Engineering Mathematics : Probability & Statistics

Probability

Probability is a fundamental concept in mathematics and statistics that quantifies the likelihood of events occurring. It serves as a powerful tool for analyzing uncertainty, making predictions, and informing decision-making processes.

Random Experiments

For any invention, a 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.

Sample Space

Each random experiment 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.

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 of only a single outcome then it is known as Simple Events. If the events consist of more than one outcome then it is known as Compound Events.

Types of Events

(i) Complementary Event – Any Event E^C is called a 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 a set of all possible outcomes, is given by {1, 2, 3, 4, 5, 6}.

An event of getting the outcome in multiple of 3 is

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

Then, E^C = {1,2,4,5}

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

Ex – The chance of a newborn baby being a boy or girl is 50% means either it can be a girl or boy.

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

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

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

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

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

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) is the probability of the event E, then P(E) will satisfy the following axioms.

Axiom 1 For every event E,

P(E) ≥ 0

The probability of an event can never be negative.

Axiom 2 In case of sure or certain event E,

P(E) = 1

The 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)

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 a probability of zero.

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

P(E^C) = 1 – P(E)

Probability Distribution

Random Variables

Suppose that to each point of a sample space, we assign a number. We then have a function defined on the sample space. This function is called a random variable or more precisely a random function.

It is usually denoted by a capital letter such as X or Y. Random variable X is associated with the outcome of an experiment which is not certain, and its value depends upon the chance.

If a random variable takes a finite set of values then it is called a Discrete random variable, whereas when a random variable takes an infinite set of values (or any value from a continuous range or graph) then it is called a Continuous Random variable.

Binomial Distribution

Suppose that we have an experiment such as tossing a coin or rolling a die repeatedly or choosing a marble from an urn repeatedly. Each toss or selection is called a trial. In any single trial, there will be a probability associated with a particular event such as head on the coin, 4 on the die, or selection of a particular colour of marble.

Poission’s Distribution

Geometric distribution

Consider a repeated trial of the Bernoulli experiment with a probability of success p, and failure q=(1-p). if the experiment is repeated until success is not achieved, then the distribution of the variable is given by geometric distribution.

If the experiment is performed “k” times, then the experiment must be failed in ‘K-1’ times.

Exponential Distribution

Continuous Uniform Distribution

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