Overview of Probability

The conditional event of A given B means the occurrence of A under the condition that B has already occurred.

It is denoted by A|B.

Let A and B be two events associated with a random experiment. Then, the probability ofthe occurrence of A under the condition that B has already occurred and P(B) ≠ 0, is called the conditional probability of A given B and is written as P(A/B).

\[P(A\cap B)=P(A).P(B/A)=P(B).P(A/B)\]

Extension of Multiplication Theorem:

\[P(A\cap B\cap C)=P(A)P(B/A).P(C/A\cap B)\]

Independent events:

 A set of events is said to be independent if the occurrence of any one of them does not, in any way, affect the occurrence of any other in the set.

Dependent events:

 Two events E and F are said to be dependent if they are not independent, i.e. if \[\mathrm{P}(\mathrm{E}\cap\mathrm{F})\neq\mathrm{P}(\mathrm{E}).\mathrm{P}(\mathrm{F})\]

Statement: 
Let S be the sample space and E₁, E₂,…, E_n be mutually exclusive and exhaustive events associated with a random experiment. Let A be any event associated with S. Then,

\[P(A)=P(E_1)P(A\mid E_1)+P(E_2)P(A\mid E_2)+\cdots+P(E_n)P(A\mid E_n)\]

or

\[P(A)=\sum P(E_i)P(A\mid E_i)\]

Statement:
Let E₁, E₂,…, E_n be mutually exclusive and exhaustive events, and A be an event such that P(A) ≠ 0. Then,

\[P(E_i\mid A)=\frac{P(E_i\cap A)}{P(E_1\cap A)+P(E_2\cap A)+\cdots+P(E_n\cap A)}\]

Second form:

\[P(E_i\mid A)=\frac{P(E_i)P(A\mid E_i)}{\sum P(E_j)P(A\mid E_j)}\]

If n = 2:

\[P(E_1\mid A)=\frac{P(E_1)P(A\mid E_1)}{P(E_1)P(A\mid E_1)+P(E_2)P(A\mid E_2)}\]

\[P(E_2\mid A)=\frac{P(E_2)P(A\mid E_2)}{P(E_1)P(A\mid E_1)+P(E_2)P(A\mid E_2)}\]

Bayes’ theorem for three events:

\[P(E_1\mid A)=\frac{P(E_1)P(A\mid E_1)}{P(E_1)P(A\mid E_1)+P(E_2)P(A\mid E_2)+P(E_3)P(A\mid E_3)}\]

Random variable:

A random variable is a variable whose values depend on chance and are the result of a random observation or experiment.

Discrete random variable:

If the set of values taken by a random variable can be counted and listed, it is called a discrete random variable.

Continuous Random Variable:

If the set of values is continuous, the variable is called a continuous random variable.

If a random variable x can take values x₁, x₂,…, x_n with probabilities p(x₁) ,p(x₂),…, p(x_n) such that p(x₁) + p(x₂) +… +  p(x_n) = 1, the function p is called the probability density function of x and is said to define the probability distribution of x.

Mean µ (Greek mu) of the above probability distribution may be defined as 

\[\mu=\frac{p_1x_1+p_2x_2+p_3x_3+.......+p_nx_n}{p_1+p_2+p_3+......+p_n}\]

\[=\frac{\sum p_ix_i}{\sum p_i}=\Sigma p_ix_i\]

\[Mean\overline{x}=\sum_{i=1}^{n}p_{i}x_{i}\],where each pi \[P_{i}\geq0\] and \[\sum p_{i}=p_{1}+p_{2}+...+p_{n}=1\]

The variance of a random variable x is denoted by σ².

First form: \[\sigma^2=\sum_{i=1}^np_i(x_i-\mu)^2\]

Second form: \[\sigma^2=\sum_{i=1}^np_ix_i^2-\mu^2\]

\[\sigma=\sqrt{\sigma^2}=\sqrt{\sum p_ix_i^2-\mu^2}\]

\[\begin{gathered}
\mu=\int_{-\infty}^{\infty}xf(x)dx \\
\sigma^2=\int_{-\infty}^\infty(x-\mu)^2f(x)dx
\end{gathered}\]

Trials of a random experiment are called Bernoulli’s trials if they satisfy the following conditions:

1. The number of trials is finite.

2. Each trial is independent of the others.

3. Each trial has exactly two outcomes: success or failure.

4. The probability of success (or failure) remains the same in each trial.

General Form: \[P(X=r)={}^nC_rp^rq^{n-r},\quad r=0,1,2,\ldots,n\]

• Probabilities are terms of (q + p)ⁿ.

• P(0) + P(1) + ⋯ + P(n) = 1.

• The binomial distribution is discrete.

• n and p are its parameters.

Special cases:

• P(0) = qⁿ

• P(1) = npqⁿ⁻¹

Statement: 

Let p be the probability of success of an event and q be the probability of failure of the event in one trial. Suppose there are n trials of the event in a binomial  experiment, then the binomial probability distribution is defined by the following table: 

Mean: μ = np

Variance: σ² = npq

Standard deviation: \[\sigma=\sqrt{npq}\]