# Basics of Probability

## What is probability?

As said by Walter Bagehot,

Life is a school of probability.

If I put it in simple words, *probability tells how likely an event will occur*.

Let's take the simplest example possible that is tossing a coin(because it has only two outcomes) and this is familiar to most people as the most favourite game of INDIA(Cricket) requires the same experiment for deciding which team will bat first. So we know that every coin has two sides HEAD and TAIL, if we toss cion then it will output either head or tail on a flat horizontal surface irrespective of it being biased or unbiased.

Now, suppose that the coin is unbiased which means the occurrence of heads and tails are *equally likely*, then we can say that the outcome may be head or tail if you toss it that is there is a 50% probability of getting either head or tail. In mathematical terms,

In the same way, we can take the dice toss experiment where suppose we need to calculate the probability of getting an even number as an outcome of tossing a dice once, that can be solved as,

some more useful examples of probability,

**Types and branches of probability:**

There are mainly three types and two branches of probability, two branches are classical probability and Bayesian probability. the latter is mainly used in machine learning.

*Theoretical probability*:

It is based on the possible chances of something to happen. The theoretical probability is mainly based on the reasoning behind probability. For example, if a coin is tossed, the theoretical probability of getting a head will be ½.

*Experimental Probability:*

It is based on the basis of the observations of an experiment. The experimental probability can be calculated based on the number of possible outcomes by the total number of trials. For example, if a coin is tossed 10 times and the heads are recorded 6 times then, the experimental probability for heads is 6/10 or, 3/5.

*Axiomatic Probability:*

In axiomatic probability, a set of rules or axioms are set which applies to all types. These axioms are set by Kolmogorov and are known as **Kolmogorov’s three axioms. **With the axiomatic approach to probability, the chances of occurrence or non-occurrence of the events can be quantified. The axiomatic probability lesson covers this concept in detail with Kolmogorov’s three rules (axioms) along with various examples.

Conditional Probability is the likelihood of an event or outcome occurring based on the occurrence of a previous event or outcome.

**Axioms of probability:**

Mathematicians avoid these tricky questions by defining the probability of an event mathematically without going into its deeper meaning. At the heart of this definition are three conditions, called the *axioms* of probability theory.

**Axiom 1:**The probability of an event is a real number greater than or equal to 0.**Axiom 2:**The probability that at least one of all the possible outcomes of a process (such as rolling a die) will occur is 1.**Axiom 3:**If two events*A*and*B*are mutually exclusive, then the probability of either*A*or*B*occurring is the probability of*A*occurring plus the probability of*B*occurring.

**Some basic properties of probability:**

1.The probability of an event E is a number P(E) such that 0 ≤ P (E) ≤ 1. Probability is always a positive number.

2. If A and B are 2 events that are mutually exclusive, then P(A⋃B) = P(A) + P(B).

3. An elementary event is an event having only one outcome. The sum of the probabilities of such events of an experiment is 1.

4. The sum of probabilities of an event and its complementary event is 1. P(A) + P(A’) = 1.

5. P(A⋃B) = P(A) + P(B) — P(A⋂B).

6. P(A⋂B) = P(A) + P(B) — P(A⋃B) .

7. If A1, A2, A3,………, An are mutually exclusive events, then P(A1 ⋃ A2 ⋃ A3… ⋃ An) = P(A1) + P(A2 ) + ………. + P(An)

**Further readings:**

Though the basics are done above, it is recommended to know the difference between some words that usually pop up in numerical questions, one such very prominent example is the difference between likelihood and probability.