Artificial Intelligence

Different Approaches to Probability Theory

Artificial Intelligence, Mathematics, Maths, Probability and Statistics, Research and Development, Statistics /

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

Probability Examples Related to Combinations

Artificial Intelligence, Counting Principle, Machine Learning, Mathematics, Maths, Permutations and Combinations, Probability and Statistics, Statistics /

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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Combinations

Theorem Related to Combinations

Artificial Intelligence, Counting Principle, Machine Learning, Mathematics, Maths, Permutations and Combinations, Probability and Statistics, Statistics /

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

Artificial Intelligence, Counting Principle, Machine Learning, Mathematics, Maths, Permutations and Combinations, Probability and Statistics, Statistics /

  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 of Permutations

Artificial Intelligence, Data Handling, Machine Learning, Mathematics, Maths, Statistics /

  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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More on Permutations

Artificial Intelligence, Counting Principle, Machine Learning, Mathematics, Permutations and Combinations, Probability and Statistics, Statistics /

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

Algebra, Artificial Intelligence, Counting Principle, Machine Learning, Mathematics, Permutations and Combinations, Probability and Statistics, Statistics /

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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Probability

Concept of Odds in Favor and Against

Artificial Intelligence, Data Handling, Machine Learning, Mathematics, Maths, Probability and Statistics, Statistics /

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

Probability Problems based on the Classical Definition of Probability

Artificial Intelligence, Data Handling, Machine Learning, Mathematics, Maths, Probability and Statistics, Statistics /

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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Throwing a Fair Die

Probability Problem: Throwing a Fair Die

Artificial Intelligence, Data Handling, Machine Learning, Mathematics, Probability and Statistics /

  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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