This document discusses probability and its key concepts. It begins by defining probability as a quantitative measure of uncertainty ranging from 0 to 1. Probability can be understood objectively based on problems or subjectively based on beliefs. Key probability concepts discussed include:
- Sample space, simple events, and compound events
- Classical, relative frequency, and subjective approaches to assigning probabilities
- Complement, intersection, and union of events
- Conditional probability and independence of events
- Rules for calculating probabilities of combined events like the multiplication rule
Examples are provided to illustrate concepts like defining sample spaces, calculating probabilities of individual and combined events, determining conditional probabilities, and assessing independence. Overall, the document provides a comprehensive overview of fundamental probability
This document discusses basic concepts of probability, including:
- The addition rule and multiplication rule for calculating probabilities of compound events.
- Events can be disjoint (mutually exclusive) or not disjoint.
- The probability of an event occurring or its complement must equal 1.
- How to calculate the probability of at least one occurrence of an event using the complement.
- When applying the multiplication rule, you must consider whether events are independent or dependent.
This document discusses types of probability and provides definitions and examples of key probability concepts. It begins with an introduction to probability theory and its applications. The document then defines terms like random experiments, sample spaces, events, favorable events, mutually exclusive events, and independent events. It describes three approaches to measuring probability: classical, frequency, and axiomatic. It concludes with theorems of probability and references.
This presentation provides an introduction to basic probability concepts. It defines probability as the study of randomness and uncertainty, and describes how probability was originally associated with games of chance. Key concepts discussed include random experiments, sample spaces, events, unions and intersections of events, and Venn diagrams. The presentation establishes the axioms of probability, including that a probability must be between 0 and 1, the probability of the sample space is 1, and probabilities of mutually exclusive events sum to the total probability. Formulas for computing probabilities of unions, intersections, and complements of events are also presented.
This document discusses key concepts in probability. It defines basic terms like experiment, sample space, event, and probability. It provides examples of calculating probability for coin tosses and dice rolls using the classical method of dividing the number of ways an event can occur by the total number of possible outcomes. The document also discusses limitations of the classical method and introduces the empirical and subjective methods of determining probability based on observed frequencies and personal judgment respectively.
- Probability theory describes the likelihood of chance outcomes and is measured on a scale from 0 to 1. Probability can be calculated classically based on equally likely outcomes or empirically based on relative frequency.
- Bayes' theorem allows updating probabilities based on new information by calculating conditional probabilities. It expresses the probability of an event A given evidence B in terms of prior probabilities and the likelihood of the evidence.
- The Monty Hall problem illustrates that switching doors in a game show scenario doubles the probability of winning the prize because it uses additional information provided by the host.
History behind the
development of the concept
In 1654, a gambler Chevalier De Metre approached the well known Mathematician Blaise Pascal for certain dice problem. Pascal became interested in these problems and discussed it further with Pierre de Fermat. Both of them solved these problems independently. Since then this concept gained limelight.
Basic Things About The Concept
Probability is used to quantify an attitude of mind towards some uncertain proposition.
The higher the probability of an event, the more certain we are that the event will occur.
This document defines key concepts related to random variables including:
- A random variable is a numerical measure of outcomes from a random phenomenon.
- Probability distributions describe the probabilities associated with random variables.
- Expected value refers to the mean or weighted average of a probability distribution.
- As the number of trials increases, the actual mean approaches the true mean due to the Law of Large Numbers.
- Binomial and geometric distributions model situations with success/failure outcomes and independence between trials.
This presentation describes some of the unsolved problems of Mathematics. These kinds of unsolved math problems create an interest among the learners to learn mathematics and its importance. It also promotes the creative abilities of the learners.
This document discusses basic concepts of probability, including:
- The addition rule and multiplication rule for calculating probabilities of compound events.
- Events can be disjoint (mutually exclusive) or not disjoint.
- The probability of an event occurring or its complement must equal 1.
- How to calculate the probability of at least one occurrence of an event using the complement.
- When applying the multiplication rule, you must consider whether events are independent or dependent.
This document discusses types of probability and provides definitions and examples of key probability concepts. It begins with an introduction to probability theory and its applications. The document then defines terms like random experiments, sample spaces, events, favorable events, mutually exclusive events, and independent events. It describes three approaches to measuring probability: classical, frequency, and axiomatic. It concludes with theorems of probability and references.
This presentation provides an introduction to basic probability concepts. It defines probability as the study of randomness and uncertainty, and describes how probability was originally associated with games of chance. Key concepts discussed include random experiments, sample spaces, events, unions and intersections of events, and Venn diagrams. The presentation establishes the axioms of probability, including that a probability must be between 0 and 1, the probability of the sample space is 1, and probabilities of mutually exclusive events sum to the total probability. Formulas for computing probabilities of unions, intersections, and complements of events are also presented.
This document discusses key concepts in probability. It defines basic terms like experiment, sample space, event, and probability. It provides examples of calculating probability for coin tosses and dice rolls using the classical method of dividing the number of ways an event can occur by the total number of possible outcomes. The document also discusses limitations of the classical method and introduces the empirical and subjective methods of determining probability based on observed frequencies and personal judgment respectively.
- Probability theory describes the likelihood of chance outcomes and is measured on a scale from 0 to 1. Probability can be calculated classically based on equally likely outcomes or empirically based on relative frequency.
- Bayes' theorem allows updating probabilities based on new information by calculating conditional probabilities. It expresses the probability of an event A given evidence B in terms of prior probabilities and the likelihood of the evidence.
- The Monty Hall problem illustrates that switching doors in a game show scenario doubles the probability of winning the prize because it uses additional information provided by the host.
History behind the
development of the concept
In 1654, a gambler Chevalier De Metre approached the well known Mathematician Blaise Pascal for certain dice problem. Pascal became interested in these problems and discussed it further with Pierre de Fermat. Both of them solved these problems independently. Since then this concept gained limelight.
Basic Things About The Concept
Probability is used to quantify an attitude of mind towards some uncertain proposition.
The higher the probability of an event, the more certain we are that the event will occur.
This document defines key concepts related to random variables including:
- A random variable is a numerical measure of outcomes from a random phenomenon.
- Probability distributions describe the probabilities associated with random variables.
- Expected value refers to the mean or weighted average of a probability distribution.
- As the number of trials increases, the actual mean approaches the true mean due to the Law of Large Numbers.
- Binomial and geometric distributions model situations with success/failure outcomes and independence between trials.
This presentation describes some of the unsolved problems of Mathematics. These kinds of unsolved math problems create an interest among the learners to learn mathematics and its importance. It also promotes the creative abilities of the learners.
The document provides an overview of probability concepts including:
- Probability is a measure of how likely an event is, defined as the number of favorable outcomes divided by the total number of possible outcomes.
- Theoretical probability predicts outcomes without performing experiments, dealing with events as combinations of elementary outcomes.
- Random experiments may have different results each time while deterministic experiments always produce the same outcome.
- Elementary events are individual outcomes, and compound events combine multiple elementary outcomes.
- Theoretical probability of an event is the number of favorable elementary events divided by the total number of possible events.
- The probabilities of an event and its negation must sum to 1.
Probability is a branch of mathematics that studies patterns of chance. It is used to quantify the likelihood of events occurring in experiments or other situations involving uncertainty. The probability of an event is expressed as a number between 0 and 1, with 0 indicating impossibility and 1 indicating certainty. Key concepts in probability include theoretical and experimental probability, sample spaces, events, mutually exclusive and exhaustive events, and rules like addition rules for calculating combined probabilities. Probability is applied in many fields including statistics, gambling, science, and machine learning.
This document provides a collection of probability questions that are often asked in aptitude tests and competitive exams. It includes 14 questions with explanations and calculations of the probability for each. The purpose is to help students prepare for exams by understanding basic probability concepts and practicing sample questions. Links are provided at the end for additional free study materials on topics like reasoning, English, mathematics and general knowledge.
This document provides an introduction to probability. It defines probability as a numerical index of the likelihood that a certain event will occur, with a value between 0 and 1. It discusses examples of using probability terms like chance and likelihood. It also covers key probability concepts such as experiments, outcomes, events, and sample spaces. It explains different types of probability including subjective, objective/classic, and empirical probabilities. It provides examples of calculating probabilities of events using various approaches.
This document provides an overview of binomial distributions including:
- Defining binomial experiments as having fixed trials with two possible outcomes, the same probability of success each trial, and counting the number of successes.
- Noting the notation used including n trials, p probability of success, q probability of failure, and x counting successes.
- Explaining how to calculate binomial probabilities using formulas, tables, and technology.
- Discussing how to graph binomial distributions and find their mean, variance, and standard deviation.
1. Probability is the study of randomness and uncertainty of outcomes from experiments or processes. It allows us to make statements about the likelihood of events occurring.
2. Events are outcomes or sets of outcomes from random experiments. The probability of an event is calculated based on the number of outcomes in the event compared to the total number of possible outcomes.
3. Conditional probability is the likelihood of one event occurring given that another event has occurred. It is calculated as the probability of both events occurring divided by the probability of the first event. Conditional probabilities are useful for problems involving dependent events.
1. The document discusses basic concepts in probability and statistics, including sample spaces, events, probability distributions, and random variables.
2. Key concepts are explained such as independent and conditional probability, Bayes' theorem, and common probability distributions like the uniform and normal distributions.
3. Statistical analysis methods are introduced including how to estimate the mean and variance from samples from a distribution.
The document discusses probability theory and provides definitions and examples of key concepts like conditional probability and Bayes' theorem. It defines probability as the ratio of favorable events to total possible events. Conditional probability is the probability of an event given that another event has occurred. Bayes' theorem provides a way to update or revise beliefs based on new evidence and relates conditional probabilities. Examples are provided to illustrate concepts like conditional probability calculations.
This document discusses several discrete probability distributions:
1. Binomial distribution - For experiments with a fixed number of trials, two possible outcomes, and constant probability of success. The probability of x successes is given by the binomial formula.
2. Geometric distribution - For experiments repeated until the first success. The probability of the first success on the xth trial is p(1-p)^(x-1).
3. Poisson distribution - For counting the number of rare, independent events occurring in an interval. The probability of x events is (e^-μ μ^x)/x!, where μ is the mean number of events.
1) The document discusses probability and provides examples to illustrate key concepts of probability, including experiments, outcomes, events, and the probability formula.
2) Tree diagrams are introduced as a way to calculate probabilities when there is more than one experiment occurring and the outcomes are not equally likely. The key rules are that probabilities are multiplied across branches and added down branches.
3) Several examples using letters in a bag, dice rolls, and colored beads in a bag are provided to demonstrate how to set up and use the probability formula and tree diagrams to calculate probabilities of events. Key concepts like mutually exclusive, independent, and dependent events are also explained.
The document discusses random variables and vectors. It defines random variables as functions that assign outcomes of random experiments to real numbers. There are two types of random variables: discrete and continuous. Random variables are characterized by their expected value, variance/standard deviation, and other moments. Random vectors are multivariate random variables. Key concepts covered include probability mass functions, probability density functions, expected value, variance, and how these properties change when random variables are scaled or combined linearly.
Subjective probability refers to a probability estimate based on personal judgment rather than calculations. The Di Finetti game is used to measure a person's subjective probability of an outcome occurring. In this game, a friend claims they will get 100% on an exam. The person is given hypothetical choices between drawing balls of different colors from a bag versus waiting for exam results. Based on the friend's choices at different ball distributions, their subjective probability of scoring 100% is estimated to be between 85-88%. Repeating the game with different ball distributions helps narrow down the range.
1) The document introduces basic concepts of probability such as sample spaces, events, outcomes, and how to calculate classical and empirical probabilities.
2) It discusses approaches to determining probability including classical, empirical, and subjective probabilities. Simulations can also be used to estimate probabilities.
3) Examples are provided to illustrate calculating probabilities using classical and empirical approaches for single and compound events with different sample spaces.
Slides for a lecture by Todd Davies on "Probability", prepared as background material for the Minds and Machines course (SYMSYS 1/PSYCH 35/LINGUIST 35/PHIL 99) at Stanford University. From a video recorded July 30, 2019, as part of a series of lectures funded by a Vice Provost for Teaching and Learning Innovation and Implementation Grant to the Symbolic Systems Program at Stanford, with post-production work by Eva Wallack. Topics include Basic Probability Theory, Conditional Probability, Independence, Philosophical Foundations, Subjective Probability Elicitation, and Heuristics and Biases in Human Probability Judgment.
LECTURE VIDEO: https://youtu.be/tqLluc36oD8
EDITED AND ENHANCED TRANSCRIPT: https://ssrn.com/abstract=3649241
The document discusses discrete and continuous random variables. It defines discrete random variables as variables that can take on countable values, like the number of heads from coin flips. Continuous random variables can take any value within a range, like height. The document explains how to calculate and interpret the mean, standard deviation, and probabilities of events for both types of random variables using examples like Apgar scores for babies and heights of young women.
This document outlines basic probability concepts, including definitions of probability, views of probability (objective and subjective), and elementary properties. It discusses calculating probabilities of events from data in tables, including unconditional/marginal probabilities, conditional probabilities, and joint probabilities. Rules of probability are presented, including the multiplicative rule that the joint probability of two events is equal to the product of the marginal probability of one event and the conditional probability of the other event given the first event. Examples are provided to illustrate key concepts.
The document discusses modern and postmodern perspectives in organization theory. It outlines key differences between the two views. The modern perspective sees organizations as real entities that can be rationally managed to achieve objectives, while the postmodern view sees organizations as social constructs that are sites of power relations without objective truths or universal principles. The document also summarizes the ideas of several theorists like Lyotard, Nietzsche, Derrida and Foucault that influenced the development of postmodern thought.
Comm120 arenivar-the organization theory ppDrew Stanford
The document outlines the "Organization Theory", which proposes that a person who is organized will be more successful than someone who is not organized. The theory is presented as an objective theory according to communication theory standards. Examples of how organization can lead to success in school, travel, and filmmaking are provided. The conclusion restates that there is a relationship between organization and success, and the theory meets the criteria of an objective theory.
The document provides an overview of probability concepts including:
- Probability is a measure of how likely an event is, defined as the number of favorable outcomes divided by the total number of possible outcomes.
- Theoretical probability predicts outcomes without performing experiments, dealing with events as combinations of elementary outcomes.
- Random experiments may have different results each time while deterministic experiments always produce the same outcome.
- Elementary events are individual outcomes, and compound events combine multiple elementary outcomes.
- Theoretical probability of an event is the number of favorable elementary events divided by the total number of possible events.
- The probabilities of an event and its negation must sum to 1.
Probability is a branch of mathematics that studies patterns of chance. It is used to quantify the likelihood of events occurring in experiments or other situations involving uncertainty. The probability of an event is expressed as a number between 0 and 1, with 0 indicating impossibility and 1 indicating certainty. Key concepts in probability include theoretical and experimental probability, sample spaces, events, mutually exclusive and exhaustive events, and rules like addition rules for calculating combined probabilities. Probability is applied in many fields including statistics, gambling, science, and machine learning.
This document provides a collection of probability questions that are often asked in aptitude tests and competitive exams. It includes 14 questions with explanations and calculations of the probability for each. The purpose is to help students prepare for exams by understanding basic probability concepts and practicing sample questions. Links are provided at the end for additional free study materials on topics like reasoning, English, mathematics and general knowledge.
This document provides an introduction to probability. It defines probability as a numerical index of the likelihood that a certain event will occur, with a value between 0 and 1. It discusses examples of using probability terms like chance and likelihood. It also covers key probability concepts such as experiments, outcomes, events, and sample spaces. It explains different types of probability including subjective, objective/classic, and empirical probabilities. It provides examples of calculating probabilities of events using various approaches.
This document provides an overview of binomial distributions including:
- Defining binomial experiments as having fixed trials with two possible outcomes, the same probability of success each trial, and counting the number of successes.
- Noting the notation used including n trials, p probability of success, q probability of failure, and x counting successes.
- Explaining how to calculate binomial probabilities using formulas, tables, and technology.
- Discussing how to graph binomial distributions and find their mean, variance, and standard deviation.
1. Probability is the study of randomness and uncertainty of outcomes from experiments or processes. It allows us to make statements about the likelihood of events occurring.
2. Events are outcomes or sets of outcomes from random experiments. The probability of an event is calculated based on the number of outcomes in the event compared to the total number of possible outcomes.
3. Conditional probability is the likelihood of one event occurring given that another event has occurred. It is calculated as the probability of both events occurring divided by the probability of the first event. Conditional probabilities are useful for problems involving dependent events.
1. The document discusses basic concepts in probability and statistics, including sample spaces, events, probability distributions, and random variables.
2. Key concepts are explained such as independent and conditional probability, Bayes' theorem, and common probability distributions like the uniform and normal distributions.
3. Statistical analysis methods are introduced including how to estimate the mean and variance from samples from a distribution.
The document discusses probability theory and provides definitions and examples of key concepts like conditional probability and Bayes' theorem. It defines probability as the ratio of favorable events to total possible events. Conditional probability is the probability of an event given that another event has occurred. Bayes' theorem provides a way to update or revise beliefs based on new evidence and relates conditional probabilities. Examples are provided to illustrate concepts like conditional probability calculations.
This document discusses several discrete probability distributions:
1. Binomial distribution - For experiments with a fixed number of trials, two possible outcomes, and constant probability of success. The probability of x successes is given by the binomial formula.
2. Geometric distribution - For experiments repeated until the first success. The probability of the first success on the xth trial is p(1-p)^(x-1).
3. Poisson distribution - For counting the number of rare, independent events occurring in an interval. The probability of x events is (e^-μ μ^x)/x!, where μ is the mean number of events.
1) The document discusses probability and provides examples to illustrate key concepts of probability, including experiments, outcomes, events, and the probability formula.
2) Tree diagrams are introduced as a way to calculate probabilities when there is more than one experiment occurring and the outcomes are not equally likely. The key rules are that probabilities are multiplied across branches and added down branches.
3) Several examples using letters in a bag, dice rolls, and colored beads in a bag are provided to demonstrate how to set up and use the probability formula and tree diagrams to calculate probabilities of events. Key concepts like mutually exclusive, independent, and dependent events are also explained.
The document discusses random variables and vectors. It defines random variables as functions that assign outcomes of random experiments to real numbers. There are two types of random variables: discrete and continuous. Random variables are characterized by their expected value, variance/standard deviation, and other moments. Random vectors are multivariate random variables. Key concepts covered include probability mass functions, probability density functions, expected value, variance, and how these properties change when random variables are scaled or combined linearly.
Subjective probability refers to a probability estimate based on personal judgment rather than calculations. The Di Finetti game is used to measure a person's subjective probability of an outcome occurring. In this game, a friend claims they will get 100% on an exam. The person is given hypothetical choices between drawing balls of different colors from a bag versus waiting for exam results. Based on the friend's choices at different ball distributions, their subjective probability of scoring 100% is estimated to be between 85-88%. Repeating the game with different ball distributions helps narrow down the range.
1) The document introduces basic concepts of probability such as sample spaces, events, outcomes, and how to calculate classical and empirical probabilities.
2) It discusses approaches to determining probability including classical, empirical, and subjective probabilities. Simulations can also be used to estimate probabilities.
3) Examples are provided to illustrate calculating probabilities using classical and empirical approaches for single and compound events with different sample spaces.
Slides for a lecture by Todd Davies on "Probability", prepared as background material for the Minds and Machines course (SYMSYS 1/PSYCH 35/LINGUIST 35/PHIL 99) at Stanford University. From a video recorded July 30, 2019, as part of a series of lectures funded by a Vice Provost for Teaching and Learning Innovation and Implementation Grant to the Symbolic Systems Program at Stanford, with post-production work by Eva Wallack. Topics include Basic Probability Theory, Conditional Probability, Independence, Philosophical Foundations, Subjective Probability Elicitation, and Heuristics and Biases in Human Probability Judgment.
LECTURE VIDEO: https://youtu.be/tqLluc36oD8
EDITED AND ENHANCED TRANSCRIPT: https://ssrn.com/abstract=3649241
The document discusses discrete and continuous random variables. It defines discrete random variables as variables that can take on countable values, like the number of heads from coin flips. Continuous random variables can take any value within a range, like height. The document explains how to calculate and interpret the mean, standard deviation, and probabilities of events for both types of random variables using examples like Apgar scores for babies and heights of young women.
This document outlines basic probability concepts, including definitions of probability, views of probability (objective and subjective), and elementary properties. It discusses calculating probabilities of events from data in tables, including unconditional/marginal probabilities, conditional probabilities, and joint probabilities. Rules of probability are presented, including the multiplicative rule that the joint probability of two events is equal to the product of the marginal probability of one event and the conditional probability of the other event given the first event. Examples are provided to illustrate key concepts.
The document discusses modern and postmodern perspectives in organization theory. It outlines key differences between the two views. The modern perspective sees organizations as real entities that can be rationally managed to achieve objectives, while the postmodern view sees organizations as social constructs that are sites of power relations without objective truths or universal principles. The document also summarizes the ideas of several theorists like Lyotard, Nietzsche, Derrida and Foucault that influenced the development of postmodern thought.
Comm120 arenivar-the organization theory ppDrew Stanford
The document outlines the "Organization Theory", which proposes that a person who is organized will be more successful than someone who is not organized. The theory is presented as an objective theory according to communication theory standards. Examples of how organization can lead to success in school, travel, and filmmaking are provided. The conclusion restates that there is a relationship between organization and success, and the theory meets the criteria of an objective theory.
Probability Theory and Mathematical Statisticsmetamath
This document provides information about a Probability Theory and Mathematical Statistics course taught at KNITU, Russia. It includes details about the course such as the number of students, preliminary courses required, distribution of working time, topics covered in lectures and workshops/laboratories. It also compares the methodology and topics studied in this course to a similar course taught at TUT, Finland. Key differences highlighted include the use of Matlab at TUT and more emphasis on practical work/tutorials versus lectures. Overall competencies covered are also summarized and compared between the two courses based on the SEFI framework.
The document discusses key dimensions of organization structure including complexity, formalization, and centralization. Complexity refers to the degree of differentiation within an organization based on factors like specialization, departmentalization, the number of hierarchical levels, and geographic dispersion. Formalization is the degree to which jobs are standardized through techniques like selection processes, role requirements, rules/procedures, training, and rituals. Centralization refers to the concentration of decision-making at a single point, while decentralization disperses decision-making throughout levels of the organization to allow for faster response to changing conditions.
The document outlines topics related to probability theory including: probability, random variables, probability distributions, expected value, variance, moments, and joint distributions. It then provides definitions and examples of these concepts. The key topics covered are random variables and their probability distributions, expected values (mean and variance), and considering two random variables jointly.
This document provides an overview of organization theory from a textbook. It defines organization theory as the study of how organizations are structured and designed. Organization theory takes a macro view of organizations, focusing on structure and effectiveness, while organizational behavior takes a micro view of individual and group behavior. The document discusses key concepts in organization theory including structure, systems perspective, life cycles, and open vs. closed systems. It also outlines the typical stages in an organization's life cycle from entrepreneurial to decline.
This document discusses organizational theory and development, defining it as using social and procedural methodologies to identify and guide corporate needs in order to define an organization's identity and enhance its ability to change and improve effectiveness. It outlines classical, humanistic, and open systems theories, emphasizing a hybrid approach. It also discusses participative management styles, elements of organizational change management, and the importance of defining organizational goals through a mission and vision. The conclusion advocates for a team-based structure allowing employee input to optimize performance and dedication through lateral communication, a strong theory, committed management, and a clear mission and vision.
The document summarizes the key contributors to classical organizational theory, including Frederick Taylor's scientific management theories, Henri Fayol's administrative management principles, Luther Gulick's expansion of Fayol's management functions, and Max Weber's ideal bureaucracy. It discusses some of their major ideas, such as Taylor's time and motion studies, Fayol's 14 management principles, Gulick's addition of budgeting as the 7th management function, and Weber's classification of authority and characteristics of rational-legal authority. The human relations movement emerged from the Hawthorne experiments in the 1920s-1930s, shifting focus to social and psychological factors.
Este documento describe un proceso de ventas en 7 pasos: 1) preparación, 2) introducción, 3) identificación de necesidades mediante preguntas, 4) presentación de la oferta, 5) manejo de objeciones, 6) cierre de la venta, y 7) seguimiento. El objetivo es aumentar la efectividad en ventas a través de una estrategia sencilla que guíe al vendedor en cada etapa del proceso.
The document discusses various types of organizational structures including functional, divisional, matrix, and network structures. It provides details on each structure type, including their advantages and disadvantages. For example, it explains that a functional structure groups people based on expertise, while a divisional structure groups them according to products, markets, or customers. A matrix structure allows dual grouping by function and product.
The document defines probability as the ratio of desired outcomes to total outcomes. It provides examples of calculating probabilities of outcomes from rolling a die or flipping a coin. It explains that probabilities of all outcomes must sum to 1. It also discusses calculating probabilities of multiple events using "and" or "or", and defines experimental probability as the ratio of outcomes to trials from an experiment.
There are several theories which explain the organization and its structure .Classical organization theory includes the scientific management approach, Weber's bureaucratic approach, and administrative theory.
This document discusses probability and its approaches. It defines probability as the likelihood of an event occurring, expressed as a number between 0 and 1. The three main probability approaches are classical, relative frequency, and subjective. Classical probability relies on assumptions like equal likelihood, relative frequency uses experimental data, and subjective is based on personal judgment. Conditional probability is the likelihood of one event given another has occurred.
The document discusses key concepts in probability theory including:
1. Probability theory provides tools to quantify uncertainties and assign probabilities using classical, relative frequency, and subjective approaches.
2. Key probability terms are defined such as experiment, event, sample space, independent and dependent events, mutually exclusive events, and union, intersection, and complement of events.
3. Basic probability rules are covered including the multiplication rule for independent events, addition rule for mutually exclusive events, and how to calculate the probability of events.
This document discusses basic probability concepts including sample spaces, events, counting rules, and probability definitions. It begins by defining a sample space as the set of all possible outcomes of an experiment. Events are defined as subsets of the sample space. Basic counting rules like the multiplication rule, permutations, and combinations are introduced. Probability is defined as a way to quantify the likelihood of events occurring. The document provides examples and explanations of these fundamental probability topics.
This document provides an introduction to probability concepts including:
- Random experiments, sample spaces, events, and set operations used to define events.
- Interpretations and axioms of probability, and examples of assigning probabilities.
- Conditional probability defined as the probability of one event occurring given that another event has occurred, and the formula for calculating conditional probabilities.
- Independence of events defined as events whose joint probability equals the product of their individual probabilities, and examples.
- The law of total probability derived from partitioning events, used to calculate probabilities of complex events.
This document provides an introduction to probability. It defines probability as a measure of how likely an event is to occur. Probability is expressed as a ratio of favorable outcomes to total possible outcomes. The key terms used in probability are defined, including event, outcome, sample space, and elementary events. The theoretical approach to probability is discussed, where probability is predicted without performing the experiment. Random experiments are described as those that may not produce the same outcome each time. Laws of probability are presented, such as a probability being between 0 and 1. Applications of probability in everyday life are mentioned, such as reliability testing of products. Two example probability problems are worked out.
The normal probability curve is a bell-shaped curve that is used to represent probability distributions of many random variables. Some key properties of the normal curve are:
1) Approximately 68% of values fall within one standard deviation of the mean, 95% within two standard deviations, and 99% within three standard deviations.
2) The curve is perfectly symmetrical, with the mean, median, and mode all being the same value.
3) It approaches but never touches the x-axis and theoretically extends from negative to positive infinity. Standard deviation is used to measure deviations from the mean.
The document provides an introduction to probability. It defines probability as a numerical index of the likelihood of an event occurring between 0 and 1. Examples are given where probability is expressed as a percentage or decimal. Key terms are defined, including experiment, outcome, event, and sample space. Common types of probability such as subjective, objective/classic, and empirical probabilities are explained. Formulas and examples are provided to demonstrate how to calculate probabilities of events.
The document provides an overview of key concepts in probability, including definitions of terms like sample space, event, and probability of an event. It also covers rules for calculating probabilities, such as the addition rule, complementary rule, and product rule for independent and dependent events. Examples are given to demonstrate calculating probabilities using these rules for events like coin tosses, card draws, and dice rolls.
This document provides an overview of key concepts in probability theory, including:
1) Empirical and theoretical probabilities, which are determined through experimentation/observation and mathematical calculations respectively.
2) Compound events and the addition rule, where the probability of two events occurring is the sum of their individual probabilities minus the probability of both occurring.
3) The multiplication rule for independent events, where the probability of two independent events occurring is the product of their individual probabilities.
4) Dependent events and Bayes' theorem, where the probability of one event is affected by another occurring.
This document introduces probability and discusses different approaches to defining it. It notes that probability is used to describe variability and uncertainty when outcomes are not certain. Three common definitions of probability are discussed - classical, relative frequency, and subjective - along with their limitations. The document advocates treating probability as a mathematical system defined by axioms rather than worrying about numerical values until a specific application. It then outlines how to construct probability models using sample spaces and assigning probabilities to events based on their composition of simple events.
This document provides an overview of key concepts in probability, including:
1) Sample spaces and events, such as mutually exclusive, independent, and complementary events.
2) Calculating probabilities of simple and compound events using classical, empirical, and subjective interpretations.
3) Determining joint, marginal, and conditional probabilities using formulas and contingency tables.
4) Applying rules of probability, such as addition and multiplication rules, to calculate probabilities of independent and dependent events.
Probability is a numerical measure of how likely an event is to occur. It is used in business to quantify uncertainty and make predictions. Some common applications in business include predicting sales based on price changes, estimating increases in productivity from new methods, and assessing the likelihood of investments being profitable. Probability is calculated on a scale of 0 to 1, with values closer to 1 indicating an event is more certain or likely to occur.
class 11th maths chapter probability cbseTanishqDada
1) The document defines probability as the ratio of favorable outcomes to total possible outcomes of a random experiment consisting of mutually exclusive, exhaustive and equally likely outcomes.
2) Key terms are defined, including random experiment, sample space, sample point, event, exhaustive outcomes, mutually exclusive outcomes, and equally likely outcomes. Independent and dependent events are also discussed.
3) Formulas for probability, including the addition theorem, multiplication theorem, and conditional probability, are presented along with illustrative examples.
The document provides an overview of the binomial distribution including its basics, prerequisites, and examples. It defines a binomial experiment as having a fixed number of independent trials where each trial results in one of two possible outcomes (success or failure) with a constant probability. The document gives examples of flipping a coin and throwing a die to illustrate binomial experiments. It also provides notation used in binomial distributions and shows how to determine if an experiment follows a binomial distribution.
This document provides an introduction to probability, conditional probability, and random variables. It defines key concepts such as sample space, simple events, probability distribution, discrete and continuous random variables, and their properties including mean, variance, and Bernoulli trials. Examples are given for each concept to illustrate their calculation and application to experiments with outcomes that are either certain or random.
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2. What do you mean by probably ?
meaning not sure ! Or are un-certain
e.g. probably it will rain today or probably
Pakistan will win the match. In such and
many other situations, there is an element of
un-certainty in our statements.
In Statistics, we have a technique or tool which is
used to measure the amount of un-certainty. i.e.
PROBABILITY.
3. The range of this numerical measure is from zero to one.
i.e. if Ei is any event, then
0 ≤ P(Ei) ≤ 1 for each i
The word Probability has two basic meanings:
1: It is the quantitative measure of un-certainty
2: Its the measure of degree of belief in a
particular statement.
Example of a melon picked from a carton.
4. The first meaning is related to the problems
of objective approach.
The second is related to the problems of
subjective approach.
5. There is a situation, where probability is an
inherent part e.g. we have:
Population Sample
Here we have full Here we have
Information and have partial information
certain results. and have un-certain
results.
6. As we know, mostly the problems are solved
using the sample data.
A statistical technique which deals with the
sample Results is statistical inference.
Un-certainty (due to sample data) is also an
inherent part of statistical inference.
7. Statistics and Probability theory
constitutes a branch of mathematics for
dealing with uncertainty
Probability theory provides a basis for the
science of statistical inference from
data
8. …a random experiment is an action or
process that leads to one of several possible
outcomes. For example:
6.
8
Experiment Outcomes
Flip a coin Heads, Tails
Exam Marks Numbers: 0, 1, 2, ..., 100
Roll a die 1,2,3,4,5,6
Course Grades F, D, C, B, A, A+
9. List the outcomes of a random experiment…
List: “Called the Sample Space”
Outcomes: “Called the Simple Events”
This list must be exhaustive, i.e. ALL
possible outcomes included.
Die roll {1,2,3,4,5} Die roll {1,2,3,4,5,6}
6.9
10. The list must be mutually exclusive,
i.e. no two outcomes can occur at the
same time:
Die roll {odd number or even number}
Die roll{ number less than 4 or even
number}
11. A list of exhaustive
[don’t leave anything out]
and
mutually exclusive outcomes
[impossible for 2 different events to occur in
the same experiment]
is called a sample space and is denoted by
S.
12. A usual six-sided die has a sample space
S={1,2,3,4,5,6}
If two dice are rolled ( or, equivalently,
if one die is rolled twice)
The sample space is shown in Figure 1.2.
13.
14. An individual outcome of a sample space is
called a simple event
[cannot break it down into several other
events],
An event is a collection or set of one or
more simple events in a sample space.
6.14
15. Roll of a die: S = {1, 2, 3, 4, 5, 6}
Simple event: the number “3” will be rolled
Compound Event: an even number
(one of 2, 4, or 6) will be rolled
16. The probability of an event is the sum of the
probabilities of the simple events that constitute
the event.
E.g. (assuming a fair die) S = {1, 2, 3, 4, 5, 6}
and
P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6
Then:
P(EVEN) = P(2) + P(4) + P(6) = 1/6 + 1/6 +1/6
= 3/6 = 1/2
6.16
17. Types of an Event
Impossible
Event
Possible
Event
Simple Event Compound Event
Certain Event
Mutually
Exclusive Equally
Likely
Exhaustive Dependent Independent
19. A desired subset of a sample space
containing at least one possible outcome.
Example
When a die is rolled once, Sample Space is
S={ 1,2,3,4,5,6}
Event E is defined as an odd number
appears on upper side of a die
E={1,3,5}
20. A desired subset of a sample space
containing only one possible outcome.
Example
Event D is defined as head will turn up on a
coin
D={ H}
21. A desired subset of a sample space
containing at least two possible outcome.
Example
Event E is defined as an odd number
appears on upper side of a die
E={1,3,5}
22. A desired sub set of sample space
containing all possible outcomes. This
event is also called certain event.
Example
Any number from 1 to 6 appears on upper
side of a die.
S={1,2,3,4,5,6}
23. Mutually Exclusive Events
Two events A and B defined in a sample
space S and have nothing common between
them are called mutually exclusive events.
Example
The events A (number 4 appear on a die)
and B( an odd number appear on a die) are
two mutually exclusive events.
24. Equally Likely Events
Two events A and B defined in simple
sample space S and their chances of
occurrences are equal are called equally
likely events.
Example
The events A (head will turn up on a coin)
and B (tail will turn up on a coin).
25. Two events A and B defined in simple sample
space S
(i) Nothing common between them.
(ii) Their union is same as the sample space.
Example
The events A(head will turn up on a coin) and B
( tail will turn up on a coin)
are two exhaustive events as A intersection B is
empty set as well as their union is S.
26. Two events A and B are independent events
if occurrence or non occurrence of one does
not affect the occurrence or non-occurrence
of the other.
27. Example: The event
A (get king card in first attempt)
and
B(get queen card in second attempt )
when two cards are to be drawn in
succession replacing the first drawn card in
pack before second draw.
28. Two events A and B are dependent events if
occurrence or non occurrence of one affect
the occurrence or non-occurrence of the
other.
29. Example : The event
A (get king card in first attempt)
and
B(get queen card in second attempt )
when two cards are to be drawn in
succession not replacing the first drawn
card in pack before second draw
30. There are three ways to assign a probability,
P(Ei), to an outcome, Ei, namely:
Classical approach:
make certain assumptions (such as equally
likely, independence) about situation.
Relative frequency:
assigning probabilities based on
experimentation or historical data.
Subjective approach:
Assigning probabilities based on the assignor’s
judgment.
6.30
31. If an experiment has “n” possible outcomes
[all equally likely to occur], this method
would assign a probability of 1/n to each
outcome.
Experiment: Rolling a die
Sample Space: S = {1, 2, 3, 4, 5, 6}
Probabilities: Each sample point has a 1/6
chance of occurring.
6.31
32. If an experiment results in ‘n’ mutually
exclusive, equally likely and exhaustive
events and if ‘m’ of these are in favor of the
occurrence of an event ‘A’, then probability
of event A is written as:
P(A)=m / n
34. Relative frequency of occurrence is based
on actual observations, it defines probability
as the number of times, the relevant event
occurs, divided by the number of times the
experiment is performed in a large number
of trials.
35. Bits & Bytes Computer Shop tracks the number
of desktop computer systems it sells over a
month (30 days):
For example,
10 days out of 30
2 desktops were sold.
From this we can construct
the 111111111probabilities of an event
(i.e. the # of desktop sold on a given day)…
6.35
Desktops Sold # of Days
0 1
1 2
2 10
3 12
4 5
36. “There is a 40% chance Bits & Bytes will sell 3 desktops on any
given day” [Based on estimates obtained from sample of 30
days]
6.36
Deskto
ps Sold
[X]
# of
Days
Relative
frequency
Desktops Sold
0 1 1/30 = .03 .03 =P(X=0)
1 2 2/30 = .07 .07 = P(X=1)
2 10 10/30 = .33 .33 = P(X=2)
3 12 12/30 = .40 .40 = P(X=3)
4 5 5/30 = .17 .17 = P(X=4)
30 ∑ = 1.00 ∑ = 1.00
37. “In the subjective approach we define
probability as the degree of belief that we
hold in the occurrence of an event”
P(you drop this course)
P(NASA successfully land a man on the
moon)
P(girlfriend says yes when you ask her to
marry you)
6.37
38. We study methods to determine probabilities
of events that result from combining other
events in various ways.
6.38
39. There are several types of combinations and
relationships between events:
Complement of an event [everything other
than that event]
Intersection of two events [event A and event B]
or [A*B]
Union of two events [event A or event B]
or [A+B]
6.39
40. Why are some mutual fund managers more
successful than others? One possible factor
is where the manager earned his or her
MBA.
The following table compares mutual fund
performance against the ranking of the
school where the fund manager earned their
MBA: Where do we get these probabilities
from? [population or sample?]
6.40
41. Venn Diagrams
6.41
Mutual fund outperforms
the market
Mutual fund doesn’t
outperform the market
Top 20 MBA program .11 .29
Not top 20 MBA program .06 .54
E.g. This is the probability that a mutual fund
outperforms AND the manager was in a top-20 MBA
program; it’s a joint probability [intersection].
42. Alternatively, we could introduce shorthand
notation to represent the events:
A1 = Fund manager graduated from a top-20 MBA program
A2 = Fund manager did not graduate from a top-20 MBA program
B1 = Fund outperforms the market
B2 = Fund does not outperform the market
6.42
B1 B2
A1
.11 .29
A2
.06 .54
E.g. P(A2 and B1) = .06
= the probability a fund outperforms the market
and the manager isn’t from a top-20 school.
43. Marginal probabilities are computed by
adding across rows and down columns; that is
they are calculated in the margins of the table:
6.43
B1 B2 P(Ai)
A1
.11 .29 .40
A2
.06 .54 .60
P(Bj) .17 .83 1.00
P(B1) = .11 + .06
P(A2) = .06 + .54
“what’s the probability a fund
outperforms the market?”
“what’s the probability a fund
manager isn’t from a top school?”
BOTH margins must add to 1
(useful error check)
44. Conditional probability is used to determine
how two events are related; that is, we can
determine the probability of one event given the
occurrence of another related event.
6.44
45. Experiment: random select one student in
class.
P(randomly selected student is male/student
is on 3rd row) =
Conditional probabilities are written as
P(A/ B) and read as “the probability of
A given B”
6.45
46. Again, the probability of an event given that
another event has occurred is called a
conditional probability…
P( A and B) = P(A)*P(B/A) = P(B)*P(A/B)
both are true
6.46
47. Events:
A1 = Fund manager graduated from a top-20 MBA
program
A2 = Fund manager did not graduate from a top-20
MBA program
B1 = Fund outperforms the market
B2 = Fund does not outperform the market
6.47
B1 B2
A1
.11 .29
A2
.06 .54
48. Example 2 • What’s the probability that a fund will
outperform the market given that the manager
graduated from a top-20 MBA program?
Recall:
A1 = Fund manager graduated from a top-20 MBA program
A2 = Fund manager did not graduate from a top-20 MBA
program
B1 = Fund outperforms the market
B2 = Fund does not outperform the market
Thus, we want to know “what is P(B1 | A1) ?”
6.48
49. We want to calculate P(B1 | A1)
6.49
Thus, there is a 27.5% chance that that a fund will outperform the
market given that the manager graduated from a top-20 MBA
program.
B1 B2 P(Ai)
A1
.11 .29 .40
A2
.06 .54 .60
P(Bj) .17 .83 1.00
50. One of the objectives of calculating conditional
probability is to determine whether two events are
related.
In particular, we would like to know whether they are
independent, that is, if the probability of one event
is not affected by the occurrence of the other event.
Two events A and B are said to be independent if
P(A|B) = P(A)
and
P(B|A) = P(B)
6.50
51. For example, we saw that
P(B1 | A1) = .275
The marginal probability for B1 is: P(B1) = 0.17
Since P(B1|A1) ≠ P(B1), B1 and A1 are not
independent events.
Stated another way, they are dependent. That
is, the probability of one event (B1) is affected
by the occurrence of the other event (A1).
6.51
52. We introduce three rules that enable us to calculate
the probability of more complex events from the
probability of simpler events…
The Complement Rule – May be easier to calculate
the probability of the complement of an event and
then substract it from 1.0 to get the probability of the
event.
P(at least one head when you flip coin 100 times)
= 1 – P(0 heads when you flip coin 100 times)
The Multiplication Rule: P(A*B)
The Addition Rule: P(A+B)
6.52
53. Multiplication Rule for Independent Events: Let A and B be two
independent events, then
( ) ( ) ( ).P A B P A P B
Examples:
• Flip a coin twice. What is the probability of observing two heads?
• Flip a coin twice. What is the probability of getting a head and then a
tail? A tail and then a head? One head?
• Three computers are ordered. If the probability of getting a “working”
computer is .7, what is the probability that all three are “working” ?
54. A graduate statistics course has seven male and
three female students. The professor wants to select
two students at random to help her conduct a
research project. What is the probability that the two
students chosen are female?
P(F1 * F2) = ???
Let F1 represent the event that the first student is
female
P(F1) = 3/10 = .30
What about the second student?
P(F2 /F1) = 2/9 = .22
P(F1 * F2) = P(F1) * P(F2 /F1) = (.30)*(.22) = 0.066
NOTE: 2 events are NOT independent.
6.54
55. The professor in Example is unavailable. Her
replacement will teach two classes. His style is
to select one student at random and pick on
him or her in the class. What is the probability
that the two students chosen are female?
Both classes have 3 female and 7 male
students.
P(F1 * F2) = P(F1) * P(F2 /F1) = P(F1) * P(F2)
= (3/10) * (3/10) = 9/100 = 0.09
NOTE: 2 events ARE independent.
6.55
56. Addition rule provides a way to compute the
probability of event A or B or both A and B
occurring; i.e. the union of A and B.
P(A or B) = P(A + B) = P(A) + P(B) –
P(A and B)
Why do we subtract the joint probability P(A
and B) from the sum of the probabilities of A
and B?
6.56
P(A or B) = P(A) + P(B) – P(A and B)
57. P(A1) = .11 + .29 = .40
P(B1) = .11 + .06 = .17
By adding P(A) plus P(B) we add P(A and B) twice. To
correct we subtract P(A and B) from P(A) + P(B)
6.57
B1 B2 P(Ai)
A1
.11 .29 .40
A2
.06 .54 .60
P(Bj) .17 .83 1.00
P(A1 or B1) = P(A) + P(B) –P(A and B) = .40 + .17 - .11 = .46
B1
A1
58. If and A and B are mutually exclusive the
occurrence of one event makes the other one
impossible. This means that
P(A and B) = P(A * B) = 0
The addition rule for mutually exclusive events
is
P(A or B) = P(A) + P(B)
Only if A and B are Mutually Exclusive.
6.58
59. In a large city, two newspapers are published, the Sun
and the Post. The circulation departments report that
22% of the city’s households have a subscription to the
Sun and 35% subscribe to the Post. A survey reveals
that 6% of all households subscribe to both newspapers.
What proportion of the city’s households subscribe to
either newspaper?
That is, what is the probability of selecting a household at
random that subscribes to the Sun or the Post or both?
P(Sun or Post) = P(Sun) + P(Post) – P(Sun and Post)
= .22 + .35 – .06 = .51
6.59