Joint and Conditional Probability Study Notes for Mechanical and Civil Engineering

By Akhil Gupta|Updated : November 16th, 2020

1. PROBABILITY

1.1.   DEFITITION

  1. Random Experiments-

For any invention, number of experiments are done. Consider an experiment whose results is not predictable under almost similar working condition then these experiments are known as Random Experiments.

     b. Sample Space –

Each random experiments of some possible outcomes, if we make a set of all the possible outcomes of random experiments then Set ‘S’ is known as the Sample Space & each possible outcome is Sample Point.

      c. Event

An event is a subset A of the sample space S, i.e., it is a set of possible outcomes.

An Event is a set of consisting some of the possible outcomes from the sample space of the experiment.

If the event consists only single outcome then it is known as Simple Events.

If the events consist of more than one outcome then its is known as Compound Events.

1.2 Types of Events-

(i)Complementary Event – Any Event EC is called complementary event of event E if it consists of all possible outcomes of sample space which is not present in E.

Ex - If we roll a die, then set of all possible outcomes, is given by {1, 2, 3, 4, 5, 6}.

An event of getting outcome in multiple of 3 is

E (multiples of 3) = {3,6}

Then, EC = {1,2,4,5}

(ii) Equally Likely Event – if any two event of sample space are in such a way that the chance of both the events are equal, then this type of events is known as Equally likely events.

Ex – Chances of a new born baby to be a boy or girl is 50% means either it can be a girl or boy.

(iii) Mutually Exclusive Events – Two events are called as mutually exclusive when occurring of both the simultaneously is not possible.

If E1 & E2 are mutually exclusive then E1 ⋂ E2 = ϕ

(iv) Collectively Exhaustive Events - Two events are called as Collectively exclusive when sample points of both the events incudes all the possible outcomes.

If E1 & E2 are mutually exclusive then E1 ⋃ E2 = S

(v) Independent Events – Two events are called as independent when occurring of 1st event does not affect the occurrence of 2nd.

 

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1.3.   The Axioms of Probability

Consider an Experiment whose sample space is S. For each event E of the sample space, we associate a real number P(E). Then P is called a probability function, and P(E) the probability of the event E, then P(E) will satisfies the following axioms.

Axiom 1 For every event E,

P(E) ≥ 0

Probability of an event can never be negative.

Axiom 2 In case of sure or certain event E,

P(E) = 1

Probability of an event with 100% surety is 1.

Axiom 3 For any number of mutually exclusive events E1, E2, ….,

P (E 1∪E 2∪E3…) = P (E 1) + P(E2) + p(E3) …..

In particular, for two mutually exclusive events E1, E2,

P (E 1∪E 2) = P (E 1) + P (E 2)

1.4.   Some Important Theorems on Probability

From the above axioms we can now prove various theorems on probability

Theorem 1: For every event E,

0 ≤ P(E) ≤ 1,

i.e., a probability is between 0 and 1.

Theorem 2:              P(Φ) = 0

i.e., the impossible event has probability zero.

Theorem 3:  If EC is the complement of E i.e. that event E will not happen, then

P(EC) = 1 – P(E)

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