## What is Probability?

The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.

## Sample Space

A set of all the possible outcome of the random experiment is called Sample Space, and which is denoted by “*S*”.

For example: when tossing a coin there are two outcomes are available, there are head and tail.

So, the sample space is Head and tail.

That is $S=\left\{H,\text{}T\right\}$

Where S= sample space

$\begin{array}{c}S=\text{Samplespace}\\ H=\text{Head}\\ T=\text{Tail}\end{array}$

### Sample Points

Sample points is the elements of samples space.

## Events

The subset of the sample space is called event.

### Probability of an Event

The probability of an event B is the sum the probabilities of all individual outcomes of which it is composed. And which is denoted by $P\left(B\right)$.

### For example

If we have an event B is $B=\left\{{b}_{1},{b}_{2,}\mathrm{...}{b}_{n}\right\}$

Then the probability of an event is

$$P\left(B\right)=P\left({b}_{1}\right)+P\left({b}_{2}\right)+\dots +P\left({b}_{n}\right)$$## Random Experiment

An experiment is called a random experiment if when the experiment has repeated under the same conditions, it is such that the outcome cannot be predicted with certainty. But all the possible outcomes can be determined prior to the performance of the experiment.

### Trial

Each performance of the random experiment is said to be a trial.

### Random Variable

A random variable is a real valued function which takes real numbers associated with the outcomes of the random experiment.

## Probability Distribution

Probability distribution gives the possible outcome of the random events. Probability distribution of the random variable defines that the probability of its unknown values. The random variables may be discrete or continuous or both. It is clear that, the random variables is finite or countable number of values provided with a probability mass function features of probability distribution of the random variables, In other words, It can takes any numerical value in the intervals or set of intervals. In the same way, the random variable is infinite or uncountable number of values in the interval provided with a probability density function features of probability distribution of the random variables and it is the combination of discrete or continuous.

### Types of Probability Distribution

There are two types of probability distribution. These two types of probability distribution are used for different purposes. There are

- Normal or Cumulative probability distribution
- Binomial or discrete probability distribution

Normal Probability Distribution:

Normal probability distribution is also known as a continuous probability distribution. In this normal distribution, a set of all possible outcomes can take on a take on a continuous range. Also, it gives all the possible outcomes of real numbers.

## Formulas

The probability density function (PDF) of Normal distribution is,

$$P(x)=\frac{1}{\sigma \sqrt{2\pi}}{e}^{-\frac{1}{2}{\left(\frac{x-\mu}{\sigma}\right)}^{2}}$$where,

Population mean=$\mu $

Standard deviation=$\sigma $

Random variable= x

If the mean value is zero and the standard deviation is one then it is called standard normal distribution. So, the probability density function of the standard normal distribution is,

$$P(x)=\frac{1}{\sqrt{2\pi}}{e}^{-\frac{1}{2}{\left(x\right)}^{2}}$$For example: If we are rolling a die, the outcomes are {1,2,3,4,5,6}, In the same way the dice are rolling multiple times.

Discrete probability distribution:

Here we have to see some discrete probability distribution and the probability mass function.

The discrete probability distributions are:

- Bernoulli distribution
- Binomial distribution
- Poisson distribution

Bernoulli Distribution:

It is known that the Bernoulli distribution is the discrete distribution. It is also having two possible outcomes that is success or failure in one single trial. The success is denoted by “p” (x=1) and the failure is denoted by “q”(x=0).

## Formulas

The probability mass function of Bernoulli distribution is,

$$p\left(x\right)={p}^{x}{(1-p)}^{1-x}\text{}(since\text{}q=1-p)$$For example:

If tossing a coin there are two possible outcomes are available. There are head and tail.

So, the value of p and q is $\frac{1}{2}\text{or}0.5$.

### Binomial Distribution

Binomial distribution is known as discrete distribution. This is the probability of success or failure of a random experiment. That is repeated in multiple number of times (n independent trials) and also there are two possible outcomes are available. The success is denoted by “p” and the failure is denoted by “q”.

## Formulas

The probability mass function of Binomial distribution is,

Where,

$\begin{array}{c}\text{Mean}=\text{}np\\ \text{Variance}=npq\end{array}$

For example: we have tossing a coin, there are two outcomes are available one is head and other is tail. In the same way the coin will toss an independent number of trials

Poisson distribution:

It is known that, the Poisson distribution is the discrete distribution. In Poisson distribution measure, how many times an event occurs within a specified period of time. The important thing of Poisson distribution is, it is based on time. The mean and variance of the poison distribution is same the probability mass function of the Poisson distribution is,

$$P\left(x\right)=\frac{{e}^{-\lambda}{\lambda}^{x}}{x!};x=0,1,\mathrm{2...}$$Where, x is the number of occurrences of an event.

The Poisson distribution parameter is $\lambda $ .

That is mean and variance are equal to $\lambda $.

For Example: The number of students arriving in the school between 8.00 to 9.00 AM.

## Conditional Probability

Conditional probability calculates the experiment, that the outcome of trial affects the outcome of subsequent trials. We take there are two events, event A and event B. Now we calculate the second event given the first event, but the first event is already happened. Then the conditional probability is

$$P\left(B/A\right)=\frac{P\left(B\cap A\right)}{P\left(A\right)}$$If the two events are independent then it is must satisfied the condition of $P\left(A/B\right)=P\left(A\right)$ or $P\left(B/A\right)=P\left(B\right)$.

The condition is, If the event A and B are independent then $P\left(A\cap B\right)=P\left(A\right)P\left(B\right)$.

Example diagram:

In condition probability mostly used Bayes condition,

$P\left(B/A\right)=\frac{P\left(B\cap A\right)P\left(B\right)}{P\left(A\right)}$Here $P\left(A\right)$is not equal to zero.

These are all the basic concept of probability and the distribution functions.

## Context and Application

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for

- Bachelor of Science Mathematics
- Master of Science Mathematics

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