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Machine Learning Basics for a Newbie
6 Steps of Machine learning Lifecycle Introduction to Predictive Modeling
Introduction to Exploratory Data Analysis & Data Insights Descriptive Statistics Inferential Statistics How to Understand Population Distributions?
Reading Data Files into Python Different Variable Datatypes
Probability for Data Science Basic Concepts of Probability Axioms of Probability Conditional Probability
Central Tendencies for Continuous Variables Spread of Data KDE plots for Continuous Variable Overview of Distribution for Continuous variables Normal Distribution Skewed Distribution Skeweness and Kurtosis Distribution for Continuous Variable
Central Tendencies for Categorical Variables Understanding Discrete Distributions Performing EDA on Categorical Variables
Dealing with Missing Values Understanding Outliers Identifying Outliers in Data Outlier Detection in Python Outliers Detection Using IQR, Z-score, LOF and DBSCAN
Sample and Population Central Limit Theorem Confidence Interval and Margin of Error
Bivariate Analysis Introduction
Covariance Pearson Correlation Spearman's Correlation & Kendall's Tau Correlation versus Causation Tabular and Graphical methods for Bivariate Analysis Performing Bivariate Analysis on Continuous-Continuous Variables
Tabular and Graphical methods for Continuous-Categorical Variables Introduction to Hypothesis Testing P-value Two sample Z-test T-test T-test vs Z-test Performing Bivariate Analysis on Continuous-Catagorical variables
Chi-Squares Test Bivariate Analysis on Categorical Categorical Variables
Multivariate Analysis A Comprehensive Guide to Data Exploration The Data Science behind IPL
Supervised Learning vs Unsupervised Learning Reinforcement Learning Generative and Descriminative Models Parametric and Non Parametric model
Machine Learning Pipeline Preparing Dataset Build a Benchmark Model: Regression Build a Benchmark Model: Classification
Evaluation Metrics for Machine Learning Everyone should know Confusion Matrix Accuracy Precision and Recall AUC-ROC Log Loss R2 and Adjusted R2
Dealing with Missing Values Replacing Missing Values Imputing Missing Values in Data Working with Categorical Variables Working with Outliers Preprocessing Data for Model Building
Understanding Cost Function Understanding Gradient Descent Math Behind Gradient Descent Assumptions of Linear Regression Implement Linear Regression from Scratch Train Linear Regression in Python Implementing Linear Regression in R Diagnosing Residual Plots in Linear Regression Models Generalized Linear Models Introduction to Logistic Regression Odds Ratio Implementing Logistic Regression from Scratch Introduction to Scikit-learn in Python Train Logistic Regression in python Multiclass using Logistic Regression How to use Multinomial and Ordinal Logistic Regression in R ? Challenges with Linear Regression Introduction to Regularisation Implementing Regularisation Ridge Regression Lasso Regression
Introduction to K Nearest Neighbours Determining the Right Value of K in KNN Implement KNN from Scratch Implement KNN in Python
Bias Variance Tradeoff Introduction to Overfitting and Underfitting Visualizing Overfitting and Underfitting Selecting the Right Model What is Validation? Hold-Out Validation Understanding K Fold Cross Validation
Introduction to Feature Selection Feature Selection Algorithms Missing Value Ratio Low Variance Filter High Correlation Filter Backward Feature Elimination Forward Feature Selection Implement Feature Selection in Python Implement Feature Selection in R
Introduction to Decision Tree Purity in Decision Tree Terminologies Related to Decision Tree How to Select Best Split Point in Decision Tree? Chi-Squares Information Gain Reduction in Variance Optimizing Performance of Decision Tree Train Decision Tree using Scikit Learn Pruning of Decision Trees
Introduction to Feature Engineering Feature Transformation Feature Scaling Feature Engineering Frequency Encoding Automated Feature Engineering: Feature Tools
Introduction to Naive Bayes Conditional Probability and Bayes Theorem Introduction to Bayesian Adjustment Rating: The Incredible Concept Behind Online Ratings! Working of Naive Bayes Math behind Naive Bayes Types of Naive Bayes Implementation of Naive Bayes
Understanding how to solve Multiclass and Multilabled Classification Problem Evaluation Metrics: Multi Class Classification
Introduction to Ensemble Techniques Basic Ensemble Techniques Implementing Basic Ensemble Techniques Finding Optimal Weights of Ensemble Learner using Neural Network Why Ensemble Models Work well?
Introduction to Stacking Implementing Stacking Variants of Stacking Implementing Variants of Stacking Introduction to Blending Bootstrap Sampling Introduction to Random Sampling Hyper-parameters of Random Forest Implementing Random Forest Out-of-Bag (OOB) Score in the Random Forest IPL Team Win Prediction Project Using Machine Learning Introduction to Boosting Gradient Boosting Algorithm Math behind GBM Implementing GBM in python Regularized Greedy Forests Extreme Gradient Boosting Implementing XGBM in python Tuning Hyperparameters of XGBoost in Python Implement XGBM in R/H2O Adaptive Boosting Implementing Adaptive Boosing LightGBM Implementing LightGBM in Python Catboost Implementing Catboost in Python
Different Hyperparameter Tuning methods Implementing Different Hyperparameter Tuning methods GridsearchCV RandomizedsearchCV Bayesian Optimization for Hyperparameter Tuning Hyperopt
Understanding SVM Algorithm SVM Kernels In-depth Intuition and Practical Implementation SVM Kernel Tricks Kernels and Hyperparameters in SVM Implementing SVM from Scratch in Python and R
Introduction to Principal Component Analysis Steps to Perform Principal Compound Analysis Computation of Covariance Matrix Finding Eigenvectors and Eigenvalues Implementing PCA in python Visualizing PCA A Brief Introduction to Linear Discriminant Analysis Introduction to Factor Analysis
Introduction to Clustering Applications of Clustering Evaluation Metrics for Clustering Understanding K-Means Implementation of K-Means in Python Implementation of K-Means in R Choosing Right Value for K Profiling Market Segments using K-Means Clustering Hierarchical Clustering Implementation of Hierarchial Clustering DBSCAN Defining Similarity between clusters Build Better and Accurate Clusters with Gaussian Mixture Models
Understand Basics of Recommendation Engine with Case Study
8 Ways to Improve Accuracy of Machine Learning Models
Introduction to Dask Working with CuML
Introduction to Machine Learning Interpretability Framework and Interpretable Models model Agnostic Methods for Interpretability Implementing Interpretable Model Understanding SHAP Out-of-Core ML Introduction to Interpretable Machine Learning Models Model Agnostic Methods for Interpretability Game Theory & Shapley Values
Introduction to Machine Learning Interpretability Framework and Interpretable Models model Agnostic Methods for Interpretability Implementing Interpretable Model Understanding SHAP Out-of-Core ML Introduction to Interpretable Machine Learning Models Model Agnostic Methods for Interpretability Game Theory & Shapley Values
Introduction to AutoML Implementation of MLBox Introduction to PyCaret TPOT Auto-Sklearn EvalML
Pickle and Joblib Introduction to Model Deployment
Deploying Machine Learning Model using Streamlit Deploying ML Models in Docker Deploy Using Streamlit Deploy on Heroku Deploy Using Netlify Introduction to Amazon Sagemaker Setting up Amazon SageMaker Using SageMaker Endpoint to Generate Inference Deploy on Microsoft Azure Cloud Introduction to Flask for Model Deploying ML model using Flask
Model Deployment in Android Model Deployment in Iphone

Axioms of Probability Every Data Scientist Should Know!

Himanshi Singh
Himanshi Singh Last Updated : 12 Mar, 2021
6 min read

Introduction

We frequently use the term Probability but don’t realize how powerful this concept is. In simple terms, the probability is the likelihood or chance of something happening. And one of the fundamental concepts of probability is the Axioms of probability, which are essential for statistics and Exploratory Data Analysis.

Axioms mean a rule a principle that most people believe to be true. It is the premise on the basis of which we do further reasoning

In this article, I’m going to cover the Three Axioms of probability in detail.

Note: If you are more interested in learning concepts in an Audio-Visual format, We have this entire article explained in the video below. If not, you may continue reading.

 

 

Axioms of Probability

There are three axioms of probability that make the foundation of probability theory-

Axiom 1: Probability of Event

The first one is that the probability of an event is always between 0 and 1. 1 indicates definite action of any of the outcome of an event and 0 indicates no outcome of the event is possible.

Axiom 2: Probability of Sample Space

For sample space, the probability of the entire sample space is 1.

Axiom 3: Mutually Exclusive Events

And the third one is- the probability of the event containing any possible outcome of two mutually disjoint is the summation of their individual probability.

Now let’s look at each one of them in detail!

 

1. Probability of Event

The first axiom of probability is that the probability of any event is between 0 and 1.

Axiom 1: Probability of Event

As we know the formula of probability is that we divide the total number of outcomes in the event by the total number of outcomes in sample space.

Axiom 1: Probability of Event formula

And the event is a subset of sample space, so the event cannot have more outcome than the sample space. Clearly, this value is going to be between 0 and 1 since the denominator is always greater than the numerator.

 

2. Probability of Sample Space

The second axiom is that the probability for the entire sample space equals 1.

Axiom 2: Probability of Sample Space

Let’s take an example from the dataset. Suppose we need to find out the probability of churning for the female customers by their occupation type.

Axiom 2: Probability of Sample Space example

In our data-set, we have 4 female customers, one of them is Salaried and three of them are self-employed. The salaried female is going to churn. Since we have only one salaried female who is going to churn, the number of salaried female customers who are not going to churn is 0. Amongst 3 self-employed female customers, two are going to churn and we can see that one self-employed female is not going to churn. This is the complete dataset:

Axiom 2: Probability of Sample Space data

So the probability of the churning status of female customer by profession, in the sample space of the problem we actually have:

Salaried Churn, Salaried Not churn, Self-employed Churn, Self-employed Not churn

And as we discussed their distribution earlier, in this sample space of female customer:

Salaried Churn = 1

Salaried Not churn = 0

Self-employed Churn = 2

Self-employed Not churn = 1

If you were to find out the probability that a person who is a female is salaried and is churning it will be equal to:

P(person who is a female is salaried and is churning)

Similarly, the probability of Salaried Not churn is:

P(Salaried Not churn)

Then we have Self-employed Churn:

P(Self-employed Churn)

And finally Self-employed Not Churn:

Probability of Self-employed Not Churn

And if we sum all of them up we get 1:

sample space

So essentially saying that this is our entire sample space and the total probability that we get here is equals to 1. This brings us to axiom 3 which is related to mutually exclusive events.

 

3. Mutually Exclusive Event

Axiom 3: Mutually Exclusive Events

If you remember the union formula you will recall that the intersection term is not here, which means there is nothing common between A and B. Let us understand these particular type of events which is called Mutually Exclusive Events.

These Mutually exclusive events mean that such events cannot occur together or in other words, they don’t have common values or we can say their intersection is zero/null. We can also represent such events as follows:

P intersection

This means that the intersection is zero or they do not have any common value. For example, if the

Event A:  is getting a number greater than 4 after rolling a die, the possible outcomes would be 5 and 6.

Event A:  is getting a number greater than 4 after rolling a die, the possible outcomes would be 5 and 6.

Even B:  is getting a number less than 3 on rolling a die. Here the possible outcomes would be 1 and 2.

Even B:  is getting a number less than 3 on rolling a die. Here the possible outcomes would be 1 and 2.

Clearly, both these events cannot have any common outcome. An interesting thing to note here is that events A and B are not complemented of each other but yet they’re mutually exclusive.

 

Mutually Exhaustive

One more important concept is Mutually Exhaustive Event which is often confused with mutually inclusive events. Mutually exhaustive events mean that such events together make up everything that can possibly happen in a random experiment. That means the union of these events makes the sample space:

Mutually Exhaustive Event

Let’s understand this with an example-

Event A: Getting a number greater than 2 after a die, possible outcomes would be-

Event A: Getting a number greater than 2 after a die, possible outcomes would be

Event B: Getting a number less than 4 after rolling a die. Here the possible outcomes would be:

Event B: Getting a number less than 4 after rolling a die. Here the possible outcomes would be

Clearly, both these events together make up all the outcomes that can possibly take place after rolling a die.

 

How Mutually Exhaustive events differ from Mutually Exclusive events?

In the previous example getting a number 3 was common between both events. So these CANNOT be mutually exclusive but they definitely are mutually exhaustive. On the other hand, if we have another event:

Event C: Getting a number less than 3 after rolling a die, the possible outcomes would be:

Event C: Getting a number less than 3 after rolling a die, the possible outcomes would be:

Now we can say that Event A and Event C are mutually exclusive since they have nothing in common.

 

End Notes

In this article, we covered the axioms of probability and the difference between mutually exclusive and mutually exhaustive events.

If you are looking to kick start your Data Science Journey and want every topic under one roof, your search stops here. Check out Analytics Vidhya’s Certified AI & ML BlackBelt Plus Program

Let us know in the comments if you have any queries!

axioms of probability
Himanshi Singh
Himanshi Singh

I’m a data lover who enjoys finding hidden patterns and turning them into useful insights. As the Manager - Content and Growth at Analytics Vidhya, I help data enthusiasts learn, share, and grow together. 

Thanks for stopping by my profile - hope you found something you liked :)

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Elem Fredrick
Elem Fredrick

It has very good information thanks May God bless you

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