The document discusses key theorems and concepts in probability, including:
1) The addition and multiplication theorems, which describe how to calculate the probability of multiple events occurring based on whether the events are mutually exclusive or independent.
2) It provides examples of how to calculate marginal, joint, and conditional probabilities under conditions of both statistical independence and dependence.
3) Bayes' theorem is introduced as a way to revise prior probability estimates based on new information using conditional probabilities. The theorem has various applications in business decision making.
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.
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 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 provides a lesson on conditional probability that includes:
1. Examples and formulas for calculating conditional probability
2. Practice problems solving for conditional probabilities in situations involving cards, dice, families, and committees
3. A discussion of how conditional probability can inform decisions about driving while using a cell phone, health, and sports.
The document defines key probability terms like union, intersection, mutually exclusive events, and complement. It provides examples of calculating probabilities of compound events, such as the probability of event A or B. It explains that if events are mutually exclusive, the probability of their union is the sum of their individual probabilities. The document ends with practice problems calculating probabilities of compound events using these concepts.
This document discusses key concepts in probability including experiments, outcomes, sample spaces, classical probability, empirical probability, subjective probability, complementary events, and the law of large numbers. Probability can be calculated classically by considering the number of outcomes in an event over the total number of outcomes, empirically by observing frequencies, or subjectively based on estimates. Understanding probability is important for properly evaluating risks and uncertainties.
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.
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 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 provides a lesson on conditional probability that includes:
1. Examples and formulas for calculating conditional probability
2. Practice problems solving for conditional probabilities in situations involving cards, dice, families, and committees
3. A discussion of how conditional probability can inform decisions about driving while using a cell phone, health, and sports.
The document defines key probability terms like union, intersection, mutually exclusive events, and complement. It provides examples of calculating probabilities of compound events, such as the probability of event A or B. It explains that if events are mutually exclusive, the probability of their union is the sum of their individual probabilities. The document ends with practice problems calculating probabilities of compound events using these concepts.
This document discusses key concepts in probability including experiments, outcomes, sample spaces, classical probability, empirical probability, subjective probability, complementary events, and the law of large numbers. Probability can be calculated classically by considering the number of outcomes in an event over the total number of outcomes, empirically by observing frequencies, or subjectively based on estimates. Understanding probability is important for properly evaluating risks and uncertainties.
Mutually exclusive events cannot occur at the same time. The probability of event A or B occurring is calculated as P(A) + P(B). Independent events do not influence each other, so P(A and B) = P(A) × P(B). Dependent events influence each other, so P(A and B) = P(A) × P(B after A). Examples are provided to demonstrate calculating probabilities for mutually exclusive, independent, and dependent events.
Ratios and proportions can be used to solve problems involving comparisons of quantities. A ratio compares two numbers using division, while a proportion states that two ratios are equal. To solve proportions, the cross product property is used, setting up equivalent fractions and solving with cross multiplication. Similar polygons have corresponding angles that are congruent and side lengths that are proportional, allowing their similarities to be described through statements listing corresponding vertices.
This document provides information on deductive and inductive reasoning. It defines key logic and reasoning terms such as premise, conclusion, argument, and syllogism. It explains deductive reasoning as deriving logical conclusions from general statements and premises. An example is provided of the classic "All men are mortal, Socrates is a man, therefore Socrates is mortal" syllogism. Inductive reasoning is explained as deriving probable conclusions from specific observations, rather than guaranteed conclusions. Examples of both deductive and inductive reasoning are given from geometry, sequences, and the movie The Princess Bride to illustrate the differences between the two types of reasoning.
The document defines conditional probability as the probability of an event occurring given that another event has already occurred. It provides the formula for conditional probability as P(B|A) = P(A and B) / P(A). Several examples are worked through applying this formula to calculate conditional probabilities in different contexts like selecting chips from a box, survey responses, exam scores, and dice rolls. Exercises at the end provide additional practice problems for calculating conditional probabilities.
The document discusses mutually exclusive and non-mutually exclusive events. It provides examples to illustrate the difference, including examples involving drawing balls from a jar numbered 1-15 and rolling a die. It discusses how to calculate the probability of unions of events depending on whether they are mutually exclusive or not. Key points are that for mutually exclusive events, the probability of their union is the sum of their individual probabilities, while for non-mutually exclusive events it is the sum of their probabilities minus their intersection.
theorems on tangents, Secants and segments of a circles 1.pptxPeejayOAntonio
The document discusses theorems related to circles, secants, tangents, and segments. It begins by defining theorems and postulates. It then presents several theorems about angles formed between secants and tangents, relationships between intercepted arcs and angles, congruent tangent segments, and properties of secant segments drawn from an exterior point. Examples are provided to demonstrate how to use the theorems to solve problems involving lengths and angle measures in circle geometry.
This document provides an introduction to probability. It defines probability as a numerical index of the likelihood that a certain event will occur, with a value between 0 and 1. It discusses examples of using probability terms like chance and likelihood. It also covers key probability concepts such as experiments, outcomes, events, and sample spaces. It explains different types of probability including subjective, objective/classic, and empirical probabilities. It provides examples of calculating probabilities of events using various approaches.
This document provides instructions on factoring quadratic trinomials of the form Ax^2 + Bx + C. It explains that trinomials of this form can be factored using the grouping method by finding two factors of the leading term times the constant term whose sum is the middle term. Examples are provided to demonstrate factoring trinomials with various coefficients of the leading term. Students are given practice problems to factor trinomials as well as determine when an expression is not factorable.
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.
- The document discusses quadratic functions and their graphs. It explains that the graph of a quadratic function is a parabola, which is a U-shaped curve.
- It describes how to write quadratic functions in standard form and use that form to sketch the graph and find features like the vertex and axis of symmetry.
- Examples are provided to demonstrate how to graph quadratic functions in standard form and how to find the minimum or maximum value of a quadratic function by setting its derivative equal to zero.
This document defines and provides examples of simple, compound, mutually exclusive and inclusive events. It explains that for mutually exclusive events, the probability of A or B equals the sum of the individual probabilities, while for inclusive events it equals the sum of the individual probabilities minus the probability they both occur. It then gives examples of calculating probabilities for various card draws and die rolls.
This document provides information on solving problems involving right triangles using trigonometry, including the Law of Sines and Law of Cosines. It includes examples of using trigonometry to solve problems involving angles of elevation/depression, finding areas and volumes, and determining distances. Tables of trigonometric function values are presented along with explanations of evaluating functions in different quadrants using reference angles. Proofs and examples are given for applying the Law of Sines and Law of Cosines to find missing sides and angles of triangles.
This document provides information about graphing quadratic functions. It defines the standard form of a quadratic function as y = ax^2 + bx + c and explains that the sign of a determines if the parabola opens up or down. It describes how to find the axis of symmetry, vertex, domain and range. The steps to graph a quadratic function are given as finding the axis of symmetry, the vertex, and then two other points to reflect across the axis and connect with a smooth curve. Methods for finding the axis of symmetry, vertex, y-intercept and solutions are demonstrated through examples.
This document provides an overview of Chapter 14 on rational expressions from a developmental mathematics textbook. It covers simplifying, multiplying, dividing, adding, and subtracting rational expressions. It also discusses finding least common denominators and changing rational expressions to equivalent forms with a common denominator in order to add or subtract expressions. The chapter is divided into sections covering specific topics like simplifying rational expressions, multiplying and dividing, adding and subtracting with the same or different denominators, and solving equations with rational expressions. Examples are provided throughout to illustrate the concepts and procedures.
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
Here are the steps to solve this problem:
1) Since ED is tangent at C, ∠ECD = 90° (property of tangents)
2) ∆OCD ~ ∆OED (AA similarity)
3) Corresponding parts are proportional:
OC/OD = OC/OE
Cross multiply: OC×OE = OD×OC
Divide both sides by OC: OE = OD
4) Given DE = 20, OE = OD = 20/2 = 10
5) ∠ACD = ∠AED (inscribed angles on the same arc)
6) ∆ACD ~ ∆AED (AA similarity)
7) Corresponding parts are proportional
Parts of quadratic function and transforming to general form to vertex form a...rowenaCARINO
This document contains notes and materials for a Math 9 class covering quadratic functions for the 6th week of the 1st quarter. It includes:
1. A math prayer to work problems and trust in God's help.
2. An overview of topics to be covered: illustrating quadratic functions, transforming between general and vertex forms, and determining parts of quadratic graphs.
3. A review question asking students to identify which situation represents a quadratic function.
4. Examples of transforming between the general and vertex forms of quadratic functions.
5. Key parts of graphs of quadratic functions including the vertex, axis of symmetry, x- and y-intercepts, and the effect of the leading coefficient on
This document provides information about trigonometric ratios of some special angles. It defines the trigonometric ratios of 30°, 45°, and 60° angles using right triangle geometry. It also lists the trigonometric ratios of 0° and 90° angles without proof. Examples are provided to demonstrate using trigonometric ratios to evaluate expressions involving angles such as 30°, 45°, 60°, 0°, and 90°.
Probability concepts-applications-1235015791722176-2satysun1990
The document introduces basic probability concepts like objective and subjective probabilities. It provides examples to illustrate mutually exclusive, collectively exhaustive events and independent vs dependent events. The key formulas for probability of single, joint and conditional probabilities are defined. It also discusses Bayes' theorem and how it can be used to update probabilities based on new information. An example is provided to show how probabilities are revised after obtaining additional data points.
The document introduces basic probability concepts and provides examples to illustrate them. It discusses the key properties of probability, types of probability (objective and subjective), mutually exclusive and collectively exhaustive events, and probabilities of independent and dependent events. It also explains Bayes' theorem and how it can be used to update probabilities as new information becomes available.
Mutually exclusive events cannot occur at the same time. The probability of event A or B occurring is calculated as P(A) + P(B). Independent events do not influence each other, so P(A and B) = P(A) × P(B). Dependent events influence each other, so P(A and B) = P(A) × P(B after A). Examples are provided to demonstrate calculating probabilities for mutually exclusive, independent, and dependent events.
Ratios and proportions can be used to solve problems involving comparisons of quantities. A ratio compares two numbers using division, while a proportion states that two ratios are equal. To solve proportions, the cross product property is used, setting up equivalent fractions and solving with cross multiplication. Similar polygons have corresponding angles that are congruent and side lengths that are proportional, allowing their similarities to be described through statements listing corresponding vertices.
This document provides information on deductive and inductive reasoning. It defines key logic and reasoning terms such as premise, conclusion, argument, and syllogism. It explains deductive reasoning as deriving logical conclusions from general statements and premises. An example is provided of the classic "All men are mortal, Socrates is a man, therefore Socrates is mortal" syllogism. Inductive reasoning is explained as deriving probable conclusions from specific observations, rather than guaranteed conclusions. Examples of both deductive and inductive reasoning are given from geometry, sequences, and the movie The Princess Bride to illustrate the differences between the two types of reasoning.
The document defines conditional probability as the probability of an event occurring given that another event has already occurred. It provides the formula for conditional probability as P(B|A) = P(A and B) / P(A). Several examples are worked through applying this formula to calculate conditional probabilities in different contexts like selecting chips from a box, survey responses, exam scores, and dice rolls. Exercises at the end provide additional practice problems for calculating conditional probabilities.
The document discusses mutually exclusive and non-mutually exclusive events. It provides examples to illustrate the difference, including examples involving drawing balls from a jar numbered 1-15 and rolling a die. It discusses how to calculate the probability of unions of events depending on whether they are mutually exclusive or not. Key points are that for mutually exclusive events, the probability of their union is the sum of their individual probabilities, while for non-mutually exclusive events it is the sum of their probabilities minus their intersection.
theorems on tangents, Secants and segments of a circles 1.pptxPeejayOAntonio
The document discusses theorems related to circles, secants, tangents, and segments. It begins by defining theorems and postulates. It then presents several theorems about angles formed between secants and tangents, relationships between intercepted arcs and angles, congruent tangent segments, and properties of secant segments drawn from an exterior point. Examples are provided to demonstrate how to use the theorems to solve problems involving lengths and angle measures in circle geometry.
This document provides an introduction to probability. It defines probability as a numerical index of the likelihood that a certain event will occur, with a value between 0 and 1. It discusses examples of using probability terms like chance and likelihood. It also covers key probability concepts such as experiments, outcomes, events, and sample spaces. It explains different types of probability including subjective, objective/classic, and empirical probabilities. It provides examples of calculating probabilities of events using various approaches.
This document provides instructions on factoring quadratic trinomials of the form Ax^2 + Bx + C. It explains that trinomials of this form can be factored using the grouping method by finding two factors of the leading term times the constant term whose sum is the middle term. Examples are provided to demonstrate factoring trinomials with various coefficients of the leading term. Students are given practice problems to factor trinomials as well as determine when an expression is not factorable.
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.
- The document discusses quadratic functions and their graphs. It explains that the graph of a quadratic function is a parabola, which is a U-shaped curve.
- It describes how to write quadratic functions in standard form and use that form to sketch the graph and find features like the vertex and axis of symmetry.
- Examples are provided to demonstrate how to graph quadratic functions in standard form and how to find the minimum or maximum value of a quadratic function by setting its derivative equal to zero.
This document defines and provides examples of simple, compound, mutually exclusive and inclusive events. It explains that for mutually exclusive events, the probability of A or B equals the sum of the individual probabilities, while for inclusive events it equals the sum of the individual probabilities minus the probability they both occur. It then gives examples of calculating probabilities for various card draws and die rolls.
This document provides information on solving problems involving right triangles using trigonometry, including the Law of Sines and Law of Cosines. It includes examples of using trigonometry to solve problems involving angles of elevation/depression, finding areas and volumes, and determining distances. Tables of trigonometric function values are presented along with explanations of evaluating functions in different quadrants using reference angles. Proofs and examples are given for applying the Law of Sines and Law of Cosines to find missing sides and angles of triangles.
This document provides information about graphing quadratic functions. It defines the standard form of a quadratic function as y = ax^2 + bx + c and explains that the sign of a determines if the parabola opens up or down. It describes how to find the axis of symmetry, vertex, domain and range. The steps to graph a quadratic function are given as finding the axis of symmetry, the vertex, and then two other points to reflect across the axis and connect with a smooth curve. Methods for finding the axis of symmetry, vertex, y-intercept and solutions are demonstrated through examples.
This document provides an overview of Chapter 14 on rational expressions from a developmental mathematics textbook. It covers simplifying, multiplying, dividing, adding, and subtracting rational expressions. It also discusses finding least common denominators and changing rational expressions to equivalent forms with a common denominator in order to add or subtract expressions. The chapter is divided into sections covering specific topics like simplifying rational expressions, multiplying and dividing, adding and subtracting with the same or different denominators, and solving equations with rational expressions. Examples are provided throughout to illustrate the concepts and procedures.
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
Here are the steps to solve this problem:
1) Since ED is tangent at C, ∠ECD = 90° (property of tangents)
2) ∆OCD ~ ∆OED (AA similarity)
3) Corresponding parts are proportional:
OC/OD = OC/OE
Cross multiply: OC×OE = OD×OC
Divide both sides by OC: OE = OD
4) Given DE = 20, OE = OD = 20/2 = 10
5) ∠ACD = ∠AED (inscribed angles on the same arc)
6) ∆ACD ~ ∆AED (AA similarity)
7) Corresponding parts are proportional
Parts of quadratic function and transforming to general form to vertex form a...rowenaCARINO
This document contains notes and materials for a Math 9 class covering quadratic functions for the 6th week of the 1st quarter. It includes:
1. A math prayer to work problems and trust in God's help.
2. An overview of topics to be covered: illustrating quadratic functions, transforming between general and vertex forms, and determining parts of quadratic graphs.
3. A review question asking students to identify which situation represents a quadratic function.
4. Examples of transforming between the general and vertex forms of quadratic functions.
5. Key parts of graphs of quadratic functions including the vertex, axis of symmetry, x- and y-intercepts, and the effect of the leading coefficient on
This document provides information about trigonometric ratios of some special angles. It defines the trigonometric ratios of 30°, 45°, and 60° angles using right triangle geometry. It also lists the trigonometric ratios of 0° and 90° angles without proof. Examples are provided to demonstrate using trigonometric ratios to evaluate expressions involving angles such as 30°, 45°, 60°, 0°, and 90°.
Probability concepts-applications-1235015791722176-2satysun1990
The document introduces basic probability concepts like objective and subjective probabilities. It provides examples to illustrate mutually exclusive, collectively exhaustive events and independent vs dependent events. The key formulas for probability of single, joint and conditional probabilities are defined. It also discusses Bayes' theorem and how it can be used to update probabilities based on new information. An example is provided to show how probabilities are revised after obtaining additional data points.
The document introduces basic probability concepts and provides examples to illustrate them. It discusses the key properties of probability, types of probability (objective and subjective), mutually exclusive and collectively exhaustive events, and probabilities of independent and dependent events. It also explains Bayes' theorem and how it can be used to update probabilities as new information becomes available.
The document defines classical probability and key probability concepts. Probability is defined as the number of ways an event can occur divided by the total number of possible outcomes. There are three types of events: independent events where the occurrence of one does not impact the other, dependent events where the occurrence of one affects the other, and mutually exclusive events that have no outcomes in common. The document also outlines laws of probability, including the addition rule for independent events, the multiplication rule for independent events, and the general addition rule.
Bba 3274 qm week 2 probability conceptsStephen Ong
This document provides an introduction and overview of probability concepts. It begins with learning objectives about understanding probability foundations, dependent and independent events, and using Bayes' theorem. It then outlines topics to be covered, including fundamental probability concepts like events having probabilities between 0 and 1 and the sum of all outcomes equaling 1. The document gives examples of determining probabilities from relative frequencies and the classical method. It discusses mutually exclusive and collectively exhaustive events, independent and dependent events, and how to calculate joint, marginal, and conditional probabilities. Finally, it introduces Bayes' theorem and how it can be used to revise probabilities with new information.
- 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.
The normal probability curve is a bell-shaped curve that is used to represent probability distributions of many random variables. Some key properties of the normal curve are:
1) Approximately 68% of values fall within one standard deviation of the mean, 95% within two standard deviations, and 99% within three standard deviations.
2) The curve is perfectly symmetrical, with the mean, median, and mode all being the same value.
3) It approaches but never touches the x-axis and theoretically extends from negative to positive infinity. Standard deviation is used to measure deviations from the mean.
Statistics is a branch of applied mathematics that involves the collection, description, analysis, and inference of conclusions from quantitative data. The mathematical theories behind statistics rely heavily on differential and integral calculus, linear algebra, and probability theory.
Statisticians, people who do statistics, are particularly concerned with determining how to draw reliable conclusions about large groups and general events from the behavior and other observable characteristics of small samples. These small samples represent a portion of the large group or a limited number of instances of a general phenomenon.
KEY TAKEAWAYSStatistics is a branch of applied mathematics that involves the collection, description, analysis, and inference of conclusions from quantitative data. The mathematical theories behind statistics rely heavily on differential and integral calculus, linear algebra, and probability theory.
Statisticians, people who do statistics, are particularly concerned with determining how to draw reliable conclusions about large groups and general events from the behavior and other observable characteristics of small samples. These small samples represent a portion of the large group or a limited number of instances of a general phenomenon.
Statistics is a branch of applied mathematics that involves the collection, description, analysis, and inference of conclusions from quantitative data. The mathematical theories behind statistics rely heavily on differential and integral calculus, linear algebra, and probability theory.
Statisticians, people who do statistics, are particularly concerned with determining how to draw reliable conclusions about large groups and general events from the behavior and other observable characteristics of small samples. These small samples represent a portion of the large group or a limited number of instances of a general phenomenon.
KEY TAKEAWAYS
Statistics is a branch of applied mathematics that involves the collection, description, analysis, and inference of conclusions from quantitative data. The mathematical theories behind statistics rely heavily on differential and integral calculus, linear algebra, and probability theory.
Statisticians, people who do statistics, are particularly concerned with determining how to draw reliable conclusions about large groups and general events from the behavior and other observable characteristics of small samples. These small samples represent a portion of the large group or a limited number of instances of a general phenomenon.
KEY TAKEAWAYS
Statistics is a branch of applied mathematics that involves the collection, description, analysis, and inference of conclusions from quantitative data. The mathematical theories behind statistics rely heavily on differential and integral calculus, linear algebra, and probability theory.
Statisticians, people who do statistics, are particularly concerned with determining how to draw reliable conclusions about ls
This document summarizes key concepts from Chapter 4 of Allan G. Bluman's textbook "Elementary Statistics: A Step-by Step Approach" including:
- Defining probability experiments, outcomes, events, and sample spaces
- Calculating classical and empirical probabilities
- Using addition and multiplication rules to find probabilities of compound events
- Defining and calculating conditional probability
- Using fundamental counting rules and permutations/combinations to calculate number of possible outcomes
The document provides examples for each concept to illustrate their application.
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 independent and dependent events in probability. It defines independent events as those whose occurrence is not affected by other events, while dependent events are those whose occurrence is affected by other events. The probability of independent events occurring can be calculated by multiplying the individual probabilities, while the probability of dependent events requires considering conditional probabilities. Several examples are provided to illustrate calculating probabilities of independent and dependent events.
Probability theory provides the foundation for statistical inference and allows conclusions to be drawn about populations based on sample data. There are two main categories of probability - objective and subjective. Objective probability includes classical and relative frequency probabilities. Probability distributions describe the possible outcomes of random variables and include discrete distributions like the binomial. Probability theory is used in understanding distributions, sampling, estimation, hypothesis testing, and advanced statistical analysis.
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.
The document discusses key concepts in probability theory including:
1. Probability theory provides tools to quantify uncertainties and assign probabilities using classical, relative frequency, and subjective approaches.
2. Key probability terms are defined such as experiment, event, sample space, independent and dependent events, mutually exclusive events, and union, intersection, and complement of events.
3. Basic probability rules are covered including the multiplication rule for independent events, addition rule for mutually exclusive events, and how to calculate the probability of events.
This document defines key probability concepts and summarizes different approaches to assigning probabilities:
1. It defines classical, empirical, and subjective probability, and explains concepts like experiments, events, outcomes, and rules for computing probabilities.
2. Empirical probability is based on observed frequencies over many trials, while subjective probability is used when past data is limited.
3. Tools for organizing and calculating probabilities are discussed, including tree diagrams, contingency tables, conditional probability, Bayes' theorem, and counting rules.
[Junoon - E - Jee] - Probability - 13th Nov.pdfPrakashPatra7
This document provides information about the probability concepts of random experiments, sample space, events, classical definition of probability, odds in favor and against an event, problems on double dice, and conditional probability. It defines key terms like random experiment, sample space, event, complement of an event, classical probability, odds in favor and against, double dice problems, and conditional probability. Examples are given for each concept to illustrate the definitions. Formulas for classical probability, odds, addition theorem on probability, and multiplication theorem on probability are also presented.
The document discusses several probability concepts:
1) Addition and multiplication theorems of probability define how to calculate the probability of events occurring together or separately.
2) Conditional probability is the probability of an event occurring given that another event has occurred.
3) The multiplication theorem of probability states that the probability of two events occurring together is equal to the probability of one event multiplied by the probability of the other event given the first has occurred.
4) Bayes' theorem provides a formula for calculating the probability of an event given a condition. It relates the probability of two events occurring together to their individual probabilities.
The document discusses probability theory and provides definitions and examples of key concepts like conditional probability and Bayes' theorem. It defines probability as the ratio of favorable events to total possible events. Conditional probability is the probability of an event given that another event has occurred. Bayes' theorem provides a way to update or revise beliefs based on new evidence and relates conditional probabilities. Examples are provided to illustrate concepts like conditional probability calculations.
This document 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.
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.
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2. Theorems of Probability
There are 2 important theorems of probability
which are as follows:
2
The Addition Theorem and
The Multiplication Theorem
3. Addition theorem when events are Mutually
Exclusive
Definition: - It states that if 2 events A and B are mutually
exclusive then the probability of the occurrence of either
A or B is the sum of the individual probability of A and B.
Symbolically
3
P(A or B) or P(A U B) = P(A) + P(B)
P(A or B or C) = P(A) + P(B) + P(C)
The theorem can be extended to three or more mutually
exclusive events. Thus,
4. Addition theorem when events are not Mutually Exclusive
(Overlapping or Intersection Events)
Definition: - It states that if 2 events A and B are not
mutually exclusive then the probability of the occurrence
of either A or B is the sum of the individual probability of
A and B minus the probability of occurrence of both A
and B.
Symbolically
4
P(A or B) or P(A U B) = P(A) + P(B) – P(A ∩ B)
5. Mutually Exclusive Events
Two events are mutually exclusive if
they cannot occur at the same time
(i.e., they have no outcomes in
common).
In the Venn Diagram above,
the probabilities of events A
and B are represented by
two disjoint sets (i.e., they
have no elements in
common).
Non-Mutually Exclusive Events
Two events are non-mutually exclusive if
they have one or more outcomes in
common.
In the Venn Diagram above, the
probabilities of events A and B
are represented by two
intersecting sets (i.e., they have
some elements in common).
6. The Addition Rule: Mutually Exclusive
P(A or B) = P(A) + P(B)
The Addition Rule: Non-mutually Exclusive
P(A or B) = P(A)+P(B) - P(A and B)
Probability of A and B
happening together
Probability of B
happening
Probability of A
happening
Probability of A or B
happening when and B are
not Mutually exclusive
Probability of either A or B happening
7. Multiplication theorem
Definition: States that if 2 events A and B are independent,
then the probability of the occurrence of both of them (A &
B) is the product of the individual probability of A and B.
Symbolically,
Probability of happening of both the events:
P(A and B) or P(A ∩ B) = P(A) x P(B)
P(A, B and C) or P(A ∩ B ∩ C) = P(A) x P(B) x P(C)
Theorem can be extended to 3 or more independent events.
Thus,
8. How to calculate probability in case of Dependent
Events
Case Formula
1. Probability of occurrence of at least A or B
1. When events are mutually
2. When events are not mutually exclusive
2. Probability of occurrence of both A & B
3. Probability of occurrence of A & not B
4. Probability of occurrence of B & not A
5. Probability of non-occurrence of both A & B
6. Probability of non-occurrence of atleast A or B
P(A U B) = P(A) + P(B)
P(A U B) = P(A) + P(B) – P(A ∩ B)
P(A ∩ B) = P(A) + P(B) – P(A U B)
P(A ∩ B) = P(A) - P(A ∩ B)
P(A ∩ B) = P(B) - P(A ∩ B)
P(A ∩ B) = 1 - P(A U B)
P(A U B) = 1 - P(A ∩ B)
8
9. How to calculate probability in case of
Independent Events
Case Formula
1. Probability of occurrence of both A & B
2. Probability of non-occurrence of both A &
B
3. Probability of occurrence of A & not B
4. Probability of occurrence of B & not A
5. Probability of occurrence of atleast one
event
6. Probability of non-occurrence of atleast
one event
7. Probability of occurrence of only one
event
P(A ∩ B) = P(A) x P(B)
P(A ∩ B) = P(A) x P(B)
P(A ∩ B) = P(A) x P(B)
P(A ∩ B) = P(A) x P(B)
P(A U B) = 1 - P(A ∩ B) = 1 – [P(A) x P(B)]
P(A U B) = 1 - P(A ∩ B) = 1 – [P(A) x P(B)]
P(A ∩ B) + P(A ∩ B) = [P(A) x P(B)] +
[P(A) x P(B)]
10. Problem
An inspector of the Alaska Pipeline has the task of
comparing the reliability of 2 pumping stations. Each
station is susceptible to 2 kinds of failure: Pump failure &
leakage. When either (or both) occur, the station must be
shut down. The data at hand indicate that the following
probabilities prevail:
Station P(Pump failure) P(Leakage) P(Both)
1 0.07 0.10 0
2 0.09 0.12 0.06
Which station has the higher probability of being shut
down.
10
11. Solution
P(Pump failure or Leakage)
= P(Pump Failure) + P(Leakage Failure)
– P(Pump Failure ∩ Leakage Failure)
11
Station 1: = 0.07 + 0.10 – 0
= 0.17
Station 2: = 0.09 + 0.12 – 0.06
= 0.15
Thus, station 1 has the higher
probability of being shut down.
13. Probabilities under conditions of
Statistical Independence
Statistically Independent Events: - The occurrence of
one event has no effect on the probability of the
occurrence of any other event
Most managers who use probabilities are
concerned with 2 conditions.
1. The case where one event or another will occur.
2. The situation where 2 or more. Events will both occur.
14. There are 3 types of probabilities under
statistical independence.
Marginal
Joint
Conditional
Marginal/ Unconditional Probability:
- A single probability where only one event can take
place.
.
Joint probability:
- Probability of 2 or more events occurring together or in
succession.
Conditional probability:
- Probability that a second event (B) will occur if a first
event (A) has already happened
15. Example: Marginal Probability - Statistical Independence
A single probability where only one event can
take place.
Marginal Probability of an Event
P(A) = P(A)
Example 1: - On each individual toss of an biased or unfair
coin, P(H) = 0.90 & P(T) = 0.10. The outcomes of several
tosses of this coin are statistically independent events too,
even tough the coin is biased.
Example 2: - 50 students of a school drew lottery to see
which student would get a free trip to the Carnival at Goa.
Any one of the students can calculate his/ her chances of
winning as:
P(Winning) = 1/50 = 0.02
16. Example: Joint Probability - Statistical Independence
The probability of 2 or more independent events occurring
together or in succession is the product of their marginal
probabilities.
Joint Probability of 2 Independent Events
P(AB) = P(A) * P(B)
Example: - What is the probability of heads on 2
successive tosses?
P(H1H2) = P(H1) * P(H2)
= 0.5 * 0.5 = 0.25
The probability of heads on 2 successive tosses is
0.25, since the probability of any outcome is not
affected by any preceding outcome.
17. We can make the probabilities of events even more
explicit using a Probabilistic Tree.
1 Toss 2 Toss 3 Toss
H1 0.5 H1H2 0.25 H1H2H3 0.125
T1 0.5 H1T2 0.25 H1H2T3 0.125
T1H2 0.25 H1T2H3 0.125
T1T2 0.25 H1T2T3 0.125
T1H2H3 0.125
T1H2T3 0.125
T1T2H3 0.125
T1T2T3 0.125
18. Example: Conditional Probability - Statistical Independence
For statistically independent events, conditional probability of
event B given that event A has occurred is simply the
probability of event B.
Conditional Probability for 2 Independent Events
P(B|A) = P(B)
Example: - What is the probability that the second toss
of a fair coin will result in heads, given that heads
resulted on the first toss?
P(H2|H1) = 0.5
For 2 independent events, the result of the first toss
have absolutely no effect on the results of the second toss.
19. Probabilities under conditions of Statistical
Dependence
Statistical Dependence exists when the probability of
some event is dependent on or affected by the
occurrence of some other event.
The types of probabilities under statistical dependence
are:
• Marginal
• Joint
• Conditional
20. Example
Assume that a box contains 10 balls distributed as follows: -
3 are colored & dotted
1 is colored & striped
2 are gray & dotted
4 are gray & striped
Event Probability of Event
1 0.1
Colored & Dotted
2 0.1
3 0.1
4 0.1 Colored & Striped
5 0.1
Gray & Dotted
6 0.1
7 0.1
Gray & Striped
8 0.1
9 0.1
10 0.1
21. Example: Marginal Probability - Statistically Dependent
It can be computed by summing up all the joint events in
which the simple event occurs.
Compute the marginal probability of the event colored.
It can be computed by summing up the probabilities of the
two joint events in which colored occurred:
P(C) = P(CD) + P(CS)
= 0.3 + 0.1
= 0.4
22. Example: Joint Probability - Statistically Dependent
Joint probabilities under conditions of statistical
dependence is given by
Joint probability for Statistically Dependent Events
P(BA) = P(B|A) * P(A)
•What is the probability that this ball is dotted and
colored?
Probability of colored & dotted balls =
P(DC) = P(D|C) * P(D)
= (0.3/0.4) * 0.5
= 0.375
23. Example: Conditional Probability - Statistically Dependent
Given A & B to be the 2 events then,
Conditional probability for Statistically Dependent Events
P(BA)
P(B|A) = ----------
P(A)
Probability of event B given that event has occurred
P(B|A)
24. What is the probability that this
ball is dotted, given that it is
colored?
The probability of drawing any
one of the ball from this box is
0.1 (1/10) [Total no. of balls in
the box = 10].
25. We know that there are 4 colored balls, 3 of which
are dotted & one of it striped.
P(DC) 0.3
P(D|C) = --------- = ------
P(C) 0.4
= 0.75
P(DC) = Probability of colored & dotted balls
(3 out of 10 --- 3/10)
P(C) = 4 out of 10 --- 4/10
27. Revising Prior Estimates of Probabilities: Bayes’
Theorem
A very important & useful application of conditional
probability is the computation of unknown probabilities,
based on past data or information.
When an event occurs through one of the various
mutually disjoint events, then the conditional probability
that this event has occurred due to a particular reason or
event is termed as Inverse Probability or Posterior
Probability.
Has wide ranging applications in Business & its
Management.
28. Since it is a concept of revision of probability based on
some additional information, it shows the improvement
towards certainty level of the event.
Example 1: - If a manager of a boutique finds that most
of the purple & white jackets that she thought would sell
so well are hanging on the rack, she must revise her prior
probabilities & order a different color combination or
have a sale.
Certain probabilities were altered after the people got
additional information. New probabilities are known as
revised, or Posterior probabilities.
29. Bayes Theorem
If an event A can occur only in conjunction with n mutually
exclusive & exhaustive events B1, B2, …, Bn, & if A actually
happens, then the probability that it was preceded by an
event Bi (for a conditional probabilities of A given B1, A given
B2 … A given Bn are known) & if marginal probabilities P(Bi) are
also known, then the posterior probability of event Bi given
that event A has occurred is given by:
P(A | Bi). P(Bi)
P(Bi | A) = ----------------------
∑ P(A | Bi). P(Bi)
30. Remarks: -
The probabilities P(B1), P(B2), … , P(Bn) are termed as
the ‘a priori probabilities’ because they exist before
we gain any information from the experiment itself.
The probabilities P(A | Bi), i=1,2,…,n are called
‘Likelihoods’ because they indicate how likely the event
A under consideration is to occur, given each & every a
priori probability.
The probabilities P(Bi | A), i=1, 2, …,n are called
‘Posterior probabilities’ because they are determined
after the results of the experiment are known.
32. Problem
In a bolt factory machines A, B, & C manufacture
respectively 25%, 35%, & 40% of the total. Of their
output 5%, 4%, 2% are defective bolts. A bolt is drawn at
random from the product & Is found to be defective.
What are the probabilities that it was manufactured by
machines A, B & C?
33. Solution
Let E1, E2, E3 denote the events manufactured by
machines A, B & C respectively.
Let E denote the event of its being defective.
P(E1) = 0.25; P(E2) = 0.35; P(E3) = 0.40;
Probability of drawing a defective bolt manufactured by
machine A is P(E|E1) = 0.05
Similarly P(E|E2) = 0.04; P(E|E3) = 0.02
Probability that defective bolt selected at random is
manufactured by machine A is given by
35. Suppose that one person in 100, 000 has a particular
rare disease for which there is a fairly accurate
diagnostic test. This test is correct 99% of the time
when to someone with the disease; it is correct
99.5% of the time when given to someone who does
not have the disease. Given this information can we
find
(a) the probability that someone who tests positive
for the disease has the disease?
(b) the probability that someone who tests negative
for the disease does not have the disease?
Should someone who tests positive be very
concerned that he or she has the disease?
36. Glossary of terms
Classical Probability: It is based on the idea that certain
occurrences are equally likely.
Example: - Numbers 1, 2, 3, 4, 5, & 6 on a fair die are
each equally likely to occur.
Conditional Probability: The probability that an event occurs
given the outcome of some other event.
Independent Events: Events are independent if the
occurrence of one event does not affect the occurrence of
another event.
Joint Probability: Is the likelihood that 2 or more events will
happen at the same time.
Multiplication Formula: If there are m ways of doing one
thing and n ways of doing another thing, there are m x n
ways of doing both.
37. Mutually exclusive events: A property of a set of categories such that
an individual, object, or measurement is included in only one
category.
Objective Probability: It is based on symmetry of games of chance or
similar situations.
Outcome: Observation or measurement of an experiment.
Posterior Probability: A revised probability based on additional
information.
Prior Probability: The initial probability based on the present level of
information.
Probability: A value between 0 and 1, inclusive, describing the
relative possibility (chance or likelihood) an event will occur.
Subjective Probability: Synonym for personal probability. Involves
personal judgment, information, intuition, & other subjective
evaluation criteria.
Example: - A physician assessing the probability of a patient’s
recovery is making a personal judgment based on what they
know and feel about the situation.