The document discusses key concepts in probability such as random experiments, outcomes, sample spaces, events, and the axiomatic approach to probability. It provides examples of random experiments like tossing a coin or rolling a die. An outcome is a possible result of an experiment, and a sample space is the set of all possible outcomes. Events can be simple, compound, impossible, or sure depending on the number of outcomes they include. The document also discusses mutually exclusive and exhaustive events and how probability can be defined through axioms about events and their probabilities.
The document discusses functions and their inverse functions. It provides examples of functions and their inverses using the square function f(x) = x^2 and explains that the inverse of a function "undoes" what the original function did. It also shows graphically that the inverse function mirrors the original function across the line y = x, switching the x and y values of points. The document demonstrates finding the inverse of two example functions step-by-step.
1. The document provides instructions and examples for using the change of base formula to convert between logarithmic bases and to solve exponential equations.
2. It introduces the change of base formula and shows examples of using it to convert log bases from base-3 to base-10 using a calculator.
3. Examples are given of using the change of base formula to solve exponential equations by taking the log of both sides and applying the formula.
The document is a trigonometric functions worksheet from Sthitpragya Science Classes in Gandhidham for Class XI. It contains 40 questions on trigonometric functions and was prepared by Mishal Chauhan, who has an M.Tech from IIT Delhi. The worksheet provides practice problems for entrance exams like JEE, BITSAT, GUJCET, and Olympiads. Contact information for the classes is included at the bottom.
The document provides an overview of quadratic equations, including:
1) Quadratic equations take the form of ax^2 + bx + c = 0 and involve an unknown (x) and known coefficients (a, b, c).
2) Quadratic equations can be solved through factoring, completing the square, using the quadratic formula, or graphing.
3) The quadratic formula provides the general solution for quadratic equations as x = (-b ± √(b^2 - 4ac))/2a.
Presentation during the Bureau of Agricultural Research (BAR) Seminar Series on July 25, 2019 at RDMIC Bldg., cor. Visayas Ave., Elliptical Rd., Diliman, Quezon City
This document discusses topics in partial differentiation including:
1) The geometrical meaning of partial derivatives as the slope of the tangent line to a surface.
2) Finding the equation of the tangent plane and normal line to a surface.
3) Taylor's theorem and Maclaurin's theorem for functions with two variables, which can be used to approximate functions and calculate errors.
This document defines and provides examples of different geometric transformations including translations, reflections, rotations, and dilations. Translations involve sliding a figure along a vector. Reflections flip a figure across a line. Rotations turn a figure around a center point by a number of degrees. Dilations enlarge or shrink a figure proportionally away or towards a center point. Key terms like preimage, image, and scale factor are introduced and transformations are demonstrated on examples like triangles, quadrilaterals, and pentagons. Real-world connections are made to mirrors, clock hands, and the pupil of the eye.
The document is from a Holt McDougal Algebra 1 textbook and covers exponential functions. It includes examples of evaluating, identifying, and graphing exponential functions. It also provides applications involving modeling population growth, money in a bank account, and car value depreciation using exponential functions. The key aspects of exponential functions discussed are that the independent variable appears as an exponent and the y-values are multiplied by a constant ratio as the x-values increase by a constant amount.
The document discusses functions and their inverse functions. It provides examples of functions and their inverses using the square function f(x) = x^2 and explains that the inverse of a function "undoes" what the original function did. It also shows graphically that the inverse function mirrors the original function across the line y = x, switching the x and y values of points. The document demonstrates finding the inverse of two example functions step-by-step.
1. The document provides instructions and examples for using the change of base formula to convert between logarithmic bases and to solve exponential equations.
2. It introduces the change of base formula and shows examples of using it to convert log bases from base-3 to base-10 using a calculator.
3. Examples are given of using the change of base formula to solve exponential equations by taking the log of both sides and applying the formula.
The document is a trigonometric functions worksheet from Sthitpragya Science Classes in Gandhidham for Class XI. It contains 40 questions on trigonometric functions and was prepared by Mishal Chauhan, who has an M.Tech from IIT Delhi. The worksheet provides practice problems for entrance exams like JEE, BITSAT, GUJCET, and Olympiads. Contact information for the classes is included at the bottom.
The document provides an overview of quadratic equations, including:
1) Quadratic equations take the form of ax^2 + bx + c = 0 and involve an unknown (x) and known coefficients (a, b, c).
2) Quadratic equations can be solved through factoring, completing the square, using the quadratic formula, or graphing.
3) The quadratic formula provides the general solution for quadratic equations as x = (-b ± √(b^2 - 4ac))/2a.
Presentation during the Bureau of Agricultural Research (BAR) Seminar Series on July 25, 2019 at RDMIC Bldg., cor. Visayas Ave., Elliptical Rd., Diliman, Quezon City
This document discusses topics in partial differentiation including:
1) The geometrical meaning of partial derivatives as the slope of the tangent line to a surface.
2) Finding the equation of the tangent plane and normal line to a surface.
3) Taylor's theorem and Maclaurin's theorem for functions with two variables, which can be used to approximate functions and calculate errors.
This document defines and provides examples of different geometric transformations including translations, reflections, rotations, and dilations. Translations involve sliding a figure along a vector. Reflections flip a figure across a line. Rotations turn a figure around a center point by a number of degrees. Dilations enlarge or shrink a figure proportionally away or towards a center point. Key terms like preimage, image, and scale factor are introduced and transformations are demonstrated on examples like triangles, quadrilaterals, and pentagons. Real-world connections are made to mirrors, clock hands, and the pupil of the eye.
The document is from a Holt McDougal Algebra 1 textbook and covers exponential functions. It includes examples of evaluating, identifying, and graphing exponential functions. It also provides applications involving modeling population growth, money in a bank account, and car value depreciation using exponential functions. The key aspects of exponential functions discussed are that the independent variable appears as an exponent and the y-values are multiplied by a constant ratio as the x-values increase by a constant amount.
Piecewise functions combine different equations depending on the domain or x-values. They are useful for modeling real-world problems like shipping costs, taxes, and ordering items. To write a piecewise function, identify the domains where the graph is cut, then determine the slopes and y-intercepts of each piece and write the corresponding equation for each domain section. Evaluating piecewise functions involves identifying which equation to use based on the domain that contains the given x-value.
The document discusses function transformations including translations, reflections, dilations, and compressions. It defines these transformations and provides examples of how they affect the graph of a function. Translations slide the graph left or right without changing its shape or orientation. Reflections create a mirror image of the graph across an axis, flipping it. Compressions squeeze the graph towards or away from an axis. Dilations stretch or shrink the graph away from an axis. The document explains how to interpret various function notation and applies the transformations to example graphs.
The document discusses second order derivatives. The second derivative of a function is the derivative of the first derivative. It can tell us whether a function is concave up or down at a point. If the second derivative is zero at a point, it does not tell us the slope. The point where a function changes from concave up to down is called the point of inflection. The second derivative test can determine if a point is a local minimum or maximum.
A shape is defined by its external boundary or outline rather than other properties. 2D shapes can be laid flat while 3D shapes occupy their own space. There are standard 3D shapes like spheres, cubes, cones and pyramids. Solids have different views - the front view shows length and height, the top view length and width, and the side view width and height. A solid is a 3D object with length, breadth and thickness bounded by surfaces, and can be classified as polyhedrons or solids of revolution. Regular polyhedra are the most symmetrical shapes and include the five Platonic solids.
This document provides definitions and formulas for trigonometric functions. It defines the trig functions using right triangles and the unit circle. It lists important properties like domain, range, period, and formulas for sums, differences, inverses, and the laws of sines, cosines, and tangents.
The document discusses Fourier series and their application to functions defined over intervals. It defines the Fourier sine and cosine series for functions on [-L,L] by extending the functions to the full interval [-π,π] in an odd or even way. The Fourier sine series results from the odd extension, using sine terms, while the Fourier cosine series uses the even extension and cosine terms. Examples are provided of calculating the Fourier sine and cosine series for basic functions over [-1,1]. The approach generalizes to 2L-periodic functions defined on [-L,L].
The perpendicular bisectors and angle bisectors of a triangle intersect at points that are equidistant from the triangle's vertices and sides, respectively. The perpendicular bisectors intersect at the triangle's circumcenter, which is equidistant from the vertices. The angle bisectors intersect at the triangle's incentre, which is equidistant from the sides. These properties are proven using theorems about congruent triangles and corresponding parts of congruent triangles.
Numerical integration;Gaussian integration one point, two point and three poi...vaibhav tailor
The document discusses numerical integration using Gaussian quadrature. It describes one-point, two-point, and three-point Gaussian quadrature rules. For each rule, it provides the formula used to approximate a definite integral of a function over an interval by calculating a weighted sum of the function values at specified points. Examples are included to demonstrate applying the one-point, two-point, and three-point rules to evaluate definite integrals.
The document discusses key concepts in probability such as random experiments, outcomes, sample spaces, events, types of events including impossible, sure, simple, and compound events. It also covers algebra of events including unions, intersections, complements and mutually exclusive events. The document defines mutually exclusive and exhaustive events. Finally, it introduces the axiomatic approach to defining probability as a function that satisfies three axioms.
What does it mean for an event to have occurred? This slide builds on the previous slide deck on event and explains the above question with an example.
This presentation will clarify all your basic concepts of Probability. It includes Random Experiment, Sample Space, Event, Complementary event, Union - Intersection and difference of events, favorable cases, probability definitions, conditional probability, Bayes theorem
06 Probability Simple and Compound EventsDhruvSethi28
Simple and compound events are defined and explored with the help of an example of two coin tossing examples. These two types of events are fundamental to the understanding of probability theory
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.
Chapter – 15 probability maths || CLASS 9 || The World Of presentation youtub...NishitGajjar7
This document defines probability and provides examples of calculating probability through experiments involving coin tossing and dice rolling. It states that probability is a measure between 0 and 1 of the likelihood of an event occurring, with 0 being impossible and 1 being certain. The basic probability formula is defined as the number of favorable outcomes divided by the total number of possible outcomes. Experiments are defined as procedures that can be repeated with a set of possible outcomes, with each repetition being a trial. The probability of an event is calculated by taking the number of outcomes where the event occurs and dividing by the total number of outcomes.
Probability is a branch of mathematics that studies patterns of chance. It is used to quantify the likelihood of events occurring in experiments or other situations involving uncertainty. The probability of an event is expressed as a number between 0 and 1, with 0 indicating impossibility and 1 indicating certainty. Key concepts in probability include theoretical and experimental probability, sample spaces, events, mutually exclusive and exhaustive events, and rules like addition rules for calculating combined probabilities. Probability is applied in many fields including statistics, gambling, science, and machine learning.
The document 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.
Piecewise functions combine different equations depending on the domain or x-values. They are useful for modeling real-world problems like shipping costs, taxes, and ordering items. To write a piecewise function, identify the domains where the graph is cut, then determine the slopes and y-intercepts of each piece and write the corresponding equation for each domain section. Evaluating piecewise functions involves identifying which equation to use based on the domain that contains the given x-value.
The document discusses function transformations including translations, reflections, dilations, and compressions. It defines these transformations and provides examples of how they affect the graph of a function. Translations slide the graph left or right without changing its shape or orientation. Reflections create a mirror image of the graph across an axis, flipping it. Compressions squeeze the graph towards or away from an axis. Dilations stretch or shrink the graph away from an axis. The document explains how to interpret various function notation and applies the transformations to example graphs.
The document discusses second order derivatives. The second derivative of a function is the derivative of the first derivative. It can tell us whether a function is concave up or down at a point. If the second derivative is zero at a point, it does not tell us the slope. The point where a function changes from concave up to down is called the point of inflection. The second derivative test can determine if a point is a local minimum or maximum.
A shape is defined by its external boundary or outline rather than other properties. 2D shapes can be laid flat while 3D shapes occupy their own space. There are standard 3D shapes like spheres, cubes, cones and pyramids. Solids have different views - the front view shows length and height, the top view length and width, and the side view width and height. A solid is a 3D object with length, breadth and thickness bounded by surfaces, and can be classified as polyhedrons or solids of revolution. Regular polyhedra are the most symmetrical shapes and include the five Platonic solids.
This document provides definitions and formulas for trigonometric functions. It defines the trig functions using right triangles and the unit circle. It lists important properties like domain, range, period, and formulas for sums, differences, inverses, and the laws of sines, cosines, and tangents.
The document discusses Fourier series and their application to functions defined over intervals. It defines the Fourier sine and cosine series for functions on [-L,L] by extending the functions to the full interval [-π,π] in an odd or even way. The Fourier sine series results from the odd extension, using sine terms, while the Fourier cosine series uses the even extension and cosine terms. Examples are provided of calculating the Fourier sine and cosine series for basic functions over [-1,1]. The approach generalizes to 2L-periodic functions defined on [-L,L].
The perpendicular bisectors and angle bisectors of a triangle intersect at points that are equidistant from the triangle's vertices and sides, respectively. The perpendicular bisectors intersect at the triangle's circumcenter, which is equidistant from the vertices. The angle bisectors intersect at the triangle's incentre, which is equidistant from the sides. These properties are proven using theorems about congruent triangles and corresponding parts of congruent triangles.
Numerical integration;Gaussian integration one point, two point and three poi...vaibhav tailor
The document discusses numerical integration using Gaussian quadrature. It describes one-point, two-point, and three-point Gaussian quadrature rules. For each rule, it provides the formula used to approximate a definite integral of a function over an interval by calculating a weighted sum of the function values at specified points. Examples are included to demonstrate applying the one-point, two-point, and three-point rules to evaluate definite integrals.
The document discusses key concepts in probability such as random experiments, outcomes, sample spaces, events, types of events including impossible, sure, simple, and compound events. It also covers algebra of events including unions, intersections, complements and mutually exclusive events. The document defines mutually exclusive and exhaustive events. Finally, it introduces the axiomatic approach to defining probability as a function that satisfies three axioms.
What does it mean for an event to have occurred? This slide builds on the previous slide deck on event and explains the above question with an example.
This presentation will clarify all your basic concepts of Probability. It includes Random Experiment, Sample Space, Event, Complementary event, Union - Intersection and difference of events, favorable cases, probability definitions, conditional probability, Bayes theorem
06 Probability Simple and Compound EventsDhruvSethi28
Simple and compound events are defined and explored with the help of an example of two coin tossing examples. These two types of events are fundamental to the understanding of probability theory
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.
Chapter – 15 probability maths || CLASS 9 || The World Of presentation youtub...NishitGajjar7
This document defines probability and provides examples of calculating probability through experiments involving coin tossing and dice rolling. It states that probability is a measure between 0 and 1 of the likelihood of an event occurring, with 0 being impossible and 1 being certain. The basic probability formula is defined as the number of favorable outcomes divided by the total number of possible outcomes. Experiments are defined as procedures that can be repeated with a set of possible outcomes, with each repetition being a trial. The probability of an event is calculated by taking the number of outcomes where the event occurs and dividing by the total number of outcomes.
Probability is a branch of mathematics that studies patterns of chance. It is used to quantify the likelihood of events occurring in experiments or other situations involving uncertainty. The probability of an event is expressed as a number between 0 and 1, with 0 indicating impossibility and 1 indicating certainty. Key concepts in probability include theoretical and experimental probability, sample spaces, events, mutually exclusive and exhaustive events, and rules like addition rules for calculating combined probabilities. Probability is applied in many fields including statistics, gambling, science, and machine learning.
The document 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.
3 PROBABILITY TOPICSFigure 3.1 Meteor showers are rare, .docxtamicawaysmith
This document provides an introduction to probability topics including terminology, rules, and examples. It defines key terms like sample space, event, probability, mutually exclusive events, independent events, and conditional probability. Examples are given to demonstrate calculating probabilities of events from sample spaces using the addition and multiplication rules. The document also discusses representing sample spaces and events using Venn diagrams, contingency tables, and tree diagrams.
This document defines probability and provides examples of calculating probabilities of events. It begins by defining probability as a measure of the likelihood of an event occurring between 0 and 1. Examples are then given to demonstrate calculating probabilities using the formula: Probability = Number of favorable outcomes / Total possible outcomes. The document concludes by noting the sum of probabilities of all outcomes is 1, and the probability of an event not occurring is 1 minus the probability of it occurring.
Basic Concepts of Probability - PowerPoint Presentation For Teachingbanot10262002
I apologize, but I do not have personal opinions or preferences about classmates. I am an AI assistant created by Anthropic to be helpful, harmless, and honest.
This document provides an introduction to probability and some key probability concepts. It discusses how probability can be used to measure the chance of outcomes in random experiments and defines key terms like sample space, events, equally likely events, unions and intersections of events. It also presents the classical approach to defining probability mathematically as the ratio of favorable outcomes to total possible outcomes when conditions like equally likely outcomes are met. Several examples are provided to illustrate probability calculations.
This document provides an introduction to probability. It defines probability as a measure of how likely an event is to occur. Probability is expressed as a ratio of favorable outcomes to total possible outcomes. The key terms used in probability are defined, including event, outcome, sample space, and elementary events. The theoretical approach to probability is discussed, where probability is predicted without performing the experiment. Random experiments are described as those that may not produce the same outcome each time. Laws of probability are presented, such as a probability being between 0 and 1. Applications of probability in everyday life are mentioned, such as reliability testing of products. Two example probability problems are worked out.
This document defines key terminology related to probability, including:
- Sample space: The set of all possible outcomes of a random experiment.
- Events: One or more possible outcomes of an experiment.
- Equally likely events: Events with an equal chance of occurring.
- Mutually exclusive events: Events that cannot occur together.
- Exhaustive events: A set of events where one of the events must occur.
It also provides examples and introduces the classical definition of probability as the number of favorable outcomes divided by the total number of outcomes.
The document discusses probability and probability distributions. It defines key probability terms like experiment, outcome, sample space, event, and elementary event. It discusses concepts like equally likely events, independent events, and complementary events. It also covers set theory concepts like subsets, unions, and intersections as they relate to probability. Counting rules like addition, multiplication, permutations, and combinations are explained with examples. Finally, it discusses the classical, frequentist, and axiomatic approaches to measuring probability.
This document introduces the principles of mathematical induction. It explains that induction can be used to prove statements for all natural numbers if (1) the statement is true for n=1, and (2) if the statement is true for an integer k, then it is also true for k+1. The document provides an example to prove the formula for the sum of squares from 1 to n using induction. It shows that the formula is true for the base case of n=1, and assumes the formula is true for an integer k to prove it is also true for k+1.
The definition of probability of an event is explored here as a part of the axiomatic approach to probability. We also take a look at probability of equally likely events occurring.
08 probability mutually exclusive eventsDhruvSethi28
Here we explore mutually exclusive events starting with its definition and exploring the concept with an example. The example used is the rolling of a die
Here various operations which are available in set theory are performed on events. Here we can combine different events with a union, perform an intersection between different events, explore their complement etc
Here we explore definitions of the impossible event, the sure event, the simple event and the compound event. To understand these events deeper please look at subsequent slides
An event E is a subset of the sample space S. In these slides, I define an event and give examples of different types of events along with their corresponding subsets of S
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
How to Build a Module in Odoo 17 Using the Scaffold MethodCeline George
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.
The simplified electron and muon model, Oscillating Spacetime: The Foundation...RitikBhardwaj56
Discover the Simplified Electron and Muon Model: A New Wave-Based Approach to Understanding Particles delves into a groundbreaking theory that presents electrons and muons as rotating soliton waves within oscillating spacetime. Geared towards students, researchers, and science buffs, this book breaks down complex ideas into simple explanations. It covers topics such as electron waves, temporal dynamics, and the implications of this model on particle physics. With clear illustrations and easy-to-follow explanations, readers will gain a new outlook on the universe's fundamental nature.
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
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.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
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Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
2. Random experiments
• In our day to day life, we perform many activities which have a fixed result
no matter any number of times they have been repeated. For example given
any triangle, without knowing the three angles, we can definitely say that
the sum of measure of angles is 180 degrees
• We also perform many experimental activities, where the result may not be
same when they are repeated under identical conditions. For example when
a coin is tossed it may turn up a head or a tail, but we are not sure which
one of these results will actually be obtained. Such experiments are called
random experiments
3. Random experiments
• An experiment is called a random experiment if it satisfies the following
two conditions:
• (i) It has more than one possible outcome
• (ii) It is not possible to predict the outcome in advance
4. Random experiments
• What examples of random experiments can you think of?
• Tossing a fair coin, rolling an unbiased die, drawing a card from a well-
shuffled pack of cards.
6. Outcomes & sample spaces
• A possible result of a random experiment is
called its outcome.
• The set of all possible outcomes is called the
sample space of the experiment
7. Outcomes & sample space Example
• Consider the experiment of rolling a die.
• The outcomes of this experiment are 1, 2, 3, 4, 5 and 6
• The set of outcomes {1,2,3,4,5,6} is called the sample space of the
experiment
8. Outcomes & sample space Example
• In general, the set of all possible outcomes of a random experiment is
called the sample space associated with the experiment.
• Sample space is denoted by the letter S.
• Each element of the sample space is called a sample point.
• In other words, each outcome of the random experiment is also called a
sample point.
10. occurrence of an event
• Consider the experiment of throwing a die. Let E denotes the event “a
number less than 4 appears”.
• Hence the sample space S of the experiment is : S = { 1, 2, 3 }
• If actually ‘1’ had appeared on the die then we say that event E has
occurred. As a matter of fact if outcomes are 2 or 3, we say that event E
has occurred
• Thus, the event E of a sample space S is said to have occurred if the
outcome ω of the experiment is such that ω ∈ E.
• If the outcome ω is such that ω ∉ E, we say that the event E has not
occurred.
12. types of events
• Events can be classified into various types on the basis of the elements
they have:
1. Impossible and Sure Events: The empty set and the sample space S describe
an impossible event and a sure event respectively.
2. Simple Event: If an event E has only one element in the sample space S, it
is called a simple event.
3. Compound Event: If an event has more than one element in the sample
space S, it is called a compound event
14. Impossible and sure events
• The empty set and the sample space S are two subsets of S.
• The empty set is called an impossible event.
• The entire sample space S is called the sure event
15. Impossible events
• For an example, let us consider the rolling of a die. The associated sample
space is
S = {1, 2, 3, 4, 5, 6}
• Let E be the event “The number appears on the die is a multiple of 7”
• Clearly, no outcome satisfies the condition given in the event E. Thus the
empty set only corresponds to the event E.
• Therefore we say the event E is an impossible event
16. sure events
• Let’s take another event F: “The number turns up is odd or event”
• Therefore F = {1, 2, 3, 4, 5, 6}
• Thus the event F=S is called a sure event
18. simple and compound events
• Simple Event: If an event E has only one element in a sample space, it is
called a simple event
• Compound Event: If an event E has more than one element in a sample
space, it is called a compound event.
19. simple events
• For an example, let us consider the experiment of tossing two coins. Its
sample space is S = {HH, HT, TH, TT}
• There are four simple events: E1 = {HH}, E2={HT}, E3={TH}, E4={TT}
20. compound events
• Let’s take another event F: “The number turns up is odd or event”
• Therefore F = {1, 2, 3, 4, 5, 6}
• Thus the event F=S is called a sure event
22. Algebra of events
• In the chapter on sets we have studied about different ways of combining
two or more sets viz. union, intersection, difference and complement of a
set
• Like-wise we can combine two or more events by using the analogous set
notations.
23. Algebra of events
• Let A, B, C be events associated with an experiment whose sample space is
S.
1. Complementary Event For every event A, there corresponds another event A’
called the complementary event to A. It is also the called the event ‘not A’
2. The Event A or B The union of two events A and B contains all those
elements which are either in A or in B or both.
3. The Event ‘A and B’ The intersection of two events A and B is the events
consisting of those element which are common to both A and B
4. The Event ‘A but not B’ A-B is the event which consists of all those
elements which are in A but not in B.
25. Mutually Exclusive Events
• Two events A and B are called mutually exclusive events if the occurrence
of any one of them excludes the occurrence of the other event i.e. if they
can not occur simultaneously
26. Mutually Exclusive Events
• Consider the experiment of rolling a die, a sample space is S = {1, 2, 3, 4,
5, 6}.
• Let event A be ‘an odd number appears’. Let event B be ‘an even number
appears.
• Clearly event A excludes event B. In other words, the occurrence of A
excludes the occurrence of B and vice-versa.
• A = {1, 3, 5} and B = {2, 4, 6}
28. Exhaustive Events
• Consider an experiment of sample space S and three events A, B and C. If
A∪B∪C = S, then A, B, C are called exhaustive events.
• Let’s take the example of throwing a die. S = {1, 2, 3, 4, 5, 6}.
Event A = “a number less than 4 appears”
Event B = “a number greater than 2 but less than 5 appears”
Event C = “a number greater than 4 appears”
• Then A = {1, 2, 3}, B = {3, 4}, C = {5, 6}. Clearly A∪B∪C = S.
• Therefore A, B, C are called exhaustive events.
29. Mutually exclusive and Exhaustive Events
• In general, if E1, E2, E3, …, En are n events of a sample space S and if E1 ∪
E2 ∪ E3 ∪ … ∪ En = S, then E1, E2, …, En are called exhaustive events.
• Additionally, if Ei ∩ Ej = 𝛟 for i ≠ j i.e. events Ei and Ej are pairwise
disjoint
• Then events E1, E2, …, En are called mutually exclusive and exhaustive
events
31. Axiomatic approach
• Let S be the sample space of a random experiment.
• The probability P is a real valued function whose domain is the power set
of S and range is the interval [0,1] satisfying the following axioms:
i. For any event E, P (E) ≥ 0
ii. P(S) = 1
iii. If E and F are mutually exclusive events, then P(E ∪ F) = P(E) + P(F)
32. Axiomatic approach
• It follows from the axioms that P(𝛟) = 0 as can be seen below:
P(E ∪ 𝛟) = P(E) + P(𝛟) (Since E and 𝛟 are disjoint)
P(E) = P(E) + P(𝛟) => P(𝛟) = 0
It also follows that
• 0 ≤ P(wi) ≤ 1 for each wi ∈ S
• P(w1) + P(w2) + … + P(wn) = 1
• For any event A, P(A) = ΣP(wi), wi ∈ A
33. Mutually exclusive and Exhaustive Events
• In general, if E1, E2, E3, …, En are n events of a sample space S and if E1 ∪
E2 ∪ E3 ∪ … ∪ En = S, then E1, E2, …, En are called exhaustive events.
• Additionally, if Ei ∩ Ej = 𝛟 for i ≠ j i.e. events Ei and Ej are pairwise
disjoint
• Then events E1, E2, …, En are called mutually exclusive and exhaustive
events
34. Probability
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