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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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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Permutations and Their Theorems Data Science and A.I. Lecture Series Author: Bindeshwar Singh Kushwaha Institute: PostNetwork Academy Theorem 1: Permutations of Distinct Objects The number of permutations of n different objects taken r at a time is: \[ P(n, r) = \frac{n!}{(n-r)!} \] Explanation: First position: n choices. Second position: n-1 choices. Continue until the
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Understanding Permutations Data Science and A.I. Lecture Series Author: Bindeshwar Singh Kushwaha PostNetwork Academy Introduction to Permutations A permutation is an arrangement of objects in a specific order. The order of arrangement is crucial in permutations. Example: Arranging the letters of the word “ABC”. Total permutations = $3! = 6$. Key Formula for Permutations The
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