math 2nd sem.devraj.pptxwgjwgjweghgwggdg
- 1. STUDENT DETAILS • Name - Devraj Maji • Name of Department - Department of Computer science and Engineering • Program Name - B. Tech. CSE(AIML)2024 • Semester/Year - 2nd semester/1st Year • Crouse Code - BSCM202 • Course Name - Probability & Statistics • Student Code - BWU/BTA/23/428 • Section - H
- 2. CONTENTS 1. Introduction to Probability 2. Sample Spaces and Events 3. Probability Axioms and Rules 4. Conditional probability 5. Independence and Dependence 6. Bayes' Theorem 7. Discrete Probability Distributions 8. Continuous Probability Distributions
- 3. INTRODUCTION TO PROBABILITY Probability is the mathematical study of uncertainty and the likelihood of events occurring. It helps us make informed decisions in the face of incomplete information or randomness.
- 4. DEFINITIONS AND TERMINOLOGY • Sample Space: The set of all possible outcomes in a probability experiment. • Event: A subset of the sample space that we are interested in observing or measuring. • Probability: A numerical measure of the likelihood that an event will occur, expressed as a value between 0 and 1.
- 5. SAMPLE SPACES AND EVENTS • A sample space is the set of all possible outcomes of an experiment. Events are subsets of the sample space, representing specific outcomes of interest. • Understanding sample spaces and events is fundamental to probability theory, enabling us to precisely define and analyze the likelihood of different outcomes.
- 6. PROBABILITY AXIOMS AND RULES • Axiom 1: Non-Negativity --->> The probability of any event must be a non-negative number, meaning it cannot be less than zero. • Axiom 2: Certainty --->> The probability of the entire sample space, or the certain event, is equal to 1. • Axiom 3: Additivity --->> The probability of the union of two mutually exclusive events is the sum of their individual probabilities.
- 8. INDEPENDENCE AND DEPENDENCE Independence Events are independent if the occurrence of one event does not affect the probability of the other event. Independent events have no influence on each other. Dependence Dependent events are those where the outcome of one event affects the probability of the other event. The occurrence of one event changes the likelihood of the other event. Conditional Probability The probability of an event given that another event has occurred is called conditional probability. It describes the likelihood of one event happening given the occurrence of another event. Examples Flipping a fair coin twice is an example of independent events. Drawing a card from a deck and then drawing another card is an example of dependent events.
- 9. BAYES' THEOREM 1 2 >> Prior Probability Initial belief about the likelihood of an event. >> Conditional Probability Probability of an event given additional information. >> Posterior Probability Updated belief about the likelihood of an event. 3
- 10. BAYES' THEOREM DEFINATION>> Bayes' Theorem is a fundamental concept in probability theory that describes the relationship between conditional probabilities. It allows us to update our beliefs about the likelihood of an event based on new information. This powerful statistical tool has applications in fields like machine learning, medical diagnosis, and decision-making.
- 11. DISCRETE PROBABILITY DISTRIBUTIONS Binomial Distribution Describes the probability of a fixed number of successes in a series of independent Bernoulli trials. Useful for modeling events with two possible outcomes, like coin flips or product defects Models the probability of a given number of events occurring in a fixed interval of time or space, assuming these events happen with a known rate and independently of the time since the last event. Poisson Distribution