This presentation is about the topic PROBABILITY. Details of this topic, starting from basic level and slowly moving towards advanced level , has been discussed in this presentation.
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
Math 1300: Section 8-1 Sample Spaces, Events, and ProbabilityJason Aubrey
The document discusses probability theory and its application to random experiments. It defines key terms like sample space, event, simple event, and compound event. The sample space is the set of all possible outcomes of an experiment. Events are subsets of the sample space and can be simple (contain one outcome) or compound (contain multiple outcomes). An example experiment is rolling two dice, where the sample space is the set of all possible (number on die 1, number on die 2) outcomes.
This document provides definitions and concepts related to probability:
- An experiment is an uncertain process with outcomes making up a sample space. An event is a subset of outcomes.
- Theoretical probability is the ratio of outcomes in an event to the total possible outcomes. Relative frequency is the ratio of times an event occurs in trials to the total number of trials.
- Venn diagrams represent relationships between sets graphically with overlapping regions.
- The union of sets contains elements in either set. The intersection contains elements in both sets. Probability of a union equals the sum of individual probabilities minus the intersection probability.
- Mutually exclusive events cannot occur together. Complementary events cover the sample space and have probabilities that
Introduction to Probability and Probability DistributionsJezhabeth Villegas
This document provides an overview of teaching basic probability and probability distributions to tertiary level teachers. It introduces key concepts such as random experiments, sample spaces, events, assigning probabilities, conditional probability, independent events, and random variables. Examples are provided for each concept to illustrate the definitions and computations. The goal is to explain the necessary probability foundations for teachers to understand sampling distributions and assessing the reliability of statistical estimates from samples.
This document provides examples and explanations of key concepts in probability, including:
1) Probability is a number between 0 and 1 that indicates the likelihood of an event. Experimental probability is calculated from observations, while theoretical probability uses the composition of a sample space.
2) Tree diagrams and the fundamental counting principle can be used to determine the number of possible outcomes and probabilities in multi-stage experiments.
3) Union, intersection, and complements of events are probability concepts used to calculate probabilities of combined events.
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.
The document discusses elementary theorems and concepts related to conditional probability, including:
1. Theorems for calculating the probability of unions and intersections of events.
2. The definition of conditional probability as the probability of an event A given that another event B has occurred.
3. Bayes' theorem, which provides a formula for calculating the probability of an event A given event B in terms of probabilities of events B given A.
Mathematics for Language Technology: Introduction to Probability TheoryMarina Santini
This document provides an introduction to probability theory. It begins with an outline of key topics like events, axioms, and theorems of probability. Probability theory analyzes random phenomena using mathematical models based on these concepts. It is important for developing computational models in natural language processing, which often rely on probabilistic reasoning. The document provides examples of defining events and calculating probabilities, as well as quizzes and an activity to illustrate probability concepts like addition rules.
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
Math 1300: Section 8-1 Sample Spaces, Events, and ProbabilityJason Aubrey
The document discusses probability theory and its application to random experiments. It defines key terms like sample space, event, simple event, and compound event. The sample space is the set of all possible outcomes of an experiment. Events are subsets of the sample space and can be simple (contain one outcome) or compound (contain multiple outcomes). An example experiment is rolling two dice, where the sample space is the set of all possible (number on die 1, number on die 2) outcomes.
This document provides definitions and concepts related to probability:
- An experiment is an uncertain process with outcomes making up a sample space. An event is a subset of outcomes.
- Theoretical probability is the ratio of outcomes in an event to the total possible outcomes. Relative frequency is the ratio of times an event occurs in trials to the total number of trials.
- Venn diagrams represent relationships between sets graphically with overlapping regions.
- The union of sets contains elements in either set. The intersection contains elements in both sets. Probability of a union equals the sum of individual probabilities minus the intersection probability.
- Mutually exclusive events cannot occur together. Complementary events cover the sample space and have probabilities that
Introduction to Probability and Probability DistributionsJezhabeth Villegas
This document provides an overview of teaching basic probability and probability distributions to tertiary level teachers. It introduces key concepts such as random experiments, sample spaces, events, assigning probabilities, conditional probability, independent events, and random variables. Examples are provided for each concept to illustrate the definitions and computations. The goal is to explain the necessary probability foundations for teachers to understand sampling distributions and assessing the reliability of statistical estimates from samples.
This document provides examples and explanations of key concepts in probability, including:
1) Probability is a number between 0 and 1 that indicates the likelihood of an event. Experimental probability is calculated from observations, while theoretical probability uses the composition of a sample space.
2) Tree diagrams and the fundamental counting principle can be used to determine the number of possible outcomes and probabilities in multi-stage experiments.
3) Union, intersection, and complements of events are probability concepts used to calculate probabilities of combined events.
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.
The document discusses elementary theorems and concepts related to conditional probability, including:
1. Theorems for calculating the probability of unions and intersections of events.
2. The definition of conditional probability as the probability of an event A given that another event B has occurred.
3. Bayes' theorem, which provides a formula for calculating the probability of an event A given event B in terms of probabilities of events B given A.
Mathematics for Language Technology: Introduction to Probability TheoryMarina Santini
This document provides an introduction to probability theory. It begins with an outline of key topics like events, axioms, and theorems of probability. Probability theory analyzes random phenomena using mathematical models based on these concepts. It is important for developing computational models in natural language processing, which often rely on probabilistic reasoning. The document provides examples of defining events and calculating probabilities, as well as quizzes and an activity to illustrate probability concepts like addition rules.
This document introduces key concepts in probability including:
- Random events have uncertain outcomes but a regular distribution appears with large numbers of trials.
- Probability is the proportion of times an outcome would occur with many trials.
- Set theory concepts like unions, intersections, and complements are used to define sample spaces and calculate probabilities.
- The three basic probability rules are that probabilities lie between 0 and 1, the probabilities of all outcomes sum to 1, and the probability of an event's complement is 1 minus the probability of the event.
Lecture: Joint, Conditional and Marginal Probabilities Marina Santini
The document discusses joint, conditional, and marginal probabilities. It begins with an introduction to joint and conditional probabilities, defining conditional probability as the probability of event A given event B. It then presents the multiplication rule for calculating joint probabilities from conditional probabilities and marginal probabilities. The document provides examples and calculations to illustrate these probability concepts. It concludes with short quizzes to test understanding of applying the multiplication rule.
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.
The document provides a history of the development of probability theory from its origins in the 16th century to modern applications. Some of the key contributors and advances mentioned include:
- Cardan wrote one of the earliest works on probability in dice rolls and games of chance in 1550.
- Pascal and Fermat laid the foundations of probability theory in correspondence solving gambling problems in 1654.
- Graunt analyzed mortality data and made predictions, gaining access to the Royal Society of London.
- Huygens published the first text on probability theory in 1657 introducing mathematical expectation.
- Laplace's 1812 work outlined the evolution of probability theory and presented key theorems, establishing it as a rigorous
1) A probabilistic experiment is one where more than one outcome is possible and the outcome is uncertain. The sample space is the set of all possible outcomes.
2) Elementary events are the individual outcomes, while compound events are unions of elementary events. Probability axioms state that probabilities of events must be between 0 and 1, the probability of the sample space is 1, and the probabilities of disjoint events sum to the probability of their union.
3) There are three fundamental theorems of probability: the probability of the empty set is 0; the probability of the entire sample space is 1; and the probability of the union of two events equals the sum of their probabilities minus the probability of their intersection.
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.
This document describes a course on mathematical foundations for communication engineering. The course objectives are to develop understanding of probability theory, random variables, and sequences of random variables. Key topics covered include probability, random variables, functions of random variables, special distributions, sequences of random variables, and random processes. Assessment is based on assignments, tests, exams, attendance, and a final exam weighting 60%. The course aims to enable students to apply probability concepts in practical problems and communication systems analysis.
Probability And Probability Distributions Sahil Nagpal
This document provides an overview of key concepts in probability and probability distributions. It defines important terms like probability, sample space, events, mutually exclusive events, independent events, and conditional probability. It also covers rules of probability like addition rules, complement rules, and Bayes' theorem. Finally, it introduces discrete and continuous random variables and discusses properties of discrete probability distributions like expected value and standard deviation.
This document provides an overview of key concepts in probability theory:
- An experiment yields possible outcomes called a sample space. Events are subsets of outcomes. Random variables assign values to outcomes.
- Probability is a measure of certainty that an event will occur, ranging from 0 (impossible) to 1 (certain). It can be defined in different ways.
- The frequentist definition is the limit of relative frequencies of an event over many trials. The Bayesian definition is a degree of belief in an event. The Laplacian definition assumes all outcomes are equally likely initially.
- Examples demonstrate random variables, events, and calculating probabilities based on the sample space and outcomes of an experiment. Key terms like sample space, event,
The document discusses the concept of probability, including defining it as a measure of likelihood between 0 and 1, and how to calculate probabilities using concepts like sample spaces, favorable outcomes, and total possible outcomes. It provides examples of calculating probabilities from experiments involving dice, cards, and coins. The document also outlines some applications of probability theory in areas like risk assessment, commodity markets, product reliability, and environmental regulation.
The document provides an introduction to probability. It defines probability as a measure of how likely an event is. Probability is expressed as the ratio of favorable outcomes to total possible outcomes. The concept of probability originated in the 16th century and has been developed by many mathematicians. Today, probability theory has applications in fields like science, economics, and engineering. The document also defines key probability terms like events, outcomes, sample space, and theoretical probability. It provides examples of calculating probability for experiments like tossing coins, rolling dice, and drawing cards.
This document defines key concepts in probability and provides examples. It discusses probability vocabulary like sample space, outcome, trial, and event. It defines probability as the number of times a desired outcome occurs over total trials. Events are independent if the outcome of one does not impact others, and mutually exclusive if they cannot occur together. The addition and multiplication rules for probability are explained. Conditional probability describes the probability of a second event depending on the first occurring. Counting techniques are discussed for finding total possible outcomes of combined experiments. Review questions are provided to test understanding of the material.
This document discusses probability and related concepts such as sample spaces, classical probability, empirical probability, mutually exclusive and non-mutually exclusive events, independent and dependent events, and conditional probability. It provides examples to illustrate these concepts using problems involving coins, dice, and manufacturing companies. Bayes' theorem is also introduced as a way to calculate reverse conditional probabilities.
Probability can be calculated using three approaches: a priori, empirical, and subjective. A priori probability is based on prior knowledge, empirical on observed outcomes, and subjective on personal analysis. Simple events have a single characteristic while joint events have multiple. The sample space includes all possible outcomes. Conditional probability is the probability of one event given another. Independence means one event does not affect another's probability.
The document defines key concepts in probability theory, such as probability experiments, outcomes, sample spaces, events, classical probability, empirical probability, and subjective probability. It provides examples of how to calculate probabilities of simple and compound events using classical probability methods, including determining probabilities using fractions or decimals and interpreting "and" and "or" probabilities.
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 provides an overview of key concepts in probability, including classical, frequentist, and subjective definitions of probability. It discusses sample spaces, events, mutually exclusive and independent events, and the rules of addition, multiplication, total probability, and Bayes' theorem. It also covers applications of probability, such as screening tests, and how concepts like sensitivity, specificity, and predictive values are used to evaluate screening tests using Bayesian reasoning.
This document provides an overview of key probability concepts:
1. It defines posterior probability, Bayes' theorem, subjective and objective probability, and the multiplication and addition rules of probability.
2. It explains key probability terms like experiments, events, outcomes, permutations, and combinations, and describes classical, empirical and subjective approaches to probability.
3. It provides examples of how to calculate probabilities using the rules of addition, multiplication, and Bayes' theorem, as well as how to apply concepts like conditional probability, joint probability, and tree diagrams.
The document discusses probability and experiments with random outcomes. It defines key probability concepts like sample space, events, and probability functions. It provides examples of common experiments with finite sample spaces like coin tosses, die rolls, and card draws. It also discusses experiments with infinite discrete sample spaces like repeated coin tosses until the first tail. The document establishes the basic properties and rules of probability, including that it is a function between 0 and 1, that probabilities of disjoint events add, and that probabilities of subsets are less than the original set.
This document introduces key concepts in probability including:
- Random events have uncertain outcomes but a regular distribution appears with large numbers of trials.
- Probability is the proportion of times an outcome would occur with many trials.
- Set theory concepts like unions, intersections, and complements are used to define sample spaces and calculate probabilities.
- The three basic probability rules are that probabilities lie between 0 and 1, the probabilities of all outcomes sum to 1, and the probability of an event's complement is 1 minus the probability of the event.
Lecture: Joint, Conditional and Marginal Probabilities Marina Santini
The document discusses joint, conditional, and marginal probabilities. It begins with an introduction to joint and conditional probabilities, defining conditional probability as the probability of event A given event B. It then presents the multiplication rule for calculating joint probabilities from conditional probabilities and marginal probabilities. The document provides examples and calculations to illustrate these probability concepts. It concludes with short quizzes to test understanding of applying the multiplication rule.
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.
The document provides a history of the development of probability theory from its origins in the 16th century to modern applications. Some of the key contributors and advances mentioned include:
- Cardan wrote one of the earliest works on probability in dice rolls and games of chance in 1550.
- Pascal and Fermat laid the foundations of probability theory in correspondence solving gambling problems in 1654.
- Graunt analyzed mortality data and made predictions, gaining access to the Royal Society of London.
- Huygens published the first text on probability theory in 1657 introducing mathematical expectation.
- Laplace's 1812 work outlined the evolution of probability theory and presented key theorems, establishing it as a rigorous
1) A probabilistic experiment is one where more than one outcome is possible and the outcome is uncertain. The sample space is the set of all possible outcomes.
2) Elementary events are the individual outcomes, while compound events are unions of elementary events. Probability axioms state that probabilities of events must be between 0 and 1, the probability of the sample space is 1, and the probabilities of disjoint events sum to the probability of their union.
3) There are three fundamental theorems of probability: the probability of the empty set is 0; the probability of the entire sample space is 1; and the probability of the union of two events equals the sum of their probabilities minus the probability of their intersection.
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.
This document describes a course on mathematical foundations for communication engineering. The course objectives are to develop understanding of probability theory, random variables, and sequences of random variables. Key topics covered include probability, random variables, functions of random variables, special distributions, sequences of random variables, and random processes. Assessment is based on assignments, tests, exams, attendance, and a final exam weighting 60%. The course aims to enable students to apply probability concepts in practical problems and communication systems analysis.
Probability And Probability Distributions Sahil Nagpal
This document provides an overview of key concepts in probability and probability distributions. It defines important terms like probability, sample space, events, mutually exclusive events, independent events, and conditional probability. It also covers rules of probability like addition rules, complement rules, and Bayes' theorem. Finally, it introduces discrete and continuous random variables and discusses properties of discrete probability distributions like expected value and standard deviation.
This document provides an overview of key concepts in probability theory:
- An experiment yields possible outcomes called a sample space. Events are subsets of outcomes. Random variables assign values to outcomes.
- Probability is a measure of certainty that an event will occur, ranging from 0 (impossible) to 1 (certain). It can be defined in different ways.
- The frequentist definition is the limit of relative frequencies of an event over many trials. The Bayesian definition is a degree of belief in an event. The Laplacian definition assumes all outcomes are equally likely initially.
- Examples demonstrate random variables, events, and calculating probabilities based on the sample space and outcomes of an experiment. Key terms like sample space, event,
The document discusses the concept of probability, including defining it as a measure of likelihood between 0 and 1, and how to calculate probabilities using concepts like sample spaces, favorable outcomes, and total possible outcomes. It provides examples of calculating probabilities from experiments involving dice, cards, and coins. The document also outlines some applications of probability theory in areas like risk assessment, commodity markets, product reliability, and environmental regulation.
The document provides an introduction to probability. It defines probability as a measure of how likely an event is. Probability is expressed as the ratio of favorable outcomes to total possible outcomes. The concept of probability originated in the 16th century and has been developed by many mathematicians. Today, probability theory has applications in fields like science, economics, and engineering. The document also defines key probability terms like events, outcomes, sample space, and theoretical probability. It provides examples of calculating probability for experiments like tossing coins, rolling dice, and drawing cards.
This document defines key concepts in probability and provides examples. It discusses probability vocabulary like sample space, outcome, trial, and event. It defines probability as the number of times a desired outcome occurs over total trials. Events are independent if the outcome of one does not impact others, and mutually exclusive if they cannot occur together. The addition and multiplication rules for probability are explained. Conditional probability describes the probability of a second event depending on the first occurring. Counting techniques are discussed for finding total possible outcomes of combined experiments. Review questions are provided to test understanding of the material.
This document discusses probability and related concepts such as sample spaces, classical probability, empirical probability, mutually exclusive and non-mutually exclusive events, independent and dependent events, and conditional probability. It provides examples to illustrate these concepts using problems involving coins, dice, and manufacturing companies. Bayes' theorem is also introduced as a way to calculate reverse conditional probabilities.
Probability can be calculated using three approaches: a priori, empirical, and subjective. A priori probability is based on prior knowledge, empirical on observed outcomes, and subjective on personal analysis. Simple events have a single characteristic while joint events have multiple. The sample space includes all possible outcomes. Conditional probability is the probability of one event given another. Independence means one event does not affect another's probability.
The document defines key concepts in probability theory, such as probability experiments, outcomes, sample spaces, events, classical probability, empirical probability, and subjective probability. It provides examples of how to calculate probabilities of simple and compound events using classical probability methods, including determining probabilities using fractions or decimals and interpreting "and" and "or" probabilities.
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 provides an overview of key concepts in probability, including classical, frequentist, and subjective definitions of probability. It discusses sample spaces, events, mutually exclusive and independent events, and the rules of addition, multiplication, total probability, and Bayes' theorem. It also covers applications of probability, such as screening tests, and how concepts like sensitivity, specificity, and predictive values are used to evaluate screening tests using Bayesian reasoning.
This document provides an overview of key probability concepts:
1. It defines posterior probability, Bayes' theorem, subjective and objective probability, and the multiplication and addition rules of probability.
2. It explains key probability terms like experiments, events, outcomes, permutations, and combinations, and describes classical, empirical and subjective approaches to probability.
3. It provides examples of how to calculate probabilities using the rules of addition, multiplication, and Bayes' theorem, as well as how to apply concepts like conditional probability, joint probability, and tree diagrams.
The document discusses probability and experiments with random outcomes. It defines key probability concepts like sample space, events, and probability functions. It provides examples of common experiments with finite sample spaces like coin tosses, die rolls, and card draws. It also discusses experiments with infinite discrete sample spaces like repeated coin tosses until the first tail. The document establishes the basic properties and rules of probability, including that it is a function between 0 and 1, that probabilities of disjoint events add, and that probabilities of subsets are less than the original set.
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.
This document provides an introduction to probability theory, including key concepts such as:
- The foundational definitions of probability put forth by Pascal and Fermat.
- Key terms like sample space, trial, random experiment, and classical definition of probability.
- Important probability rules including addition rule, mutually exclusive events, complements of events, conditional probability, and multiplication theorem.
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Probability is the one of the most important topics in engineering because it helps us to understand some aspects of the future of an event. Probability is not only used in mathematics but also is various domains of engineering.
This document provides an introduction to probability and some key probability concepts. It discusses how probability can be used to measure the chance of outcomes in random experiments and defines key terms like sample space, events, equally likely events, unions and intersections of events. It also presents the classical approach to defining probability mathematically as the ratio of favorable outcomes to total possible outcomes when conditions like equally likely outcomes are met. Several examples are provided to illustrate probability calculations.
1. The document discusses probability and chance experiments. It provides examples to illustrate key concepts such as sample space, events, and how to calculate probabilities.
2. One example examines student food preferences in a cafeteria, with the sample space consisting of all possible combinations of student gender and food line choice.
3. The document also covers conditional probability, explaining how to calculate the probability of an event given that another event has occurred. An example calculates the probability of nausea given being seated in the front of a bus.
PROBABILITY
Defn:
Probability is a branch of mathematics which deals with and shows how to measure these uncertainties of events in every day life. It provides a quantitative occurrences and situations. In other words. It is a measure of chances.
MATHS PRESENTATION OF STATISTICS AND PROBABILITY.pptxpavantech57
This document provides an introduction to probability and related concepts. It includes the following sections:
1. An introduction to probability, including the probability formula.
2. Methods for assigning probabilities, including classical, empirical, and subjective approaches.
3. Discussion of events and their probabilities, including independent/dependent events and impossible/certain events.
4. An explanation of conditional probability and examples calculating conditional probabilities.
5. A definition and derivation of Bayes' Theorem, which describes the probability of an event based on prior knowledge of conditions related to that event.
This document defines key concepts in probability, including experiments, outcomes, sample spaces, events, unions and intersections of events, complements of events, mutually exclusive events, and the probability of independent and dependent events. It provides examples to illustrate these concepts, such as calculating the probability of drawing cards from a deck or marbles from a box. Formulas are given for calculating probabilities of unions, intersections, complements, mutually exclusive and inclusive events.
This document discusses key concepts in probability theory, including:
- Probability models random phenomena that may have deterministic or non-deterministic outcomes.
- The sample space defines all possible outcomes, and an event is any subset of outcomes.
- Probability is defined as the number of outcomes in an event divided by the total number of outcomes, if the sample space is finite and all outcomes are equally likely.
- Rules of probability include addition for mutually exclusive events and complement rules. Conditional probability adjusts probabilities based on additional information. Independence means events do not impact each other's probabilities.
This document provides an overview of probability concepts including chance experiments, sample spaces, events, Venn diagrams, independence, conditional probability, and Bayes' rule. Key points covered include defining probability as a limit of relative frequency, and how to calculate probabilities of events using formulas like the addition rule, multiplication rule, and Bayes' rule. Examples are provided to illustrate concepts like conditional probability and working through word problems step-by-step.
The document provides information about probability and statistics concepts including:
1) Mathematical, statistical, and axiomatic definitions of probability are given along with examples of mutually exclusive, equally likely, and independent events.
2) Laws of probability such as addition law, multiplication law, and total probability theorem are defined and formulas are provided.
3) Concepts of random variables, discrete and continuous random variables, probability mass functions, probability density functions, and expected value are introduced.
This document provides an introduction to probability and statistical concepts using R. It defines key terms like random variables, sample space, events, and probability. It discusses definitions of probability, conditional probability, independent and dependent events. It provides examples of calculating probabilities for things like coin tosses, dice rolls, and card draws. It also introduces Bayes' theorem and provides examples of how to calculate conditional probabilities using this approach. Finally, it discusses how naive Bayes classification works in machine learning by applying Bayes' theorem.
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event.
This document covers key concepts in probability, including sample spaces, events, rules of probability such as addition and multiplication, and conditional probability. It defines probability as the proportion of times an outcome occurs in a long series of repetitions. It introduces terminology like the sample space, events, and rules for calculating probabilities of individual events and combined events, depending on whether they are disjoint, independent, or conditional. Several examples demonstrate how to apply the rules of probability, addition, multiplication, and conditional probability to calculate probabilities in finite sample spaces.
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2. Salient features of this course
Briefly covers the basic concepts and makes understanding
easy and really quick
Makes you familiar with the problem solving techniques
Makes you deal easily and quickly with the term
“PROBABILITY”
Its really fun and lots more to learn here !!!
3. What are we going to do in this course???
Understanding the
word “PROBABILITY”
Step-
1 Deal with
some
formulaes
Step-
2
Solve
Examples
Step-
3
5. DEFINITION
PROBABILITY is the extent to which something is likely to happen or be the case in an event.
It is the ratio of the desired outcomes to the total outcomes.
So, if n(e) = desired outcomes , t(e) = total outcomes from an event
then we can say P(e) = n(e)/t(e)
6. Some terms related to PROBABILITY
Sample Space
In probability theory, the sample
space of an experiment or random
trial
is the set of all possible outcomes or
results of that experiment.
Event
In probability theory, an event is a
set
of outcomes of an experiment (a
subset of the sample space) to which
a probability is assigned.
7. So lets take some EXAMPLES
If I roll a dice, then total number of outcomes will be six.
The possible outcomes can be {1,2,3,4,5,6}
So, probability of getting a 1 while throwing a dice = 1/6 as desired number of
outcomes = n(e) = 1 and total number of possible outcomes = t(e) = 6.
So, according to the formulae P(e) = n(e)/t(e) , we get 1/6 .
Similarly, Probability of getting a 2 = Probability of getting a 3 = Probability of
getting a 4 = Probability of getting a 5 = Probability of getting a 6 = 1/6 .
8. LETS take a PRACTICE
If I toss a coin, then what is the probability that I will get a head?
Also calculate the probability to get a tail.
9. Answer to previous question
If I toss a coin then the possible outcomes are either head or tail.
So, total number of possible outcomes = t(e) = 2
Now, probability of getting a head = P(e) = n(e)/t(e) = ½
Similarly, Probability of getting a tail = ½.
Here, we see both of them add to 1.
Thus, our answer is correct.
10. Two or more events at the same time
In case of “AND” term
If the AND term is used while describing the outcomes of an event,
then we have to multiply the probabilities of the outcomes of the
event.
11. Practise problem
If I throw a disc and toss a coin at the same time, what is the
probability that I will get a 5 and heads??
Analysing the problem:-
Two Simultaneous events– Throwing a disc, toss a coin
Here, the term AND is used in case of outcomes
So, we have to multiply the corresponding probabilities.
Probability of getting a 5 = 1/6
Probability of getting a heads = ½
So, required probability = P(5 and heads) = 1/6*1/2 =1/12
12. Two or more events at the same time
In case of “OR” term
If the OR term is used while describing the outcomes of an event,
then we have to add the probabilities of the outcomes of the
event.
13. Practise problem
If I throw a disc and toss a coin at the same time, what is the probability
that I will get a 5 or heads??
Analysing the problem:-
Two Simultaneous events– Throwing a disc, toss a coin
Here, the term OR is used in case of outcomes
So, we have to add the corresponding probabilities.
Probability of getting a 5 = 1/6
Probability of getting a heads = ½
So, required probability = P(5 or heads) = 1/6+1/2 = 2/3
14. Discussion of Basic concepts from Set Theory
The probability that Events A and B both occur is the probability of
the intersection of A and B. The probability of the intersection of Events A and
B is denoted by P (A ∩ B).
The probability that either Event A or B occur is the probability of
the union of A and B. The probability of the union of Events A and
B is denoted by P(A U B).
The compliment of Event A is A
It is a set of all outcomes that are not in A.
When A and B have no outcomes in common, then they are said to be mutually exclusive
or mutually disjoint events.
c
15. ILLUSTRATION
Suppose, A, B and C are three events.
A = {1,5,7,9}
B = {2,3,4,6,8}
C = {5,7,6}
So, (A U B) = {1,2,3,4,5,6,7,8,9}
(A ∩ B) = {0} as they have no elements in common. So, they are mutually
exclusive events.
A = {2,3,4,6,8}, elements not present in A.
Also, (A ∩ C) = {5,7}. So, they are not mutually exclusive.
c
16. PROPERTIES of PROBABILITY
Probability of an event A is symbolized by P(A). Probability of an event A is lies between 0 ≤ P(A) ≤ 1.
The first property of probability is that the probability of the entire sample space is one. Symbolically we
write P (S ) = 1
The second property of probability deals with mutually exclusive events. If E1 and E2 are mutually exclusive
events, meaning that they have an empty intersection, then P (E1 U E2 ) = P (E1) + P(E2). If they are not
mutually exclusive then
P (E1 U E2 ) = P (E1) + P (E2) - P (E1 ∩ E2 )
As long as this occurs, the probability of the union of the events is the same as the sum of the probabilities:
P (E1 U E2 U . . . U En ) = P (E1) + P (E2) + . . . + En
17. PROPERTIES of PROBABILITY
According to third property, if we denote the complement of the event E by E C then,
E and E C have an empty intersection and are mutually exclusive.
Furthermore E U E C = S, the entire sample space.
Thus, we can say , 1 = P (S ) = P (E U E C) = P (E ) + P (E C) .
Rearranging above equation ,we get P (E ) = 1 - P (E C).
If A, B and C are not independent or mutually exclusive then the union probability is given by:
P(A∪B∪C) =P(A) + P(B) + P(C) - P(A∩B) - P(B∩C) - P(A∩C) - P(A∩B∩C)
18. PROPERTIES of PROBABILITY
INDEPENDENT EVENTS
Two events are said to be independent of each other when the
probability that one event occurs in no way affects the probability of the
other event occurring. An example of two independent events is as
follows; say you rolled a die and flipped a coin.
If A and B are two independent events, then P(A ∩ B) = P(A)*P(B)
Similarly, P(A ∩ B ∩ C) = P(A)*P(B)*P(C) provided A,B,C are independent set of
events.
19. CONDITIONAL PROBABILITY
Definition: If E and F are two events associated with the same sample space of a random
experiment, the conditional probability of the event E given that F has occurred, i.e. P (E|F) is given
by P(E|F) = P (E ∩ F) / P(F) provided P(F) ≠ 0
Example
If P(A) = 7 /13 , P(B) = 9 /13 and P(A ∩ B) = 4 /13 , evaluate P(A|B).
Solution : We have P(A|B) = P(A ∩ B) / P(B) = (4/13)/(9/13) = 4/9
20. Total Probability Theorem
Statement: Let {E1 , E2 ,...,En } be a partition of the sample space S, and suppose that each of
the events E1 , E2 ,..., En has nonzero probability of occurrence. Let A be any event associated with
S , then
P(A) = P(E1 ) P(A|E1 ) + P(E2 ) P(A|E2 ) + ... + P(En ) P(A|En ) = 𝒋=𝟏
𝒏
𝑷 𝑬𝒋 𝑷( A|Ej )
21. Problem
A person has undertaken a construction job. The probabilities are 0.65 that there will be strike, 0.80 that the
construction job will be completed on time if there is no strike, and 0.32 that the construction job will be completed
on
time if there is a strike. Determine the probability that the construction job will be completed on time.
Solution Let A be the event that the construction job will be completed on time, and B be the event that there
will be a
strike.
So, We have to find P(A).
We have P(B) = 0.65,
Hence, P(no strike) = P(B′) = 1 − P(B) = 1 − 0.65 = 0.35
Also, P(A|B) = 0.32, P(A|B′) = 0.80
Since events B and B′ form a partition of the sample space S, therefore, by theorem of total probability, we have
P(A) = P(B) P(A|B) + P(B′) P(A|B′) = 0.65 × 0.32 + 0.35 × 0.8 = 0.208 + 0.28 = 0.488
Thus, the probability that the construction job will be completed in time is 0.488.
22. BAYES’ THEOREM
If E1 , E2 ,..., En are n non-empty events which constitute a partition of
sample space S,
i.e. E1 , E2 ,..., En are pairwise disjoint and ( E1 ∪ E2 ∪ ... ∪ En ) = S and A is
any event
of non-zero probability, then
P(Ei|A)=
𝑷 𝑬𝒊 𝑷 𝑨 𝑬𝒊
𝒋=𝟏
𝒏 𝑷 𝑬𝒋 𝑷(𝑨|𝑬𝒋)
for any i = 1, 2, 3, ..., n
This is Bayes’ Theorem.
23. Example
Bag I contains 3 red and 4 black balls while another Bag II contains 5 red and 6 black balls. One ball is drawn at
random
from one of the bags and it is found to be red. Find the probability that it was drawn from Bag II.
Solution : Let E1 be the event of choosing the bag I, E2 the event of choosing the bag II and A be the event of drawing
a red
ball.
Then P(E1 ) = P(E2 ) = 1/2 .
Also P(A|E1 ) = P(drawing a red ball from Bag I) = 3/7 and
P(A|E2 ) = P(drawing a red ball from Bag II) = 5/11
Now, the probability of drawing a ball from Bag II, being given that it is red, is P(E2 |A) By using Bayes' theorem, we
have
P(Ei|A)=
𝑷 𝑬𝒊 𝑷 𝑨 𝑬𝒊
𝒋=𝟏
𝒏 𝑷 𝑬𝒋 𝑷(𝑨|𝑬𝒋)
= (½ * 5/11)/(1/2 * 3/7 + ½ * 5/11) = 35/68