Probability1. Introduction

2. Definitions of various Terms

3. Mathematical or Classical Probability,

4. Statistical or Empirical Probability,

5. Mathematical Tools,

6. Sets and Elements of sets,

7. Operations on sets,

8. axiomatic Approach to Probability,

9. Probability function,

10. Mathematical Law of Addition of Probabilities

11. Conditional Probabilities,

12. Independent Events,

13. Bayes Theorem,

14. Geometric Probability. 1

Introduction 1. The theory of probability was developed towards the end of the 18th

century and its history suggests that it developed with the study of games

and chance, such as rolling a dice, drawing a card, flipping a coin etc.

2. ‘probability’ thus suggest that there is an uncertainty about the happening

of events

3. Probability is the chance that something will happen - how likely it is

that some event will happen

4. Sometimes you can measure a probability with a number: "10% chance of

rain", or you can use words such as impossible, unlikely, possible, even

chance, likely and certain.

2

Probability- Basic Term

Trial:-A procedure or an experiment to

collect any statistical data such as rolling

a dice or flipping a coin is called a trial.

3

Probability- Basic Term

Random Trial or Random Experiment:- When the

outcome of any experiment can not be predicted

precisely then the experiment is called a random trial

or random experiment

4

Probability- Basic Term

Sample Space:- The totality of all the outcomes or

results of a random experiment is denoted by Greek

alphabet W or English alphabets and is called the

sample space

5

Probability- Basic Term

Event:- Any subset of a sample space is

called an event

6

Probability- Basic Term

Equally Likely Events:- All possible results of a

random experiment are called equally likely

outcomes and we have no reason to expect any one

rather than the other.

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Probability- Basic Term

Mutually Exclusive Events :- Events are called

mutually exclusive or disjoint or incompatible if the

occurrence of one of them precludes the occurrence

of all the others.

8

Probability- Basic Term

Exhaustive Events:- Events are exhaustive

when they include all the possibilities

associated with the same trial

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Probability Independent Events:-Two events are said to be independent

if the occurrence of any event does not affect the occurrence

of the other event.

Dependent Events:- If the occurrence or non-occurrence of

any event affects the happening of the other, then the events

are said to be dependent events

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Definitions of Probability

We shall now consider two definitions of

probability :

(1) Mathematical or a priori or classical.

(2) Statistical or empirical.

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Mathematical or a priori or classical Definitions of Probability

1. If there are ‘n’ exhaustive, mutually exclusive and equally

likely cases and m of them are favorable to an event A,

The probability of A happening is defined as the ratio m/n

Expressed as a formula :-

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2. This definition is due to ‘Laplace.’ Thus probability is a

concept which measures numerically the degree of certainty or

uncertainty of the occurrence of an event.

Example Probability

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Example Probability

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Van Mise’s Statistical (or Empirical) Definitions of Probability

1. The classical definition of probability has a disadvantage i.e. the words

‘equally likely’ are vague.

2. In fact, since these words seem to be synonymous with "equally probable".

This definition is circular as it is defining (in terms) of itself.

3. Therefore, the estimated or empirical probability of an event is taken as the

relative frequency of the occurrence of the event when the number of

observations is very large.

4. If trials are to be repeated a great number of times under essentially

the same condition then the limit of the ratio of the number of times

that an event happens to the total number of trials, as the number of

trials increases indefinitely is called the probability of the happening

of the event.

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Van Mise’s Statistical (or Empirical) Definitions of Probability

1. If trials are to be repeated a great number of times under

essentially the same condition then the limit of the ratio of the

number of times that an event happens to the total number of

trials, as the number of trials increases indefinitely is called

the probability of the happening of the event.

2. The two definitions are apparently different but both of them

can be reconciled the same sense.

3. Symbolically p (A) = p =

4. provided it is finite and unique.

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Example Probability

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Example1 Find the probability of getting heads in tossing a coin.

Solution : Experiment : Tossing a coin

Sample space : S = { H, T} n (S) = 2

Event A : getting heads

A = { H}

n (A) = 1

Therefore, p (A) =

Example Probability

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Example2 Find the probability of getting 3 or 5 in throwing a die

Solution :

Experiment : Throwing a dice

Sample space : S = {1, 2, 3, 4, 5, 6 } Þ n (S) = 2

Event A : getting 3 or 6

A = {3, 6} n (A) = 2

Therefore, p (A) =

Example Probability

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Example3 Two dice are rolled. Find the probability that the score on the

second die is greater than the score on the first die.

Solution :

Experiment : Two dice are rolled

Sample space : S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6) (2, 1), (2, 2),

(2, 3), (2, 4), (2, 6)}...

(6, 1), (6, 2) (, 3), (6, 4), (6, 5), (6, 6) }

n (S) = 6 ´ 6 = 36

Event A : The score on the second die > the score on the 1st die.

i.e. A = { (1, 2), (1, 3), (1, 4), (1, 5), (1, 6) (2, 3), (2, 4), (2, 5), (2, 6) (3, 4),

(3, 5), (3, 6) (4, 5), (4, 6) (5, 6)}

n (A) = 15

Therefore, p (A) =

Example Probability

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Example4 A ball is drawn at random from a box containing 6 red balls, 4 whiteballs and 5 blue balls. Determine the probability that the ball drawn is

red

Solution :

Let R denote the events of drawing a red ball.