Probability: Basic concept of probability, Definition of probability, Addition theorem, Multiplication theorem, Dependent and Independent events, Conditional probability, Combined use of Addition and Multiplication theorem, Bernoulli theorem, Baye's Theorem. Probability distributions: Binomial distribution, Poisson Distribution, Hypergeometric distribution, Normal distribution: Area under the normal curve, Relation between Binomial,Poisson and Normal distributions. Exponential distribution, Mean time between failure, waiting line analysis.

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Quantitative Techniques
Volume-5
(Revised)
1. Probability : Basic concept and Baye’s theorem
2. Probability distributions: Binomial, Poisson, Hypergeometric, Normal and Exponential distributions
E-Book Code : QTVOL5
by
Narender Sharma
“Save Paper, Save Trees, Save Environment”

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E-mail : shakehandwithlife@gmail.com , narender@shakehandwithlife.in
Click on Contents
Probability ................................................................................................................................................................................................................. 3
Some Basic Concept ..................................................................................................................................................................................... 3
Definition of Probability: ........................................................................................................................................................................... 4
Importance of Probability: ........................................................................................................................................................................ 6
Probability Scale ............................................................................................................................................................................................ 6
Combination in theory of Probability ................................................................................................................................................... 6
Addition Theorem ............................................................................................................................................................................................. 7
Multiplication Theorem .................................................................................................................................................................................. 9
Conditional Probability ................................................................................................................................................................................. 10
Multiplication Theorem for Dependent Events/Conditional Probability ............................................................................ 10
Combined Use of Addition and Multiplication Theorem ............................................................................................................ 11
Bernoulli’s Theorem ....................................................................................................................................................................................... 11
Bayes’ Theorem ................................................................................................................................................................................................ 12
Probability Distribution ..................................................................................................................................................................................... 14
Uses of Probability/Theoretical frequency Distributions .......................................................................................................... 14
Types of theoretical or Probability distributions .......................................................................................................................... 15
Binomial Distribution ................................................................................................................................................................................... 16
Definition and Formula............................................................................................................................................................................. 16
Assumptions to Apply Binomial Theorem ........................................................................................................................................ 16
Characteristics .............................................................................................................................................................................................. 16

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Applications ................................................................................................................................................................................................... 17
Poisson Distribution ....................................................................................................................................................................................... 20
Assumption .................................................................................................................................................................................................... 20
Limiting Form of Binomial Distribution ............................................................................................................................................ 20
Uses of Poisson Distribution .................................................................................................................................................................. 20
Application of Poisson Distribution .................................................................................................................................................... 20
Hypergeometric Distribution ...................................................................................................................................................................... 23
Assumption .................................................................................................................................................................................................... 23
Formula ........................................................................................................................................................................................................... 23
Characteristics .............................................................................................................................................................................................. 23
Application ..................................................................................................................................................................................................... 23
Normal Distribution ........................................................................................................................................................................................ 25
Introduction .................................................................................................................................................................................................. 25
Definition and Formula............................................................................................................................................................................. 25
Shape of Normal Distribution Curve ................................................................................................................................................... 25
Assumptions of Normal Distribution .................................................................................................................................................. 25
Characteristics Normal Distribution Curve ...................................................................................................................................... 26
Importance of Normal Distribution ..................................................................................................................................................... 27
Relation among Binomial, Poisson and Normal Distribution ................................................................................................... 27
Application of Normal distribution .................................................................................................................................................... 27
Exponential Distribution .............................................................................................................................................................................. 32
Introduction .................................................................................................................................................................................................. 32
Properties of Exponential Distribution ............................................................................................................................................. 32
Application of Exponential Distribution .......................................................................................................................................... 33
Table 1 : Combinations of ’ things taking ’ at a time ........................................................................................................................ 35
Table 2 : Values of for computing Poisson Probabilities ............................................................................................................ 36
Table 3 : Area Under The Normal Curve ..................................................................................................................................................... 37
Table 4 : Exponential Functions ..................................................................................................................................................................... 38
References ............................................................................................................................................................................................................... 39
Feedback .................................................................................................................................................................................................................. 39

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Probability
Some Basic Concept
1. Experiment
When we conduct a trial to obtain some statistical information , it is called an experiment. e.g.
i) Tossing of fair coin, It has two outcomes : Head (H) or Tail (T)
ii) Rolling of fair die It has six possible outcomes : appearance of 1/2/3/4/5/6
ii) Drawing of a card from a well shuffled pack of playing cards is an experiment, it has 52 possible outcomes.
2. Events
The possible outcomes of a trail experiments are called events. Events are generally denoted by capital letters A, B, C etc. e.g. if a coin tossed , the outcomes – head or tail are called events. For a rolled die 1/2/3/4/5/6 appearing up are the events.
3. Exhaustive Events
The total number of outcomes of a trial /experiments are called exhaustive events. In other words, if all the possible outcomes of an experiments are taken into consideration, then such events are called exhaustive events. e.g. in case of tossing a die the set of possible outcomes i.e. 1, 2, 3, 4, 5, 6 are exhaustive events.
In case of tossing two dice , the set of possible outcomes are which are given below
(1,1) (1,2) (1,3) (1,4) (1,5 ) (1,6) (2,1)(2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5)(6,6) H T

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4. Equally- Likely Events
The events are said to be equally-Likely if the chance of happening of each event is equal or same. In other words , events are said to be equally likely when one does not occur more often than the others. e.g. Tossing of a coin , Rolling of coin.
5. Mutually Exclusive Events
Two events are said to be mutually exclusive when they cannot happen simultaneously in a single trial. In other words, Two events are said to be mutually exclusive when the happening of one excludes the happening of the other in a single trial. e.g.
I. In tossing a coin, the events of Head and tail are mutually exclusive because both cannot happen simultaneously in a single trial. Either Head occurs or tail occurs. Both cannot occur simultaneously. The happening of head excludes the possibility of happening of tail.
II. In Rolling of die the events 1, 2, 3 4, 5, 6 are mutually exclusive.
6. Complementary Events
For Two events A and B, A is called complementary event of B and B is called the complementary event of A if A and B are mutually exclusive and exhaustive.
I. Occurrence of head and tail are complementary events.
II. In tossing of die occurrence of an even number (2,4,6) and odd number (1,3,5) are complementary events
7. Simple and Compound Events
I. In case of simple events , we consider the probability of happening or not happening of single events e.g. if a die is rolled once and A be the event that face number 5 is tuned up then A is called simple event.
II. In case of compound events, we consider the joint occurrences of two or more events. e.g. if two coins are tossed simultaneously and we shall be finding the probability of getting two heads, then we are dealing with compound events.
8. Independent Events
Two events are said to be independent if the occurrence of one does not affect and is not affected by the occurrence of the other. For Example
I. In tossing a die twice, the event of getting 4 in the 2nd throw is independent if 5 in the first throw.
II. In tossing a coin twice, the event of getting a head in the 2nd throw is independent of getting head in the 1st throw .
9. Dependent Events
Two events are said to be dependent when the occurrence of one does affect the probability of the occurrence of the other events.
Example :
I. If a card is drawn from a pack of 52 playing cards and is not replaced, this will affect the probability of the second card being drawn.
II. The probability of drawing a king from a pack of 52 cards is . But if the card drawn (king) is not replaced in the pack , the probability of drawing again a king is .
Definition of Probability:
1. Classical or Mathematical Definition
2. Empirical or Relative Frequency Definition
3. Subjective Approach
Classical or Mathematical Definition
“Probability is the ratio of the favourable cases to the total number of equally likely cases.” ------- Laplace
Symbolically, ( ) ( )
Similarly