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
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.
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 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.
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 document discusses conditional probability and independence. It defines conditional probability as P(B|A), the probability of event B given that event A has occurred. To calculate this, one considers only the outcomes where A occurred and calculates the fraction where B also occurred. Two events A and B are independent if P(B|A) = P(B), meaning the probability of B is unaffected by the occurrence of A. The general multiplication rule for any events A and B is P(A and B) = P(A) × P(B|A) or P(A) × P(B|A). Disjoint events cannot be independent as the occurrence of one rules out the other. Conditional probabilities are best understood
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.
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.
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.
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 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.
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 document discusses conditional probability and independence. It defines conditional probability as P(B|A), the probability of event B given that event A has occurred. To calculate this, one considers only the outcomes where A occurred and calculates the fraction where B also occurred. Two events A and B are independent if P(B|A) = P(B), meaning the probability of B is unaffected by the occurrence of A. The general multiplication rule for any events A and B is P(A and B) = P(A) × P(B|A) or P(A) × P(B|A). Disjoint events cannot be independent as the occurrence of one rules out the other. Conditional probabilities are best understood
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.
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.
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.
The document provides an introduction to probability. It discusses:
- What probability is and the definition of probability as a number between 0 and 1 that expresses the likelihood of an event occurring.
- A brief history of probability including its development in French society in the 1650s and key figures like James Bernoulli, Abraham De Moivre, and Pierre-Simon Laplace.
- Key terms used in probability like events, outcomes, sample space, theoretical probability, empirical probability, and subjective probability.
- The three types of probability: theoretical, empirical, and subjective probability.
- General probability rules including: the probability of impossible/certain events; the sum of all probabilities equaling 1; complements
This document provides an overview of key probability concepts including:
(1) Definitions of random experiments, sample spaces, events, and probability;
(2) The addition and multiplication theorems and conditional probability;
(3) Mathematical expectation and probability distributions including the binomial, Poisson, and normal distributions. Examples are provided to illustrate key terminology and formulas.
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.
It is a consolidation of basic probability concepts worth understanding before attempting to apply probability concepts for predictions. The material is formed from different sources. ll the sources are acknowledged.
The document provides an overview of key probability concepts including:
1. Random experiments, sample spaces, events, and the classification of events as simple, mutually exclusive, independent, and exhaustive.
2. The three main approaches to defining probability: classical, relative frequency, and subjective.
3. Important probability theorems like the addition rule, multiplication rule, and Bayes' theorem.
4. How to calculate probabilities of events using these theorems, including examples of finding probabilities of independent, dependent, mutually exclusive, and conditional events.
This document defines and explains key probability concepts such as the cumulative distribution function (CDF), expectation, mean, variance, and properties of the CDF. The CDF measures the probability that a random variable X assumes a value less than or equal to x. Expectation, also called the mean or first moment, is the expected value of a random variable X. Variance is a measure of how spread out the possible values of a random variable are around the mean.
: Random Variable, Discrete Random variable, Continuous random variable, Probability Distribution of Discrete Random variable, Mathematical Expectations and variance of a discrete random variable.
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.
1. continuous probability distribution
2. Normal Distribution
3. Application of Normal Dist
4. Characteristics of normal distribution
5.Standard Normal Distribution
The document discusses various probability distributions including the binomial, Poisson, and normal distributions. It provides definitions and key properties of each distribution. It also discusses sampling with and without replacement as well as the Monte Carlo method for simulating physical systems using random sampling. The Monte Carlo method can be used to computationally estimate values like pi by simulating the throwing of darts at a circular target.
The document provides an introduction to probability theory, including definitions of key terms like trial, event, exhaustive events, favorable events, independent events, mutually exclusive events, and equally likely events. It discusses three approaches to defining probability: classical, statistical, and axiomatic. The classical approach defines probability as the ratio of favorable cases to total possible cases. The statistical approach determines probabilities based on empirical observations over many trials. The axiomatic approach uses set theory and axioms to define probability without restrictions of previous approaches.
Conditional probability is the probability of an 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. An example is given of calculating the probability of drawing two white balls in succession from an urn without replacement. The formula for conditional probability is derived as the probability of events A and B occurring divided by the probability of A. This is demonstrated using an example of finding the percentage of friends who like chocolate that also like strawberry.
Some Basic concepts of Probability along with advanced concepts on Medical probability & Probability in Gambling. A lot of Sample Questions and Practice Questions will help you understand and apply the concepts in real life.
This presentation guide you through Basic Probability Theory and Statistics, those are Random Experiment, Sample Space, Random Variables, Probability, Conditional Probability, Variance, Probability Distribution, Joint Probability Distribution, Conditional Probability Distribution (CPD) and Factor.
For more topics stay tuned with Learnbay.
This document discusses probability theory and its applications. It begins by defining probability as a measure of how likely an event is to occur between 0 and 1. It then provides examples of calculating theoretical probability for simple events like a coin toss or dice roll. The document goes on to explain how probability theory is applied in many areas such as mathematics, statistics, science, and engineering. It provides examples of using probability for risk assessment in fields like finance, biology, and engineering reliability. Finally, it discusses how probability assessments influence decisions and have changed society.
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.
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 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.
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.
The document provides an introduction to probability. It discusses:
- What probability is and the definition of probability as a number between 0 and 1 that expresses the likelihood of an event occurring.
- A brief history of probability including its development in French society in the 1650s and key figures like James Bernoulli, Abraham De Moivre, and Pierre-Simon Laplace.
- Key terms used in probability like events, outcomes, sample space, theoretical probability, empirical probability, and subjective probability.
- The three types of probability: theoretical, empirical, and subjective probability.
- General probability rules including: the probability of impossible/certain events; the sum of all probabilities equaling 1; complements
This document provides an overview of key probability concepts including:
(1) Definitions of random experiments, sample spaces, events, and probability;
(2) The addition and multiplication theorems and conditional probability;
(3) Mathematical expectation and probability distributions including the binomial, Poisson, and normal distributions. Examples are provided to illustrate key terminology and formulas.
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.
It is a consolidation of basic probability concepts worth understanding before attempting to apply probability concepts for predictions. The material is formed from different sources. ll the sources are acknowledged.
The document provides an overview of key probability concepts including:
1. Random experiments, sample spaces, events, and the classification of events as simple, mutually exclusive, independent, and exhaustive.
2. The three main approaches to defining probability: classical, relative frequency, and subjective.
3. Important probability theorems like the addition rule, multiplication rule, and Bayes' theorem.
4. How to calculate probabilities of events using these theorems, including examples of finding probabilities of independent, dependent, mutually exclusive, and conditional events.
This document defines and explains key probability concepts such as the cumulative distribution function (CDF), expectation, mean, variance, and properties of the CDF. The CDF measures the probability that a random variable X assumes a value less than or equal to x. Expectation, also called the mean or first moment, is the expected value of a random variable X. Variance is a measure of how spread out the possible values of a random variable are around the mean.
: Random Variable, Discrete Random variable, Continuous random variable, Probability Distribution of Discrete Random variable, Mathematical Expectations and variance of a discrete random variable.
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.
1. continuous probability distribution
2. Normal Distribution
3. Application of Normal Dist
4. Characteristics of normal distribution
5.Standard Normal Distribution
The document discusses various probability distributions including the binomial, Poisson, and normal distributions. It provides definitions and key properties of each distribution. It also discusses sampling with and without replacement as well as the Monte Carlo method for simulating physical systems using random sampling. The Monte Carlo method can be used to computationally estimate values like pi by simulating the throwing of darts at a circular target.
The document provides an introduction to probability theory, including definitions of key terms like trial, event, exhaustive events, favorable events, independent events, mutually exclusive events, and equally likely events. It discusses three approaches to defining probability: classical, statistical, and axiomatic. The classical approach defines probability as the ratio of favorable cases to total possible cases. The statistical approach determines probabilities based on empirical observations over many trials. The axiomatic approach uses set theory and axioms to define probability without restrictions of previous approaches.
Conditional probability is the probability of an 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. An example is given of calculating the probability of drawing two white balls in succession from an urn without replacement. The formula for conditional probability is derived as the probability of events A and B occurring divided by the probability of A. This is demonstrated using an example of finding the percentage of friends who like chocolate that also like strawberry.
Some Basic concepts of Probability along with advanced concepts on Medical probability & Probability in Gambling. A lot of Sample Questions and Practice Questions will help you understand and apply the concepts in real life.
This presentation guide you through Basic Probability Theory and Statistics, those are Random Experiment, Sample Space, Random Variables, Probability, Conditional Probability, Variance, Probability Distribution, Joint Probability Distribution, Conditional Probability Distribution (CPD) and Factor.
For more topics stay tuned with Learnbay.
This document discusses probability theory and its applications. It begins by defining probability as a measure of how likely an event is to occur between 0 and 1. It then provides examples of calculating theoretical probability for simple events like a coin toss or dice roll. The document goes on to explain how probability theory is applied in many areas such as mathematics, statistics, science, and engineering. It provides examples of using probability for risk assessment in fields like finance, biology, and engineering reliability. Finally, it discusses how probability assessments influence decisions and have changed society.
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.
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 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.
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.
1 Probability Please read sections 3.1 – 3.3 in your .docxaryan532920
1
Probability
Please read sections 3.1 – 3.3 in your textbook
Def: An experiment is a process by which observations are generated.
Def: A variable is a quantity that is observed in the experiment.
Def: The sample space (S) for an experiment is the set of all possible outcomes.
Def: An event E is a subset of a sample space. It provides the collection of outcomes
that correspond to some classification.
Example:
Note: A sample space does not have to be finite.
Example: Pick any positive integer. The sample space is countably infinite.
A discrete sample space is one with a finite number of elements, { }1,2,3,4,5,6 or one that
has a countably infinite number of elements { }1,3,5,7,... .
A continuous sample space consists of elements forming a continuum. { }x / 2 x 5< <
2
A Venn diagram is used to show relationships between events.
A intersection B = (A ∩ B) = A and B
The outcomes in (A intersection B) belong to set A as well as to set B.
A union B = (A U B) = A alone or B alone or both
Union Formula
For any events A, B, P (A or B) = P (A) + P (B) – P (A intersection B) i.e.
P (A U B) = P (A) + P (B) – P (A ∩ B)
3
cA complement not A A ' A A = = = =
A complement consists of all outcomes outside of A.
Note: P (not A) = 1 – P (A)
Def: Two events are mutually exclusive (disjoint, incompatible) if they do not intersect,
i.e. if they do not occur at the same time. They have no outcomes in common.
When A and B are mutually exclusive, (A ∩ B) = null set = Ø, and P (A and B) = 0.
Thus, when A and B are mutually exclusive, P (A or B) = P (A) + P (B)
(This is exactly the same statement as rule 3 below)
Axioms of Probability
Def: A probability function p is a rule for calculating the probability of an event. The
function p satisfies 3 conditions:
1) 0 ≤ P (A) ≤1, for all events A in the sample space S
2) P (Sample Space S) = 1
3) If A, B, C are mutually exclusive events in the sample space S, then
P(A B C) P(A) P(B) P(C)∪ ∪ = + +
4
The Classical Probability Concept: If there are n equally likely possibilities, of which one
must occur and s are regarded as successes, then the probability of success is s
n
.
Example:
Frequency interpretation of Probability: The probability of an event E is the proportion of
times the event occurs during a long run of repeated experiments.
Example:
Def: A set function assigns a non-negative value to a set.
Ex: N (A) is a set function whose value is the number of elements in A.
Def: An additive set function f is a function for which f (A U B) = f (A) + f (B) when A and
B are mutually exclusive.
N (A) is an additive set function.
Ex: Toss 2 fair dice. Let A be the event that the sum on the two dice is 5. Let B be the
event that the sum on ...
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.
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.
1) The document provides four probability problems involving combinations and permutations to further understanding of combinatorics. It gives the solutions and explanations for each problem involving probabilities of card hands.
2) It then introduces conditional probability and uses examples like the Monty Hall problem to illustrate how conditioning on additional information can change a probability. It provides the definition of conditional probability and proves Bayes' theorem.
3) The document discusses independence of events and uses examples like coin flips and natural disasters to demonstrate independence. It also introduces the continuity of probability and uses examples like the Cantor set to illustrate how it allows calculating probabilities of infinite sets.
This document provides study material for a course on probability and statistics. It covers topics such as sample spaces, events, axioms of probability, conditional probability, Bayes' theorem, random variables, probability distributions including binomial, Poisson, normal and other continuous distributions, joint and marginal distributions, mathematical expectation, decision making, sampling distributions and statistical inference. Various examples are provided to illustrate concepts such as probability calculations for events from finite sample spaces, conditional probability, independence of events and finding probabilities of unions and intersections of events.
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.
Introduction to Discrete Probabilities with Scilab - Michaël Baudin, Consort...Scilab
This document provides an introduction to discrete probabilities with Scilab. It begins with definitions of sets, including union, intersection, complement, difference, and cross product. It then defines discrete distribution functions and probability of events. Properties of probabilities are discussed, such as the probability of a union of disjoint events being the sum of the individual probabilities. The document also covers conditional probability and Bayes' formula. Examples using a six-sided die are provided throughout to illustrate the concepts.
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This document defines key concepts in probability theory such as experiments, sample spaces, events, and conditional probability. It provides examples to illustrate these concepts like calculating the probability of drawing certain numbers when rolling a die. Formulas are given for computing probabilities of unions, intersections, and complements of events. The document also discusses classical and axiomatic approaches to defining probability and credits contributors like Laplace, Kolmogorov. In short, it serves as an introduction to fundamental probability concepts and terminology.
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.
The document provides an overview of elementary probability concepts including:
- Defining probability as the chance of an event occurring and explaining common probability notions.
- Introducing key probability terms like sample space, events, outcomes, and complementary/intersection of events.
- Explaining counting rules like the addition rule, multiplication rule, and how to calculate permutations and combinations.
- Outlining different approaches to defining probability including classical, subjective, axiomatic, and frequency-based definitions.
- Detailing several probability rules like calculating the probability of a union of events and applying the addition rule to non-mutually exclusive events.
The document provides an overview of elementary probability concepts including:
- Defining probability as the chance of an event occurring and explaining common probability notions.
- Introducing key probability terms like sample space, events, outcomes, and complementary/intersection of events.
- Explaining counting rules like the addition rule, multiplication rule, and how to calculate permutations and combinations.
- Outlining different approaches to defining probability including classical, subjective, axiomatic, and frequency-based definitions.
- Detailing several probability rules like calculating the probability of a union of events and applying the addition rule to non-mutually exclusive events.
This document discusses probability theories and formulas. It defines probability as a branch of mathematics dealing with random phenomena and introduces key concepts like experiments, sample spaces, events, and equally likely probabilities. It provides examples of experiments like coin tosses, dice rolls, and drawing balls from an urn. The document also covers basic probability formulas like addition rule, complementary events, disjoint and independent events. It gives examples of calculating probabilities and introduces the conditional probability formula.
- Probability theory studies possible outcomes of events and their likelihoods, expressed as a value from 0 to 1.
- Probability can be understood as the chance of an outcome, often expressed as a percentage between 0 and 100%.
- The analysis of data using probability models is called statistics.
This document provides information about a probability and statistics course including the textbook, reference book, instructor, and an overview of key probability concepts like sample space, events, axioms of probability, joint probability, conditional probability, Bayes' theorem, statistical independence, and an example probability problem.
I am Jacinta Lawrence Currently associated with statisticsassignmenthelp.com as a Probability and Statistics assignment helper. After completing my master's from Princeton University, USA. I was in search of an opportunity that would expand my area of knowledge hence I decided to help students with their assignments. I have written several statistics assignments to date to help students overcome numerous difficulties they face.
This document discusses various methods of proving mathematical propositions, including direct proof, indirect proof, proof by contradiction, proof by cases, and proof by mathematical induction. It provides examples to illustrate each method. Direct proof involves directly deducing the conclusion from the given statements, while indirect proof establishes an equivalent proposition. Proof by contradiction assumes the negation of the statement to be proved and arrives at a contradiction. Proof by cases considers all possible cases of the hypothesis. Mathematical induction proves a statement for all natural numbers based on proving it for the base case and assuming it is true for some arbitrary case k.
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How to Manage Your Lost Opportunities in Odoo 17 CRMCeline George
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Odoo provides an option for creating a module by using a single line command. By using this command the user can make a whole structure of a module. It is very easy for a beginner to make a module. There is no need to make each file manually. This slide will show how to create a module using the scaffold method.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
Reimagining Your Library Space: How to Increase the Vibes in Your Library No ...Diana Rendina
Librarians are leading the way in creating future-ready citizens – now we need to update our spaces to match. In this session, attendees will get inspiration for transforming their library spaces. You’ll learn how to survey students and patrons, create a focus group, and use design thinking to brainstorm ideas for your space. We’ll discuss budget friendly ways to change your space as well as how to find funding. No matter where you’re at, you’ll find ideas for reimagining your space in this session.
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Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
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Answers about how you can do more with Walmart!"
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This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
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Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
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Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
2. Lecture Outline
1. Basic Probability Theory
2. Conditional Probability
3. Independence
4. Philosophical Foundations
5. Subjective Probability Elicitation
6. Heuristics and Biases in Human Probability Judgment
5. Sample spaces
DEFINITION 1.1. Let S = {s1,s2,...,sn} be a finite set of possible outcomes in
a context C. S is a (finite) sample space for C iff exactly one outcome
among the elements of S is or will be true in C.
EXAMPLE 1.2: Let C be the particular flipping of a coin. Then…
▪ S = {Heads, Tails} is a sample space for C.
▪ Another sample space for C is S' = {Heads is observed, Tails is
observed, Cannot observe whether the coin is heads or tails}.
▪ Yet another is S'' = {Heads is observed and someone coughs, Heads is
observed and no one coughs, Tails is observed whether someone
coughs or not}.
6. Event spaces
DEFINITION 1.3. Let S be a sample space, and ⊘ ≠ E ⊆ 2S (E is a
nonempty subset of the power set of S, i.e., it is a set of subsets of S). Then
E is an event space (or algebra of events) on S iff for every A,B ∈ E:
(a) S A = AC ∈ E (the S-complement of A is in E)
and
(b) A∪B ∈ E (the union of A and B is in E).
We call the elements of E consisting of single elements of S atomic events.
EXAMPLE 1.4: If S = {Heads, Tails}, then E = {⊘, {Heads}, {Tails}, {Heads,
Tails}} is an event space on S. The atomic events are {Heads} and {Tails}.
7. Probability measures
DEFINITION 1.5. Let S be a sample space and E an event space on S.
Then a function P: E →[0,1] is a (finitely additive) probability measure on E
iff for every A,B ∈ E:
(a) P(S) = 1
and
(b) If A∩B = ⊘ (the intersection of A and B is empty, in which case we
say that A and B are disjoint events), then P(A∪B) = P(A) + P(B)
(additivity).
The triple <S,E,P> is called a (finitely additive) probability space.
A B
S
8. Basic corollaries
COROLLARY 1.6. Binary complementarity. If <S,E,P> is a finitely additive
probability space, then for all A,B ∈ E:
P(AC) = 1 – P(A).
Proof.
S = A∪AC by the definition of complementarity. Therefore P(A∪AC) = 1 by
Definition 1.5(a). A and AC are disjoint by the definition of complementarity,
so by 1.5(b), P(A∪AC) = P(A) + P(AC), so P(A) + P(AC) = 1 and the result
follows by subtracting P(A) from both sides of the equation.
AACS
9. Basic corollaries
COROLLARY 1.7. If <S,E,P> is a finitely additive probability space, then for
all A,B ∈ E:
P(⊘) = 0.
Proof.
SC = S S = ⊘. Thus P(⊘) = P(SC) = 1 – P(S) by Corollary 1.6, which by
Definition 1.5(a) is 1-1 = 0.
10. Basic corollaries
COROLLARY 1.8. If <S,E,P> is a finitely additive probability space, then for
all A,B ∈ E:
P(A∪B) = P(A) + P(B) - P(A∩B).
Proof.
From set theory, we have A = (A∩B)∪(A∩BC) and B = (B∩A)∪(B∩AC), A∪B
= (A∩B)∪(A∩BC)∪(B∩AC) = (A∩B)∪[(A∩BC)∪(B∩AC)], (A∩B)∩(A∩BC) = ⊘,
(B∩A)∩(B∩AC) = ⊘, and (A∩B)∩[(A∩BC)∩(B∩AC)] = ⊘∩⊘=⊘. Therefore,
P(A)=P[(A∩B)∪(A∩BC)]=P(A∩B)+P(A∩BC) and
P(B)=P[(B∩A)∪(B∩AC)]=P(B∩A)+P(B∩AC), and P(A∪B) =
P{(A∩B)∪{(A∩BC)∪(B∩AC)]} = P(A∩B)+P(A∩BC)+P(B∩AC). Substituting,
P(A∪B) = P(A∩B)+[P(A)-P(A∩B)]+[P(B)-P(B∩A)]. B∩A= A∩B, so P(A∪B) =
P(A)+P(B)-P(A∩B).
11. Basic corollaries
Venn diagram version of the previous proof:
COROLLARY 1.8. If <S,E,P> is a finitely additive probability space, then for
all A,B ∈ E:
P(A∪B) = P(A) + P(B) - P(A∩B).
A BS
12. Basic corollaries
COROLLARY 1.9. Conjunction rule. If <S,E,P> is a finitely additive
probability space, then for all A,B ∈ E:
P(A∩B) ≤ P(A).
Proof. From set theory, we have A = (A∩B)∪(A∩BC), and (A∩B)∩(A∩BC) =
⊘, so for a probability measure, by Definition 1.5(b),
P(A)=P[(A∩B)∪(A∩BC)]=P(A∩B)+P(A∩BC). By the definition of probability,
P(A∩BC) ≥ 0, so P(A) – P(A∩B) = P(A∩BC) ≥ 0. Therefore P(A) ≥ P(A∩B).
A B
S
13. Basic corollaries
PROPOSITION 1.10. Disjunction rule. If <S,E,P> is a finitely additive
probability space, then for all A,B ∈ E:
P(A∪B) ≥ P(A).
EXERCISE 1.11. Prove Proposition 1.10.
15. Definition of conditional probability
DEFINITION 2.1. The conditional probability P(A|B) of an event A given
event B is defined as follows: P(A|B) = P(A∩B) / P(B).
COROLLARY 2.2. P(A∩B) = P(A|B) P(B).
A BS
16. Bayes’s theorem
THEOREM 2.3. For events A and B, P(A|B) = [P(B|A)P(A)] / P(B).
Proof.
• By Definition 2.1, P(A|B) = P(A∩B) / P(B).
• A∩B = B∩A, so P(A∩B) = P(B∩A).
• From Corollary 2.2, P(B∩A) = P(B|A)P(A).
• From Corollary 2.2 again, P(B∩A) = P(A∩B)= P(A|B)P(B).
• Therefore P(A|B)P(B) = P(B|A)P(A) [*]
• The theorem follows if we divide both sides of equation [*] by P(B).
17. Applying Bayes to Medical Diagnosis
EXAMPLE 2.4. (from David Eddy ,1982) Calculation of the probability that a
breast lesion is cancerous based on a positive mammogram:
▪ Eddy estimates the probability that a lesion will be detected through a
mammogram as .792.
▪ Hence the test will turn up negative when cancer is actually present
20.8% of the time.
▪ When no cancer is present, the test produces a positive result 9.6% of
the time (and is therefore correctly negative 90.4% of the time).
▪ The prior probability that a patient who has a mammogram will have
cancer is taken to be 1%.
▪ Thus, the probability of cancer given as positive test as [(.792)(.01)] /
[(.792)(.01) + (.096)(.99)] = .077, applying Theorem 2.3.
▪ So a patient with a positive test has less than an 8% chance of having
breast cancer. Does this seem low to you?
19. Independent events
DEFINITION 3.1. Two events A and B are independent iff P(A∩B) =
P(A)P(B).
COROLLARY 3.2. Two events A and B satisfying P(B)>0 are independent
iff P(A|B) = P(A).
Proof. (a) Only if direction: Since A and B are independent, P(A∩B) =
P(A)P(B) by Definition 3.1. Since P(B)>0, P(A|B) = P(A∩B)/P(B) =
P(A)P(B)/P(B) = P(A). (b) If direction: P(A|B) = P(A), so multiplying both
sides by P(B), P(A|B)P(B) = P(A)P(B) = P(A∩B).
20. Independent coin tosses
EXAMPLE 3.3. Consider two flips of a coin.
▪ Let A={Heads on the first toss} and
B={Tails on the second toss}.
▪ The probability for tails on the second
toss is not affected by what happened
on the first toss and vice versa, so
P(B|A) = P(B) and P(A|B) = P(A).
▪ Assuming both sides of the coin have a
nonzero probability of landing on top,
the tosses are independent, and
therefore, P(A∩B) = P(A)P(B).
▪ Assuming the coin is unbiased
(P({Heads})=P({Tails})=0.5), this means
that P(A∩B) = (.5)(.5) = .25.
21. Dice rolls
EXERCISE 3.4. Consider two six-sided dice (with faces varying from 1 to 6
dots) which are rolled simultaneously, and assume each roll is independent
of the other. What is the probability that the sum of the two dice is 7?
22. Conditional independence
DEFINITION 3.5. Two events A and B are conditionally independent given
event C iff P(A∩B|C) = P(A|C)P(B|C).
PROPOSITION 3.6. Two events A and B are conditionally independent
given event C iff P(A|B∩C) = P(A|C) or P(B|C) = 0.
EXERCISE 3.7. Prove Proposition 3.6.
25. 25
Frequentist view
Toss the coin many times
Observe the proportion of times the
coin comes up Heads versus Tails
Probability for each side is therefore
the observed frequency (proportion)
of that outcome out of the total
number of tosses.
26. 26
Subjectivist view
Make a judgment of how likely you
think it is for this coin to turn up
Heads versus Tails.
Probability represents your relative
degree of belief in one outcome
relative to another.
Also known as the Bayesian view.
28. Numerical method
Q: What do you believe is the
probability that it will rain tomorrow?
A. I would put the probability of rain
tomorrow at _____%.
29. Choice method
Q: Which of the following two events
do you think is more probable?
• Rain tomorrow
• No rain tomorrow
30. Probability wheel
Q: Choose between...
(I) Receiving $50 if it rains tomorrow
(II) Receiving $50 if the arrow lands within the displayed
probability range
31. Procedural Invariance
DEFINITION 5.1. An agent’s stated confidence (elicited subjective
probability) D is procedurally invariant with respect to two procedures ! and
!’ iff the inferred inequality relations >! and >!’ are such that for all events A
and B, D(A) >! D(B) iff D(A) >!’ D(B).
Numerical
Method
Choice
Method
≅
32. Calibration
DEFINITION 5.2. The confidence D of a probability judge is perfectly
calibrated if for all values x ∈ [0,1] , and all events E, if D(E) = x, then the
observed P(E) = x.
For values x ∈ [0.5,1] , if D(E) > P(E), then the judge is said to be
overconfident. If D(E) < P(E), then the judge is said to be underconfident.
33. 6. Heuristics and
Biases of Human
Probability Judgment
Lecture: Probability
SYMSYS 1
Todd Davies
34. Using probability theory to predict probability judgments
EXPERIMENT 6.1. Test of binary complementarity. A number of
experiments reported in Tversky and Koehler (1994) and Wallsten,
Budescu, and Zwick (1992) show that subjects' confidence judgments
generally obey Corollary 1.6, so that D(AC) = 1 – D(A).
▪ The latter authors “presented subjects with 300 propositions concerning
world history and geography (e.g. 'The Monroe Doctrine was proclaimed
before the Republican Party was founded')... (cited in T&K ‘94)
▪ True and false (complementary) versions of each proposition were
presented on different days.”
▪ The average sum probability for propositions and their complements was
1.02, which is insignificantly different from 1.
35. Probability judgment heuristics
A landmark paper by Amos Tversky
and Daniel Kahneman (Science,
1974) argued that human judges
often employ heuristics in making
judgments about uncertain
prospects, in particular:
• Representativeness: A is judged
more likely than B in a context C if
A is more representative of C than
B is.
• Availability: A is judged more
likely than B in context C if
instances of A are easier than of B
to bring to mind in context C.
36. Another experiment…
EXPERIMENT 6.2. Tennis prediction. Tversky and Kahneman (1983):
▪ Subjects evaluated the relative likelihoods that Bjorn Borg (then the most
dominant male tennis player in the world), would (a) win the final match
at Wimbledon, (b) lose the first set, (c) lose the first set but win the
match, and (d) win the first set but lose the match.
>>> How would you rank events (a) through (d) by likelihood?
37. Conjunction fallacy…
EXPERIMENT 6.2. Tennis prediction. Tversky and Kahneman (1983):
▪ Subjects evaluated the relative likelihoods that Bjorn Borg (then the most
dominant male tennis player in the world), would (a) win the final match
at Wimbledon, (b) lose the first set, (c) lose the first set but win the
match, and (d) win the first set but lose the match.
▪ The average rankings (1=most probable, 2 = second most probable, etc.)
were 1.7 for a, 2.7 for b, 2.2 for c, and 3.5 for d.
▪ Thus, subjects on average ranked the conjunction of Borg losing the first
set and winning the match as more likely than that he would lose the first
set (2.2 versus 2.7).
▪ The authors' explanation is that people rank likelihoods based on the
representativeness heuristic, which makes the conjunction of Borg's
losing the first set but winning the match more representative of Borg
than is the proposition that Borg loses the first set.
38. Another conjunction fallacy
EXPERIMENT 6.3. Natural disasters. Tversky and Kahneman asked
subjects to evaluate the probability of occurrence of several events in 1983.
▪ Half of subjects evaluated a basic outcome (e.g. “A massive flood
somewhere in North America in 1983, in which more than 1,000 people
drown”) and the other half evaluated a more detailed scenario leading to
the same outcome (e.g. “An earthquake in California sometime in 1983,
causing a flood in which more than 1,000 people drown.”)
▪ The estimates of the conjunction were significantly higher than those for
the flood.
▪ Thus, scenarios that include a cause-effect story appear more plausible
than those that lack a cause, even though the latter are extensionally
more likely.
▪ The causal story makes the conjunction easier to imagine, an aspect of
the availability heuristic.
39. Disjunction fallacy?
EXERCISE 6.4. Construct an example experiment in which you would expect
subjects to violate the disjunction rule of Proposition 1.10 (and which you were
asked to prove in Exercise 1.11).
40. Is human judgment Bayesian?
This is a highly studied question.
The answer depends on the type of judgement or cognitive task being
performed, and to a lesser extent on the identity of the judge.
See separate lecture in this series, “Are people Bayesian reasoners?”
41. Gambler’s fallacy
EXPERIMENT 6.5. Tversky and Kahneman (1974):
▪ Subjects on average regard the sequence H-T-H-T-T-H of fair coin
tosses to be more likely than the sequence H-H-H-T-T-T, even though
both sequences are equally likely, a result that follows from the
generalized definition of event independence.
▪ People in general regard, for example, a heads toss as more likely after
a long run of tails tosses than after a similar run of tails or a mixed run.
This tendency has been called the “gambler's fallacy”.
▪ Tversky and Kahneman's explanation is that people expect sequences of
tosses to be representative of the process that generates them. H-T-H-T-
T-H is rated more likely than H-H-H-T-T-T because the former
sequences is more typical of fair coin toss sequences generally, in which
H and T are not grouped such that all the Hs precede all the Ts, but
rather Hs and Ts are intermingled.
42. Other findings
Belief in the “hot hand”. Both lay and expert basketball observers perceive
players as being more likely to make a shot after a hit (or a run or previous
hits) than after a miss, even when the data indicate that shots are
independent events as defined in 3.1. (Gilovich, Tversky, and Vallone,
1985)
Overconfidence. Various studies have shown that people’s probability
judgments are not well calibrated, and that on average people exhibit
overconfidence by the criteria in Definition 5.2. (see Hoffrage, 2004)
Belief reversals. Fox and Levav (2000) showed evidence that probability
judgments are affected by the elicitation method, in violation of the
procedural invariance criterion of Definition 5.1, and that choice and
numerical methods sometimes yield opposite results.
45. Further perspectives
Dual process model of judgment (Stanovich and West, 2000; Kahneman
and Frederick, 2002)
▪ System 1 – quick, intuitive
▪ System 2 – slower, deliberative
Evolutionary psychology (e.g., Gigerenzer and Todd, 1999)
Rational analysis of cognition (Oaksford and Chater, 2007)
Computational probabilistic models of cognition (e.g. Tenenbaum, Kemp,
Griffiths, and Goodman, 2011)