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.
This document provides an overview of probability concepts and calculations. It defines key terms like probability, experiment, event, and outcome. It describes three approaches to assigning probabilities: classical, empirical, and subjective. It explains how to calculate probabilities using the rules of addition and multiplication, including for mutually exclusive, independent, and conditional events. Contingency tables and tree diagrams are presented as tools for organizing probability calculations involving multiple criteria or stages. The learning objectives cover defining key terms, assigning probabilities, and calculating probabilities using various rules and representations.
This document defines key probability concepts and provides examples to illustrate them. It discusses that probability expresses the likelihood of events, and is calculated as the number of favorable outcomes divided by the total number of possible outcomes. Examples are given to demonstrate calculating probabilities of independent and conditional events. Common terms are defined, such as experiment, outcome, sample space, and event. General probability rules and how probabilities are expressed are also covered.
The document provides an overview of key concepts in probability, including definitions of terms like sample space, event, and probability of an event. It also covers rules for calculating probabilities, such as the addition rule, complementary rule, and product rule for independent and dependent events. Examples are given to demonstrate calculating probabilities using these rules for events like coin tosses, card draws, and dice rolls.
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 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.
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.
This document provides an overview of probability concepts and calculations. It defines key terms like probability, experiment, event, and outcome. It describes three approaches to assigning probabilities: classical, empirical, and subjective. It explains how to calculate probabilities using the rules of addition and multiplication, including for mutually exclusive, independent, and conditional events. Contingency tables and tree diagrams are presented as tools for organizing probability calculations involving multiple criteria or stages. The learning objectives cover defining key terms, assigning probabilities, and calculating probabilities using various rules and representations.
This document defines key probability concepts and provides examples to illustrate them. It discusses that probability expresses the likelihood of events, and is calculated as the number of favorable outcomes divided by the total number of possible outcomes. Examples are given to demonstrate calculating probabilities of independent and conditional events. Common terms are defined, such as experiment, outcome, sample space, and event. General probability rules and how probabilities are expressed are also covered.
The document provides an overview of key concepts in probability, including definitions of terms like sample space, event, and probability of an event. It also covers rules for calculating probabilities, such as the addition rule, complementary rule, and product rule for independent and dependent events. Examples are given to demonstrate calculating probabilities using these rules for events like coin tosses, card draws, and dice rolls.
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 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.
This document provides an overview of key probability concepts:
1. It defines posterior probability, Bayes' theorem, subjective and objective probability, and the multiplication and addition rules of probability.
2. It explains key probability terms like experiments, events, outcomes, permutations, and combinations, and describes classical, empirical and subjective approaches to probability.
3. It provides examples of how to calculate probabilities using the rules of addition, multiplication, and Bayes' theorem, as well as how to apply concepts like conditional probability, joint probability, and tree diagrams.
The 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.
This document discusses probability and provides examples of calculating probabilities of events. Some key points covered include:
- Probability allows quantifying the variability in outcomes of experiments with uncertain results.
- Key concepts like sample space, events, outcomes, mutually exclusive events, independent events are defined.
- Probability is calculated as the number of favorable outcomes divided by the total number of outcomes.
- Examples of probability calculations involving dice rolls, card draws, and coin tosses are provided.
- Theorems like addition rule, multiplication rule, and conditional probability are discussed.
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.
The document provides an overview of probability concepts including:
- Defining probability as a value between 0 and 1 describing the likelihood of an event.
- Explaining experiment, outcome, and event.
- Identifying the three approaches to assigning probabilities: classical, empirical, and subjective.
- Explaining rules of addition, multiplication, and conditional probability for calculating probabilities.
- Describing how to calculate probabilities using contingency tables, Bayes' theorem, and principles of counting such as permutations and combinations.
The document provides an overview of probability concepts including:
- Defining probability as a value between 0 and 1 describing the likelihood of an event.
- Explaining experiment, outcome, and event.
- Identifying the three approaches to assigning probabilities: classical, empirical, and subjective.
- Explaining rules of addition, multiplication, and conditional probability for calculating probabilities.
- Describing how to calculate probabilities using contingency tables, Bayes' theorem, and principles of counting such as permutations and combinations.
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.
The document discusses key concepts in probability, including:
- Probability is the likelihood of an event occurring, defined as favorable outcomes divided by total outcomes.
- Sample space is the set of all possible outcomes, sample points are elements of the sample space, and events are sets of outcomes.
- Conditional probability is the probability of one event given that another event has occurred, defined as the probability of the events intersecting divided by the probability of the conditioning event.
- The multiplication theorem and independence relate the probabilities of intersecting and independent events.
- Bayes' theorem gives the probability of an event given observed evidence by relating conditional and marginal probabilities.
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.
Chapter 12 Probability and Statistics.pptJoyceNolos
The document discusses probability and statistics concepts including:
1) The counting principle, which states that if there are "a" ways for one activity to occur and "b" ways for a second activity to occur, then there are (a x b) ways for both to occur.
2) Independent and dependent events, where independent events do not affect each other and dependent events do.
3) Probability is calculated as the number of desired outcomes divided by the total number of possible outcomes. Probability must be between 0 and 1.
4) Examples are provided to demonstrate calculating probabilities of independent and dependent events using formulas and tree diagrams.
BASIC PROBABILITY distribution - Copy.pptxMinilikDerseh1
This document provides an overview of key probability concepts needed for statistical inference. It defines probability as relating to outcomes that are not absolutely certain but affect the likelihood of events. Probability is quantified using experiments with sample spaces and outcomes, where classical probability assigns equal likelihood and empirical probability is based on observations. Events are defined which can be mutually exclusive, unions, or intersections. Conditional probability expresses the likelihood of one event given another. Counting rules like permutations and combinations are introduced. Random variables are defined as taking on values from a probability distribution, which can be discrete or continuous. Examples demonstrate constructing probability distributions for discrete random variables.
This document provides an overview of key probability concepts including:
- Random experiments, outcomes, events, sample spaces, mutually exclusive and independent events
- Classical, relative frequency, subjective, and axiomatic approaches to defining probability
- Probability theorems including addition, multiplication, Bayes' theorem, and their applications to dependent and independent events
- Examples are provided to illustrate concepts like conditional probability and using Bayes' theorem to update probabilities with new information
Probability Overview with Hands-on from Object AutomationObject Automation
Object Automation Software Solutions Pvt Ltd in collaboration with SRM Ramapuram delivered Workshop for Skill Development on Artificial Intelligence.
Probability Overview with Hands-on by Mr.N.Vinay, Business Manager from Object Automation.
4Probability and probability distributions (1).pptxAmanuelMerga
Draw the figure. We want the area between -2.55 and +2.55.
Look up the area to the left of -2.55 in the table. It is 0.0049. Also look up
the area to the left of +2.55. It is 0.9937. Subtract these two areas. 0.9937
- 0.0049 = 0.9888. Hence, the probability that a z picked at random will
have a value between -2.55 and +2.55 is 0.9888 or 98.88%
4Probability and probability distributions.pdfAmanuelDina
Here are the key steps to solve this problem:
1) Draw the standard normal curve
2) The probability is the area between -2.55 and 2.55
3) From the standard normal table:
P(Z ≤ 2.55) = 0.9938
P(Z ≤ -2.55) = 0.0049
4) Use the area property:
P(-2.55 ≤ Z ≤ 2.55) = P(Z ≤ 2.55) - P(Z ≤ -2.55)
= 0.9938 - 0.0049
= 0.9889
Therefore, the probability that a z value will be between -2.55 and 2
[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.
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 provides an introduction to probability concepts including:
1) Definitions of probability, sample space, outcomes, events, and interpretations of probability including classical, empirical, and subjective.
2) Examples of sample spaces for experiments like coin tosses and dice rolls.
3) Explanations of tree diagrams, Venn diagrams, and the addition rules for determining probabilities of mutually exclusive and non-mutually exclusive events.
4) Descriptions of joint, marginal, and conditional probabilities and examples of calculating each from contingency tables.
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.
Falcon stands out as a top-tier P2P Invoice Discounting platform in India, bridging esteemed blue-chip companies and eager investors. Our goal is to transform the investment landscape in India by establishing a comprehensive destination for borrowers and investors with diverse profiles and needs, all while minimizing risk. What sets Falcon apart is the elimination of intermediaries such as commercial banks and depository institutions, allowing investors to enjoy higher yields.
Discover the Future of Dogecoin with Our Comprehensive Guidance36 Crypto
Learn in-depth about Dogecoin's trajectory and stay informed with 36crypto's essential and up-to-date information about the crypto space.
Our presentation delves into Dogecoin's potential future, exploring whether it's destined to skyrocket to the moon or face a downward spiral. In addition, it highlights invaluable insights. Don't miss out on this opportunity to enhance your crypto understanding!
https://36crypto.com/the-future-of-dogecoin-how-high-can-this-cryptocurrency-reach/
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Similar to CLO2-PPT2-Probability Addition Rules.pptx
This document provides an overview of key probability concepts:
1. It defines posterior probability, Bayes' theorem, subjective and objective probability, and the multiplication and addition rules of probability.
2. It explains key probability terms like experiments, events, outcomes, permutations, and combinations, and describes classical, empirical and subjective approaches to probability.
3. It provides examples of how to calculate probabilities using the rules of addition, multiplication, and Bayes' theorem, as well as how to apply concepts like conditional probability, joint probability, and tree diagrams.
The 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.
This document discusses probability and provides examples of calculating probabilities of events. Some key points covered include:
- Probability allows quantifying the variability in outcomes of experiments with uncertain results.
- Key concepts like sample space, events, outcomes, mutually exclusive events, independent events are defined.
- Probability is calculated as the number of favorable outcomes divided by the total number of outcomes.
- Examples of probability calculations involving dice rolls, card draws, and coin tosses are provided.
- Theorems like addition rule, multiplication rule, and conditional probability are discussed.
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.
The document provides an overview of probability concepts including:
- Defining probability as a value between 0 and 1 describing the likelihood of an event.
- Explaining experiment, outcome, and event.
- Identifying the three approaches to assigning probabilities: classical, empirical, and subjective.
- Explaining rules of addition, multiplication, and conditional probability for calculating probabilities.
- Describing how to calculate probabilities using contingency tables, Bayes' theorem, and principles of counting such as permutations and combinations.
The document provides an overview of probability concepts including:
- Defining probability as a value between 0 and 1 describing the likelihood of an event.
- Explaining experiment, outcome, and event.
- Identifying the three approaches to assigning probabilities: classical, empirical, and subjective.
- Explaining rules of addition, multiplication, and conditional probability for calculating probabilities.
- Describing how to calculate probabilities using contingency tables, Bayes' theorem, and principles of counting such as permutations and combinations.
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.
The document discusses key concepts in probability, including:
- Probability is the likelihood of an event occurring, defined as favorable outcomes divided by total outcomes.
- Sample space is the set of all possible outcomes, sample points are elements of the sample space, and events are sets of outcomes.
- Conditional probability is the probability of one event given that another event has occurred, defined as the probability of the events intersecting divided by the probability of the conditioning event.
- The multiplication theorem and independence relate the probabilities of intersecting and independent events.
- Bayes' theorem gives the probability of an event given observed evidence by relating conditional and marginal probabilities.
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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The document discusses probability and statistics concepts including:
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2) Independent and dependent events, where independent events do not affect each other and dependent events do.
3) Probability is calculated as the number of desired outcomes divided by the total number of possible outcomes. Probability must be between 0 and 1.
4) Examples are provided to demonstrate calculating probabilities of independent and dependent events using formulas and tree diagrams.
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This document provides an overview of key probability concepts including:
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4Probability and probability distributions (1).pptxAmanuelMerga
Draw the figure. We want the area between -2.55 and +2.55.
Look up the area to the left of -2.55 in the table. It is 0.0049. Also look up
the area to the left of +2.55. It is 0.9937. Subtract these two areas. 0.9937
- 0.0049 = 0.9888. Hence, the probability that a z picked at random will
have a value between -2.55 and +2.55 is 0.9888 or 98.88%
4Probability and probability distributions.pdfAmanuelDina
Here are the key steps to solve this problem:
1) Draw the standard normal curve
2) The probability is the area between -2.55 and 2.55
3) From the standard normal table:
P(Z ≤ 2.55) = 0.9938
P(Z ≤ -2.55) = 0.0049
4) Use the area property:
P(-2.55 ≤ Z ≤ 2.55) = P(Z ≤ 2.55) - P(Z ≤ -2.55)
= 0.9938 - 0.0049
= 0.9889
Therefore, the probability that a z value will be between -2.55 and 2
[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.
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 provides an introduction to probability concepts including:
1) Definitions of probability, sample space, outcomes, events, and interpretations of probability including classical, empirical, and subjective.
2) Examples of sample spaces for experiments like coin tosses and dice rolls.
3) Explanations of tree diagrams, Venn diagrams, and the addition rules for determining probabilities of mutually exclusive and non-mutually exclusive events.
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Fabular Frames and the Four Ratio ProblemMajid Iqbal
Digital, interactive art showing the struggle of a society in providing for its present population while also saving planetary resources for future generations. Spread across several frames, the art is actually the rendering of real and speculative data. The stereographic projections change shape in response to prompts and provocations. Visitors interact with the model through speculative statements about how to increase savings across communities, regions, ecosystems and environments. Their fabulations combined with random noise, i.e. factors beyond control, have a dramatic effect on the societal transition. Things get better. Things get worse. The aim is to give visitors a new grasp and feel of the ongoing struggles in democracies around the world.
Stunning art in the small multiples format brings out the spatiotemporal nature of societal transitions, against backdrop issues such as energy, housing, waste, farmland and forest. In each frame we see hopeful and frightful interplays between spending and saving. Problems emerge when one of the two parts of the existential anaglyph rapidly shrinks like Arctic ice, as factors cross thresholds. Ecological wealth and intergenerational equity areFour at stake. Not enough spending could mean economic stress, social unrest and political conflict. Not enough saving and there will be climate breakdown and ‘bankruptcy’. So where does speculative design start and the gambling and betting end? Behind each fabular frame is a four ratio problem. Each ratio reflects the level of sacrifice and self-restraint a society is willing to accept, against promises of prosperity and freedom. Some values seem to stabilise a frame while others cause collapse. Get the ratios right and we can have it all. Get them wrong and things get more desperate.
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1. CLO 2: Addition Rules of Probability
CIS 2003 Probability and Statistics
2. LEARNING OBJECTIVES
At the end of this lecture, student should be able to
Classify events as mutually exclusive and not mutually exclusive
Apply addition rules of probability to both mutually exclusive and non-
mutually exclusive events.
3. 3
Probability
PROBABILITY A value between zero and one,
describing the relative possibility (chance or
likelihood) an event will occur.
A probability of 0 or 0% means the event is impossible.
A probability of 1 or 100% means the event will happen for
sure.
0 1
The sun The sun
will disappear will rise
tomorrow tomorrow.
0% 100%
4. 4
Addition Rule of Probability : Mutually Exclusive
Events
We can add probabilities as long as the events are mutually exclusive.
Mutually exclusive events cannot happen at the same time - that is, only
one can occur during the experiment.
(If you roll one die, you cannot roll a 6 and a 1. You can roll a 6 or a 1, however).
5. 5
Addition Rules of Probability
There are 2 Addition Rules- for events that are mutually exclusive or are not mutually exclusive.
For Mutually Exclusive Events - If two events A and B are mutually exclusive, the probability of one or the other
event occurring equals the sum of their probabilities.
P(A or B) = P(A) + P(B)
(Remember that events are mutually exclusive if they can’t occur at the same time. )
Example: What is the probability of rolling a number greater than 4 or a number less than
3?
We can call the first event, A (rolling 5 or 6) and the second event, B (rolling 1 or 2).
Since both these events cannot occur at the same time,
P(A) can be added to P(B). i.e. P(A or B) = P(A) + P(B)
=2/6 + 2/6 = 4/6 or 2/3
Question
6. 6
Addition Rules of Probability
For events that are not Mutually Exclusive - If two events A and B are not
mutually exclusive, the probability of one or the other event is given as:
P(A or B) = P(A) + P(B) – P(A and B)
Events are not mutually exclusive if they can occur at the same time.
Example: What is the probability of rolling 4 or more or rolling an odd
number?
Event A: rolling 4 or more ={4,5,6}
Event B: rolling an odd number= {1,3,5}
The outcome 5 is in both events so the events are not mutually
exclusive.
So, P(A or B) = 3/6 + 3/6 – 1/6 =5/6
7. 7
Example 2: What is the probability of rolling a number greater than or
equal to 4 or rolling a 6?
P(4, 5, or 6) + P(6) X We cannot add these because they are not mutually
exclusive. 6 is in both sets so we need another addition rule here!
How would you solve the problem?
Since 6 is already included in the first set, we are still only looking for P(4,5, or 6)
The probability is 3/6 or ½.
8. 8
Picking Cards From a Deck
J Q K A 2 3 4 5 6 7 8 9 10
J Q K A 2 3 4 5 6 7 8 9 10
J Q K A 2 3 4 5 6 7 8 9 10
J Q K A 2 3 4 5 6 7 8 9 10
9. 9
Example 3:What is the probability of picking a 3 or a 10 from a deck of
cards?
(Total 52 cards in a deck).
P(A) = probability of picking a 3.
P(B) = probability of picking a 10.
P(A) = 4/52 because there are 4 threes.
P(B) = 4/52 because there are 4 tens.
P(A or B) = 4/52 + 4/52 = 8/52
= 2/13.
10. 10
Joint Probability
Often events are not mutually exclusive.
Sometimes two or more events can happen at the same time.
This is called joint probability.
For example:
A student can take both English and Mathematics in the same semester.
The probability of a student taking English and Mathematics is an example
of joint probability.
11. 11
Rule of Addition for non-mutually Exclusive Events
If two events are not mutually exclusive the probability of
one or the other occurring (A or B) is calculated by taking
the sum of the individual probabilities minus the joint
probability:
P(A or B) = P(A) + P(B) – P(A and B)
12. 12
Example
What is the probability of rolling a die and observing a number less than or equal to 4,
or a number greater than 3?
P(1,2,3 or 4) or P(4, 5 or 6).
Using a Venn Diagram, 4 is in both events:
1,2 ,3 4 5,6 P(1,2,3 or 4) = 4/6
P(4,5 or 6) = 3/6
P(4) = 1/6
P(1,2,3 or 4) + P(4, 5 or 6) – P(4)
= 4/6 + 3/6 – 1/6 =
= 6/6 = 1 or 100%
13. 13
Picking Cards From a Deck
J Q K A 2 3 4 5 6 7 8 9 10
J Q K A 2 3 4 5 6 7 8 9 10
J Q K A 2 3 4 5 6 7 8 9 10
J Q K A 2 3 4 5 6 7 8 9 10
14. 14
Example 2
What is the probability that a card chosen at random from a standard deck of cards
will be either a king or a heart?
Card Probability
King 4 cards out of 52 4/52
Heart 13 cards out of 52 13/52
P(King or Heart) = P(King) + P(Heart)- P(King and Heart)
P(A or B) = P(A) + P(B) - P(A and B)
= 4/52 + 13/52 - 1/52
= 16/52, or .3077 or 30.77%
15. 15
Example 3
A survey of 1000 tourists to RAK
showed that 800 visited the Pure
Veg Restaurant and 500 visited
the Grand Restaurant. 400
tourists visited both restaurants.
What is the probability that a
selected tourist visited the
Pure Veg Restaurant or the
Grand Restaurant?
P(Pure Veg) + P(Grand) – P(Pure
Veg and Grand)
P(A) + P(B) – P(A and B)
= 800/1000 + 500/1000 -
400/1000
= 900/1000 = 9/10 = 0.9 =
90%
What is the probability that the
selected tourist did not visit
the Pure Veg or the Grand?
1 – 0.9 = 0.1 or 100% -
90% = 10%
What rule did you use to solve
this?
The Complement Rule.
What is the probability that a
tourist visits the Pure Veg and
the Grand Restaurant called?
Joint probability
Are the events of eating at the
Pure Veg and Eating at the
Grand mutually exclusive?
No, because some tourists ate
at both restaurants.
