Axiomatic Approach to Probability Data Science and A.I. Lecture Series Axiomatic Approach to Probability Definition: Let \( S \) be a sample space for a random experiment. Let \( A \) be an event that is a subset of \( S \). \( P(A) \) is called a probability function if it satisfies the […]
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Set Operations and Important Laws Data Science and A.I. Lecture Series Set Operations Union Definition: \[ A \cup B = \{x : x \in A \text{ or } x \in B\} \] Examples: \( A = \{1, 2, 3\}, B = \{3, 4, 5\} \implies A \cup B = \{1, 2, 3, 4, 5\}
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Venn Diagrams – Data Science and AI Lecture Series Welcome to our Data Science and AI Lecture Series! In this post, we’ll dive into the world of Venn Diagrams, an essential tool in set theory that simplifies understanding the relationships between sets. Whether you’re studying mathematics, data science, or AI, mastering concepts like intersections, unions,
& Hierarchy of Sets Universal Set (\( U \)) The universal set (\( U \)) contains all elements under consideration in a specific context. Example 1: If studying numbers, \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \). Example 2: If studying letters, \( U = \{\text{a, b, c, …,
Introduction to Sets and Type of Sets Data Science and A.I. Lecture Series Introduction A set is a well-defined collection of distinct objects. Examples of collections: Books in a library. Natural numbers that are factors of a given number. States in a country. Sets are fundamental in mathematics and are used in many areas,
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Subjective Approach to Probability Data Science and A.I. Lecture Series Author: Bindeshwar Singh Kushwaha What is the Subjective Approach? The subjective approach to probability is based on personal judgment, intuition, wisdom, and expertise. Unlike the classical or frequency-based approaches, it focuses on individual beliefs about the likelihood of an event. When to Use the
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Relative Frequency Approach in Probability Data Science and A.I. Lecture Series Author: Bindeshwar Singh Kushwaha Relative Frequencies and Probabilities So, in general, if \( X \) is a variable having the values \( x_1, x_2, \dots, x_n \) with frequencies \( f_1, f_2, \dots, f_n \), respectively, then: \[ \frac{f_1}{\sum f_i}, \frac{f_2}{\sum f_i}, \dots,
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Different Approaches to Probability Theory Data Science and AI Lecture Series Author: Bindeshwar Singh Kushwaha Introduction Classical probability has limitations when outcomes are not equally likely or finite. Alternative approaches are needed in situations where classical definitions fail. This unit introduces methods based on past experiences, observed data, and axioms. Topics discussed include: Relative
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Probability Examples Related to Combinations Data Science and A.I. Lecture Series Author: Bindeshwar Singh Kushwaha Example: Drawing Two Cards from a Well-Shuffled Pack of Cards Find the probability of the following scenarios: One red and one black card. Both cards of the same suit. One jack and one king. One red card and one card
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Examples and Theorem Related to Combinations Data Science and A.I. Lecture Series Author: Bindeshwar Singh Kushwaha Theorem: Relationship Between Permutations and Combinations Theorem: The number of permutations of \(n\) different objects taken \(r\) at a time is related to the number of combinations by: \[ P^n_r = C^n_r \cdot r! \] where \(0 < r
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