Probability of Happening at Least One Independent Event Data Science and A.I. Lecture Series By: Bindeshwar Singh Kushwaha Institute: PostNetwork Academy 1. Probability of Happening at Least One Independent Event If \( A \) and \( B \) are independent events, the probability of happening at least one of the events is: \[ P(A […]
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Conditional Probability and Multiplicative Law Data Science and A.I. Lecture Series Conditional Probability Conditional probability represents the likelihood of an event \( A \), given that another event \( B \) has already occurred. It is defined as: \[ P(A|B) = \frac{P(A \cap B)}{P(B)}, \quad \text{if } P(B) > 0. \] Example: Deck
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More on Axiomatic Approach to Probability Data Science and AI Lecture Series By Bindeshwar Singh Kushwaha Statement of the First Proof Prove: \( P(A \cap B^c) = P(A) – P(A \cap B) \) This formula expresses the probability of \( A \) occurring without \( B \). It uses the complement rule and properties of
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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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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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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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Understand Combinations Data Science and A.I. Lecture Series Introduction to Combinations A combination is a selection of items where the order does not matter. Example: Selecting 2 players from a group of 3 players (X, Y, Z). Possible combinations: XY, XZ, YZ. Formula for combinations: \[ \binom{n}{r} = \frac{n!}{r!(n-r)!}, \quad 0 \leq r \leq
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Examples from Permutations Data Science and A.I. Lecture Series By Bindeshwar Singh Kushwaha, PostNetwork Academy Example 1 How many 4-digit numbers can be formed by using the digits 1 to 9 if repetition of digits is not allowed? Solution: Total digits: 9 Required 4-digit numbers: \[ P(9, 4) = \frac{9!}{(9-4)!} = 9 \times 8
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