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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Addition Law of Probability and Examples Data Science and A.I. Lecture Series Introduction to Addition Law Statement: For any two events \( A \) and \( B \), the probability of their union is given by: \[ P(A \cup B) = P(A) + P(B) – P(A \cap B) \] This formula calculates the probability
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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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& 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, …,
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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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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Concept of Odds in Favor and Against Data Science and A.I. Lecture Series Author: Bindeshwar Singh Kushwaha Institute: PostNetwork Academy Concept of Odds Odds in Favor and Against Odds in Favor: Ratio of favorable cases to unfavorable cases:\[ \text{Odds in favor of } A = m : (n – m) \] Odds Against: Ratio of
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Probability Problems Based on Classical Definition of Probability Data Science and A.I. Lecture Series Questions What is the total number of outcomes (sample space)? How do we determine favorable cases? How do probability rules apply to the problem? Example: Throwing Two Dice Find the probability of: A doublet (same number on both dice) Sum
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Probability Problem: Throwing a Fair Die Data Science and A.I. Lecture Series Problem Statement A fair die is thrown. Find the probability of: A prime number An even number A number multiple of 2 or 3 A number multiple of 2 and 3 A number greater than 4 Step 1: Sample Space Sample Space:
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