## Probability for machine learning and data science – Basic Probability 2 of 6

### Probability Axioms

In this post we will look into probability axioms there are 3 basic axioms of probability and it is mentioned below

Axiom 1 : For any set A

P(A) \geq 0

Axiom 2 :

P (\Omega) = 1

Axiom 3 : if A_1, A_2, ... is any set of disjoint events, then

P\left(\bigcup\limits_{i=1}^{\infty}A_i\right) = \sum\limits_{i=1}^{\infty}P(A_i)

Definition

if A = \bigcup\limits_{i=1}^{\infty} A_i,. and A_1, A_2, .... are disjoint, then A_1, A_2, .... is said to be a partition of A

Axiom 3 also holds for finite collection of events A_1,..., A_n which is trivially true if you set A_{n+1} = \emptyset for all i \in \N

By using the above axioms we can get more axioms. Below are the results from axioms

Compliments

P(A^c) = 1-P(A)

Differences

if A is contained in B ( A \subset B), then

P(B \cap A^c) = P(B) - P(A)

Inclusion – Exclusion

P(A \cup B) = P(A) + P(B) - P(A \cap B)

### Equally likely outcomes

In some cases, we can safely assume the outcomes are equally likely like if we roll a fair dice or toss a fair coin twice or more.

Suppose

• n_A is the number of sample points in event A
• N is the number of sample points in a finite sample space \Omega

if all outcomes are equally likely in a sample space \Omega, then the probability that event A occurs is

P(A)=\frac{n_A}{N},
• n_A is the number of sample points in A
• N is the number of sample points in \Omega

So in this post we have seen the Axioms of probability and in the next post we will start with Counting

Happy Learning

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