09. Probability Foundations: Axioms of Probability, Conditional Probability & Independence • Mathematics, Statistics & Machine Learning (Big Tech Prep) • Rishabh Ki Kaksha

Mathematics, Statistics & Machine Learning (Big Tech Prep) › 09. Probability Foundations: Axioms of Probability, Conditional Probability & Independence

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Course Syllabus

Module 1: Linear Algebra for Machine Learning & Deep Learning

01. Vectors, Vector Spaces, Dot Products & Norms (L1, L2, L-Infinity) in High Dimensions 02. Matrices, Matrix Multiplication & Linear Transformations (Rotation, Scaling, Shear) 03. Eigenvalues & Eigenvectors: Geometric Intuition, Diagonalization & Spectral Theorem 04. Principal Component Analysis (PCA) & Singular Value Decomposition (SVD) Math

Module 2: Multivariable Calculus, Gradients & Optimization Algorithms

05. Partial Derivatives, Directional Derivatives & The Gradient Vector (Del Operator) 06. The Jacobian Matrix & Hessian Matrix (Second Derivative Curvature Analysis) 07. Multivariate Chain Rule & The Mathematical Backbone of Neural Backpropagation 08. Optimization Algorithms: Gradient Descent, SGD with Momentum, RMSprop & Adam Optimizer Math

Module 3: Probability Theory & Random Variables

09. Probability Foundations: Axioms of Probability, Conditional Probability & Independence 10. Bayes' Theorem & Bayesian Inference: Prior, Likelihood, Posterior & Evidence 11. Probability Distributions: Gaussian (Normal), Bernoulli, Binomial, Poisson & Exponential Distributions

Module 4: Inferential Statistics & Hypothesis Testing

12. Expectation, Variance, Covariance & Pearson Correlation Coefficient 13. Law of Large Numbers & Central Limit Theorem (CLT) with Monte Carlo Simulations 14. Hypothesis Testing: Null Hypothesis, p-Values, Z-Test, Student's t-Test & Chi-Square Test 15. Maximum Likelihood Estimation (MLE) vs Maximum A Posteriori (MAP) Estimation

Module 5: Classical Machine Learning Algorithms from Scratch (NumPy)

16. Linear Regression from Scratch: Ordinary Least Squares (OLS) Closed Form vs Gradient Descent 17. Logistic Regression & Binary Cross-Entropy Loss from Scratch with NumPy 18. Regularization Mathematics: L1 Lasso (Sparsity) vs L2 Ridge (Weight Decay) Penalties

Interactive Lesson Studio

# 09. Probability Foundations: Axioms of Probability, Conditional Probability & Independence

> Mathematics & ML Foundations Notes by Rishabh - Topic 09

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### 1. Kolmogorov's 3 Probability Axioms 🎲

1. **Non-Negativity:** For any event $E$, $P(E) \ge 0$.
2. **Total Probability:** For sample space $S$, $P(S) = 1$.
3. **Countable Additivity:** For mutually exclusive events $A \cap B = \emptyset$, $P(A \cup B) = P(A) + P(B)$.

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### 2. Conditional Probability & Statistical Independence 🎯

$$P(A \mid B) = \frac{P(A \cap B)}{P(B)} \quad (\text{Probability of } A \text{ given } B \text{ occurred})$$

*Events $A$ and $B$ are **Independent** if and only if:*
$$P(A \cap B) = P(A) \cdot P(B) \iff P(A \mid B) = P(A)$$

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### 3. Master Comparison Matrix

| Concept | Mathematical Equation | Meaning |
| :--- | :--- | :--- |
| **Joint Probability** | $P(A, B)$ | Dono events $A$ aur $B$ ek sath occur hone ka chance |
| **Marginal Probability**| $P(A) = \sum_B P(A, B)$ | Kisi doosre variable ke saare values ko sum/integrate karke $A$ nikalna |
| **Conditional Probability**| $P(A \mid B) = \frac{P(A, B)}{P(B)}$ | Given context $B$ me $A$ ka updated chance |

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### 4. Interview Ke 3 Sawal Is Topic Se

1. **Chain Rule of Probability high-dimensional joint distributions me kaise apply hota hai?** -> $P(X_1, X_2, \dots, X_n) = P(X_1) \prod_{i=2}^n P(X_i \mid X_1, \dots, X_{i-1})$ (LLM Next-Token Prediction backbone!).
2. **Conditional Independence kya hoti hai?** -> $P(A, B \mid C) = P(A \mid C) P(B \mid C)$ (Given $C$, knowing $B$ gives zero extra info about $A$ - Naive Bayes Classifier assumption).
3. **Law of Total Probability kab use hota hai?** -> Jab sample space $B_1, B_2, \dots, B_k$ mutually exclusive partitions me divide ho: $P(A) = \sum_{i=1}^k P(A \mid B_i) P(B_i)$.

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**Next Topic:** 10. Bayes' Theorem & Bayesian Inference: Prior, Likelihood, Posterior & Evidence

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