î X and " y Î Y.  Compute gf and show that (gof)  is invertible.  What is (gof)-1?
43.	Let X = Y = R, be the set of all real numbers and the functions f:X®Y and g:Y®X defined by
	f(x) = 2x3 – 1 " xÎ X; g(y) = {½(y  + 1)}1/3 " y Î Y.  Show that each of f and g is a one-to-one correspondence and that each is the inverse of the other.
44.	Define an Abelian group.  Is the set of integers Z an Abelian group?
45.	Prove that the identity element and the inverse element in a group áG, *ñ are both unique.
46.	Let G be the set of real nu;mbers and not equal to -1 and * defined by a * b = a + b +ab.  Prove that áG, *ñ is an Abelian group.
47.	Let G be a group with identity element e.  Show that if x2 = x for some x in G, then x = e
48.	In a group g having more than one element, if x2 = x for every xG, prove that G is an Abelian group
49.	Prove that a group G in which every element is its own inverse in Abelian group.
50.	Prove that a group G is Ableian if and only if (ab)2 = a2b2 for all a, b Î G
51.	Consider the additive group áZ, +ñ of all integers and the group {-1. 1}under multiplication. Define f: Z® H by -1, if n is odd                           f(n)= Prove that f is homomorphism.
52.	Prove that the set S={0, 1, 2, 3, 4, 5} forms a ring under addition and multiplication modulo 6. Is this forms a field?
53.	Define the following terms (i) Random experiments, (ii) Sample space, (iii) Mutually exclusive outcomes, (iv) Equally likely outcomes An experiment in which results are unpredictable The set of all possible outcomes of a random experiment is called sample space denoted by W. Occurrence of one event precludes or prevents the occurrence of other event in a random experiment Every outcome of a random experiment which occur will have an equal chance
54.	Define probability in the classical approach.  What are its limitations? In a random experiment, out of ‘n’ exhaustive, mutually exclusive, equally likely, independent events if ‘m’ of them are favorable to the occurrence of an event, say, ‘A’, then the probability of an event ‘A’, denoted by P(A) is m/n 1.This definition emphasizes that the events must be equally likely. Thus, fails when various  outcomes of a trail are not equally likely.  2. This definition is useful only when we deal with card games, dice games, coin tossing and the like.  It fails in situations when we try to apply it to less orderly decision problem we encounter in management. 3. It does not consider those situations that are unlikely but that could conceivably happen.  Like the occurrence of a coin landing on its edge or our room burning watching TV etc., which are extremely unlikely but not impossible. 4. In case the exhaustive number of cases in a trial is infinite, the definition fails to give the required probability. 5. In some situation there may be a difference of opinion with respect to the ways of forming the possible and favourable outcomes.
54.	Give the axiomatic definition of probability. Axiom 1: P(A) is a real number, i.e., P(A) ³ 0 for every A e S or 0£ P(A) £1. Axiom 2:  P(W) = 1. We call P(A) the probability of an event.  The axiom 3 is called countable additivity. Thus, if W is a sample space consisting of n events and if each event is equally likely to happen, then its probability is 1/n.  Now if A is an event of W containing m elements of  W, i.e., n(A) = m, then P(A) = 1/n + 1/n + 1/n +…+ up to m              terms         = m/n
55.	State and prove addition rule of probability.
56.	Prove that (a) (b)
57.	Define the conditional probability of two events.
58.	A box contain 6 red, 4 white and 5 black balls.  If 4 balls are drawn at random what is the probability that among the balls drawn there is at least one ball of each colour?
59.	Two dice, say one green and the other red, are thrown.  Let A be the event that the sum of the points on the faces shown is odd and B be the event that at most one ace (number 1). Find the     probabilities of the events: (i) P(A), (ii) P(B), (iii) P(AÈB), (iv) P(AÇB), (v) P(AÇBc), (vi) P(AcÇBc),  and(vii) P (AÈB)c
60.	In a group of 160 Engineering students, 92 are enrolled for advanced course on CSD, 63 students on VLSI and 40 are enrolled in both.  How many of these students are not enrolled in either course?
61.	For any two events A and B prove that P(AcÇ B) = P(B) - P(A Ç B)
62.	For any two events A and B prove that P(AÇBc)ÈP(AcÇB)=P(A)+P(B)-2P(AÇB)
63.	In a group of 160 Engineering students, 92 are enrolled for advanced course on CSD,  63 students on VLSI and 40 are enrolled in both.  How many of these students are not enrolled in either course?
64.	In a city it was observed that 80% of    the families owns a two wheeler and 43% owns a car. Those who owns both are 38%.  If a family is selected at random what is the probability that they own either a two wheeler or a car.
65.	Prove that the probability of an impossible event is zero, i.e., P(j) = 0. The probability of an impossible event is zero, i.e., P(j) = 0 It follows that                W È j  = W            P(W È j) = P(W)            P(W) + P(j) = P(W) from axiom 3            P(j) = 1-1 = 0 from axiom 1           \P(j) = 0

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