A probability distribution is an outline of probabilities for the estimations of an arbitrary variable. As a distribution, the planning of the estimations of an irregular variable to a likelihood has a shape when all estimations of the arbitrary variable are arranged. The distribution additionally has general properties that can be estimated.
Probability can be calculated using three approaches: a priori, empirical, and subjective. A priori probability is based on prior knowledge, empirical on observed outcomes, and subjective on personal analysis. Simple events have a single characteristic while joint events have multiple. The sample space includes all possible outcomes. Conditional probability is the probability of one event given another. Independence means one event does not affect another's probability.
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.
Stat lesson 4.2 rules of computing probabilitypipamutuc
Here are the answers to the quiz questions:
1. a) The experiment is surveying teenagers about a newly developed soft drink and asking them to compare it with their favorite drink.
b) One possible outcome is that a teenager prefers the new soft drink to their favorite drink.
c) A possible event is that a teenager selected prefers Coke to the new soft drink.
2. A contingency table classifies observations according to two or more identifiable characteristics. It allows the computation of conditional probabilities.
3. The multiplication rule for independent events states that the probability of two independent events occurring together is equal to the product of their individual probabilities. So if A and B are independent, P(A and B) = P(
This document provides an overview of key probability concepts including:
(1) Definitions of random experiments, sample spaces, events, and probability;
(2) The addition and multiplication theorems and conditional probability;
(3) Mathematical expectation and probability distributions including the binomial, Poisson, and normal distributions. Examples are provided to illustrate key terminology and formulas.
This document provides an overview of probability concepts including complements, conditional probability, and Bayes' theorem. It defines key probability terms and notation. Examples are provided to demonstrate how to calculate probabilities of events, complementary events, conditional probabilities, and use Bayes' theorem. Objectives cover determining sample spaces, finding probabilities using classical or empirical methods, and applying addition, multiplication, counting and other probability rules.
This document discusses three topics related to probability: 1) the probability of an event, which is defined as the number of desired outcomes divided by the total number of possible outcomes, 2) the probability of mutually exclusive events, which is the sum of the individual probabilities minus the probability of both occurring, and 3) the probability of independent events, which uses the multiplication rule and is the product of the individual probabilities if the events do not affect each other.
It gives detail description about probability, types of probability, difference between mutually exclusive events and independent events, difference between conditional and unconditional probability and Bayes' theorem
Solution to final exam engineering statistics 2014 2015Chenar Salam
1. The question asks to find the probability of a couple having at least 2 boys among 5 children, assuming equal probability of boys and girls and independence between children.
2. The sample space includes outcomes with 0 boys (1 outcome), 1 boy (5 outcomes), and at least 2 boys.
3. The probability of having at least 2 boys is calculated as 1 minus the probability of having less than 2 boys (0 or 1 boy). This gives a probability of 0.324 of having at least 2 boys among 5 children.
Probability can be calculated using three approaches: a priori, empirical, and subjective. A priori probability is based on prior knowledge, empirical on observed outcomes, and subjective on personal analysis. Simple events have a single characteristic while joint events have multiple. The sample space includes all possible outcomes. Conditional probability is the probability of one event given another. Independence means one event does not affect another's probability.
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.
Stat lesson 4.2 rules of computing probabilitypipamutuc
Here are the answers to the quiz questions:
1. a) The experiment is surveying teenagers about a newly developed soft drink and asking them to compare it with their favorite drink.
b) One possible outcome is that a teenager prefers the new soft drink to their favorite drink.
c) A possible event is that a teenager selected prefers Coke to the new soft drink.
2. A contingency table classifies observations according to two or more identifiable characteristics. It allows the computation of conditional probabilities.
3. The multiplication rule for independent events states that the probability of two independent events occurring together is equal to the product of their individual probabilities. So if A and B are independent, P(A and B) = P(
This document provides an overview of key probability concepts including:
(1) Definitions of random experiments, sample spaces, events, and probability;
(2) The addition and multiplication theorems and conditional probability;
(3) Mathematical expectation and probability distributions including the binomial, Poisson, and normal distributions. Examples are provided to illustrate key terminology and formulas.
This document provides an overview of probability concepts including complements, conditional probability, and Bayes' theorem. It defines key probability terms and notation. Examples are provided to demonstrate how to calculate probabilities of events, complementary events, conditional probabilities, and use Bayes' theorem. Objectives cover determining sample spaces, finding probabilities using classical or empirical methods, and applying addition, multiplication, counting and other probability rules.
This document discusses three topics related to probability: 1) the probability of an event, which is defined as the number of desired outcomes divided by the total number of possible outcomes, 2) the probability of mutually exclusive events, which is the sum of the individual probabilities minus the probability of both occurring, and 3) the probability of independent events, which uses the multiplication rule and is the product of the individual probabilities if the events do not affect each other.
It gives detail description about probability, types of probability, difference between mutually exclusive events and independent events, difference between conditional and unconditional probability and Bayes' theorem
Solution to final exam engineering statistics 2014 2015Chenar Salam
1. The question asks to find the probability of a couple having at least 2 boys among 5 children, assuming equal probability of boys and girls and independence between children.
2. The sample space includes outcomes with 0 boys (1 outcome), 1 boy (5 outcomes), and at least 2 boys.
3. The probability of having at least 2 boys is calculated as 1 minus the probability of having less than 2 boys (0 or 1 boy). This gives a probability of 0.324 of having at least 2 boys among 5 children.
This document discusses probability and its key concepts. It begins by defining probability as a quantitative measure of uncertainty ranging from 0 to 1. Probability can be understood objectively based on problems or subjectively based on beliefs. Key probability concepts discussed include:
- Sample space, simple events, and compound events
- Classical, relative frequency, and subjective approaches to assigning probabilities
- Complement, intersection, and union of events
- Conditional probability and independence of events
- Rules for calculating probabilities of combined events like the multiplication rule
Examples are provided to illustrate concepts like defining sample spaces, calculating probabilities of individual and combined events, determining conditional probabilities, and assessing independence. Overall, the document provides a comprehensive overview of fundamental probability
This document provides an introduction to probability, conditional probability, and random variables. It defines key concepts such as sample space, simple events, probability distribution, discrete and continuous random variables, and their properties including mean, variance, and Bernoulli trials. Examples are given for each concept to illustrate their calculation and application to experiments with outcomes that are either certain or random.
The document defines probability as a numerical value between 0 and 1 used to express the likelihood of an event occurring. It explains the different approaches to probability including relative frequency, classic, and subjective. Examples are provided to illustrate concepts such as sample space, events, mutually exclusive and non-mutually exclusive events, conditional probability, independent events, and counting rules for probability calculations.
This document discusses basic concepts of probability, including:
- The addition rule and multiplication rule for calculating probabilities of compound events.
- Events can be disjoint (mutually exclusive) or not disjoint.
- The probability of an event occurring or its complement must equal 1.
- How to calculate the probability of at least one occurrence of an event using the complement.
- When applying the multiplication rule, you must consider whether events are independent or dependent.
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
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 provides an overview of probability models and concepts. It discusses the Poisson probability model and how it can be used to model rare events that occur over a region of opportunity. An example is provided of a bank analyzing the number of customers arriving at its ATM during peak hours using a Poisson probability distribution based on an average arrival rate. The document also introduces the chi-square goodness of fit test for evaluating how well a probability model fits real data.
This document discusses probability and related concepts such as sample spaces, classical probability, empirical probability, mutually exclusive and non-mutually exclusive events, independent and dependent events, and conditional probability. It provides examples to illustrate these concepts using problems involving coins, dice, and manufacturing companies. Bayes' theorem is also introduced as a way to calculate reverse conditional probabilities.
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 provides guidance on teaching a lesson on probability. It outlines key concepts to define, such as experimental and theoretical probabilities. It also describes solving problems involving combined probabilities of mutually exclusive and independent events. Example problems are provided for different probability concepts. The teacher is instructed to check students' understanding through oral questions and exercises.
This presentation provides an introduction to basic probability concepts. It defines probability as the study of randomness and uncertainty, and describes how probability was originally associated with games of chance. Key concepts discussed include random experiments, sample spaces, events, unions and intersections of events, and Venn diagrams. The presentation establishes the axioms of probability, including that a probability must be between 0 and 1, the probability of the sample space is 1, and probabilities of mutually exclusive events sum to the total probability. Formulas for computing probabilities of unions, intersections, and complements of events are also presented.
Unit 4--probability and probability distribution (1).pptxakshay353895
This document provides an overview of probability concepts and probability distributions. It begins by defining basic probability terms like random experiments, sample spaces, events, and outcomes. It then discusses how to calculate probabilities of simple, joint, and compound events using formulas and diagrams like Venn diagrams and contingency tables. The document also covers conditional probability, independence, Bayes' theorem, and counting methods like permutations and combinations. Finally, it introduces discrete random variables and how to calculate the expected value, variance, and standard deviation of discrete random variables.
The document provides an introduction to probability, including:
1) Probability is a measure of how likely an event is to occur, expressed as a number between 0 and 1.
2) The development of probability was influenced by gambling in 17th century France and the later works of James Bernoulli, Abraham De Moivre, and Pierre-Simon Laplace.
3) There are three types of probability: theoretical, empirical, and subjective. Theoretical probability uses mathematical models, empirical probability is based on experimental data, and subjective probability relies on personal beliefs.
1. The document discusses basic concepts in probability and statistics, including sample spaces, events, probability distributions, and random variables.
2. Key concepts are explained such as independent and conditional probability, Bayes' theorem, and common probability distributions like the uniform and normal distributions.
3. Statistical analysis methods are introduced including how to estimate the mean and variance from samples from a distribution.
The document provides an introduction to probability. It discusses:
- What probability is and the definition of probability as a number between 0 and 1 that expresses the likelihood of an event occurring.
- A brief history of probability including its development in French society in the 1650s and key figures like James Bernoulli, Abraham De Moivre, and Pierre-Simon Laplace.
- Key terms used in probability like events, outcomes, sample space, theoretical probability, empirical probability, and subjective probability.
- The three types of probability: theoretical, empirical, and subjective probability.
- General probability rules including: the probability of impossible/certain events; the sum of all probabilities equaling 1; complements
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 ...
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.
This document introduces key concepts in probability:
- Probability is the likelihood of an event occurring, which can be measured numerically or described qualitatively.
- Events can be classified as exhaustive, favorable, mutually exclusive, equally likely, complementary, and independent.
- There are three approaches to defining probability: classical, frequency, and axiomatic. The classical approach defines probability as the number of favorable outcomes over the total number of possible outcomes. The frequency approach defines probability as the limit of the ratio of favorable outcomes to the total number of trials. The axiomatic approach defines probability based on axioms or statements assumed to be true.
- Key properties of probability include that the probability of an event is between 0
This document introduces key concepts in probability:
1. Probability is the likelihood of an event occurring, which can be expressed as a number or words like "impossible" or "likely".
2. Events can be classified as exhaustive, favorable, mutually exclusive, equally likely, complementary, and independent.
3. There are three approaches to defining probability: classical, frequency, and axiomatic. The classical approach defines probability as the number of favorable outcomes over the total number of possible outcomes. The frequency approach defines it as the limit of favorable outcomes over total trials. The axiomatic approach uses axioms like probabilities being between 0 and 1.
4. Several properties of probability are described, like the sum
- Probability theory studies possible outcomes of events and their likelihoods, expressed as a value from 0 to 1.
- Probability can be understood as the chance of an outcome, often expressed as a percentage between 0 and 100%.
- The analysis of data using probability models is called statistics.
This document discusses basic probability concepts including probability, events, sample spaces, and counting rules. It defines probability as the chance an uncertain event will occur between 0 and 1. Simple probability is the probability of a single event, while joint probability is the probability of two or more events occurring together. Conditional probability is the probability of one event given another has occurred. The document provides examples of calculating probabilities using formulas and contingency tables. It also covers independence, addition rules, and counting rules for determining the number of possible outcomes.
This document discusses probability and its key concepts. It begins by defining probability as a quantitative measure of uncertainty ranging from 0 to 1. Probability can be understood objectively based on problems or subjectively based on beliefs. Key probability concepts discussed include:
- Sample space, simple events, and compound events
- Classical, relative frequency, and subjective approaches to assigning probabilities
- Complement, intersection, and union of events
- Conditional probability and independence of events
- Rules for calculating probabilities of combined events like the multiplication rule
Examples are provided to illustrate concepts like defining sample spaces, calculating probabilities of individual and combined events, determining conditional probabilities, and assessing independence. Overall, the document provides a comprehensive overview of fundamental probability
This document provides an introduction to probability, conditional probability, and random variables. It defines key concepts such as sample space, simple events, probability distribution, discrete and continuous random variables, and their properties including mean, variance, and Bernoulli trials. Examples are given for each concept to illustrate their calculation and application to experiments with outcomes that are either certain or random.
The document defines probability as a numerical value between 0 and 1 used to express the likelihood of an event occurring. It explains the different approaches to probability including relative frequency, classic, and subjective. Examples are provided to illustrate concepts such as sample space, events, mutually exclusive and non-mutually exclusive events, conditional probability, independent events, and counting rules for probability calculations.
This document discusses basic concepts of probability, including:
- The addition rule and multiplication rule for calculating probabilities of compound events.
- Events can be disjoint (mutually exclusive) or not disjoint.
- The probability of an event occurring or its complement must equal 1.
- How to calculate the probability of at least one occurrence of an event using the complement.
- When applying the multiplication rule, you must consider whether events are independent or dependent.
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
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 provides an overview of probability models and concepts. It discusses the Poisson probability model and how it can be used to model rare events that occur over a region of opportunity. An example is provided of a bank analyzing the number of customers arriving at its ATM during peak hours using a Poisson probability distribution based on an average arrival rate. The document also introduces the chi-square goodness of fit test for evaluating how well a probability model fits real data.
This document discusses probability and related concepts such as sample spaces, classical probability, empirical probability, mutually exclusive and non-mutually exclusive events, independent and dependent events, and conditional probability. It provides examples to illustrate these concepts using problems involving coins, dice, and manufacturing companies. Bayes' theorem is also introduced as a way to calculate reverse conditional probabilities.
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 provides guidance on teaching a lesson on probability. It outlines key concepts to define, such as experimental and theoretical probabilities. It also describes solving problems involving combined probabilities of mutually exclusive and independent events. Example problems are provided for different probability concepts. The teacher is instructed to check students' understanding through oral questions and exercises.
This presentation provides an introduction to basic probability concepts. It defines probability as the study of randomness and uncertainty, and describes how probability was originally associated with games of chance. Key concepts discussed include random experiments, sample spaces, events, unions and intersections of events, and Venn diagrams. The presentation establishes the axioms of probability, including that a probability must be between 0 and 1, the probability of the sample space is 1, and probabilities of mutually exclusive events sum to the total probability. Formulas for computing probabilities of unions, intersections, and complements of events are also presented.
Unit 4--probability and probability distribution (1).pptxakshay353895
This document provides an overview of probability concepts and probability distributions. It begins by defining basic probability terms like random experiments, sample spaces, events, and outcomes. It then discusses how to calculate probabilities of simple, joint, and compound events using formulas and diagrams like Venn diagrams and contingency tables. The document also covers conditional probability, independence, Bayes' theorem, and counting methods like permutations and combinations. Finally, it introduces discrete random variables and how to calculate the expected value, variance, and standard deviation of discrete random variables.
The document provides an introduction to probability, including:
1) Probability is a measure of how likely an event is to occur, expressed as a number between 0 and 1.
2) The development of probability was influenced by gambling in 17th century France and the later works of James Bernoulli, Abraham De Moivre, and Pierre-Simon Laplace.
3) There are three types of probability: theoretical, empirical, and subjective. Theoretical probability uses mathematical models, empirical probability is based on experimental data, and subjective probability relies on personal beliefs.
1. The document discusses basic concepts in probability and statistics, including sample spaces, events, probability distributions, and random variables.
2. Key concepts are explained such as independent and conditional probability, Bayes' theorem, and common probability distributions like the uniform and normal distributions.
3. Statistical analysis methods are introduced including how to estimate the mean and variance from samples from a distribution.
The document provides an introduction to probability. It discusses:
- What probability is and the definition of probability as a number between 0 and 1 that expresses the likelihood of an event occurring.
- A brief history of probability including its development in French society in the 1650s and key figures like James Bernoulli, Abraham De Moivre, and Pierre-Simon Laplace.
- Key terms used in probability like events, outcomes, sample space, theoretical probability, empirical probability, and subjective probability.
- The three types of probability: theoretical, empirical, and subjective probability.
- General probability rules including: the probability of impossible/certain events; the sum of all probabilities equaling 1; complements
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 ...
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.
This document introduces key concepts in probability:
- Probability is the likelihood of an event occurring, which can be measured numerically or described qualitatively.
- Events can be classified as exhaustive, favorable, mutually exclusive, equally likely, complementary, and independent.
- There are three approaches to defining probability: classical, frequency, and axiomatic. The classical approach defines probability as the number of favorable outcomes over the total number of possible outcomes. The frequency approach defines probability as the limit of the ratio of favorable outcomes to the total number of trials. The axiomatic approach defines probability based on axioms or statements assumed to be true.
- Key properties of probability include that the probability of an event is between 0
This document introduces key concepts in probability:
1. Probability is the likelihood of an event occurring, which can be expressed as a number or words like "impossible" or "likely".
2. Events can be classified as exhaustive, favorable, mutually exclusive, equally likely, complementary, and independent.
3. There are three approaches to defining probability: classical, frequency, and axiomatic. The classical approach defines probability as the number of favorable outcomes over the total number of possible outcomes. The frequency approach defines it as the limit of favorable outcomes over total trials. The axiomatic approach uses axioms like probabilities being between 0 and 1.
4. Several properties of probability are described, like the sum
- Probability theory studies possible outcomes of events and their likelihoods, expressed as a value from 0 to 1.
- Probability can be understood as the chance of an outcome, often expressed as a percentage between 0 and 100%.
- The analysis of data using probability models is called statistics.
This document discusses basic probability concepts including probability, events, sample spaces, and counting rules. It defines probability as the chance an uncertain event will occur between 0 and 1. Simple probability is the probability of a single event, while joint probability is the probability of two or more events occurring together. Conditional probability is the probability of one event given another has occurred. The document provides examples of calculating probabilities using formulas and contingency tables. It also covers independence, addition rules, and counting rules for determining the number of possible outcomes.
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Chapter 4: Probability
4.3: Complements and Conditional Probability, and Bayes' Theorem
This document discusses elementary probability and its applications in medicine. It provides definitions of key probability terms like chance, outcome, experiment, and event. It also covers different approaches to measuring probability, including the classical, frequentist, subjective, and axiomatic approaches. Conditional probability and independence are explained. The document emphasizes that probability theory allows physicians to assess diagnoses and draw conclusions about patient populations despite diagnostic uncertainties.
The document provides an overview of elementary probability concepts including:
- Defining probability as the chance of an event occurring and explaining common probability notions.
- Introducing key probability terms like sample space, events, outcomes, and complementary/intersection of events.
- Explaining counting rules like the addition rule, multiplication rule, and how to calculate permutations and combinations.
- Outlining different approaches to defining probability including classical, subjective, axiomatic, and frequency-based definitions.
- Detailing several probability rules like calculating the probability of a union of events and applying the addition rule to non-mutually exclusive events.
The document provides an overview of elementary probability concepts including:
- Defining probability as the chance of an event occurring and explaining common probability notions.
- Introducing key probability terms like sample space, events, outcomes, and complementary/intersection of events.
- Explaining counting rules like the addition rule, multiplication rule, and how to calculate permutations and combinations.
- Outlining different approaches to defining probability including classical, subjective, axiomatic, and frequency-based definitions.
- Detailing several probability rules like calculating the probability of a union of events and applying the addition rule to non-mutually exclusive events.
This document discusses types of probability and provides definitions and examples of key probability concepts. It begins with an introduction to probability theory and its applications. The document then defines terms like random experiments, sample spaces, events, favorable events, mutually exclusive events, and independent events. It describes three approaches to measuring probability: classical, frequency, and axiomatic. It concludes with theorems of probability and references.
Probability and probability distributions ppt @ bec domsBabasab Patil
This document provides an overview of key concepts in probability and probability distributions, including:
- Defining probability, experiments, sample spaces, and events
- Common probability rules such as addition and multiplication rules
- Discrete and continuous random variables and their associated probability distributions
- Key metrics for probability distributions like expected value and standard deviation
- Conditional probability and Bayes' Theorem
The document aims to explain fundamental probability concepts and prepare the reader to compute and apply common probability measures.
The document discusses conditional probability and how odds ratios and risk ratios can be interpreted as conditional probabilities. It provides examples of how to calculate odds ratios from case-control study data and risk ratios from cohort study data. It also discusses how odds ratios from studies where the outcome is common should be interpreted as increased odds rather than risk, and how odds ratios can be converted to risk ratios using a simple formula.
This document discusses probability and related concepts. It covers:
- Defining probability and methods of assigning probabilities.
- Classical probability and how it assigns probabilities based on outcomes.
- Key probability terms like sample space, events, experiments, and trials.
- Laws of probability like addition, multiplication, and conditional probability.
- Examples are provided to illustrate concepts like independent events and finding probabilities.
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Chapter 4: Probability
4.1: Basic Concepts of Probability
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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
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Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
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2. Random Variable
• A random variable x takes on a defined set of
values with different probabilities.
• For example, if you roll a die, the outcome is random (not
fixed) and there are 6 possible outcomes, each of which
occur with probability one-sixth.
For example, if you poll people about their voting
preferences, the percentage of the sample that responds
“Yes on Proposition 100” is a also a random variable (the
percentage will be slightly differently every time you poll).
•
• Roughly, probability is how frequently we
expect different outcomes to occur if we
repeat the experiment over and over
(“frequentist” view)
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3. Random variables can be
discrete or continuous
n Discrete random variables have a countable
number of outcomes
n
Examples: Dead/alive, treatment/placebo, dice,
counts, etc.
Continuous random variables have an
infinite continuum of possible values.
n
n
Examples: blood pressure, weight, the speed of a
car, the real numbers from 1 to 6.
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4. Probability functions
n A probability function maps the possible
values of x against their respective
probabilities of occurrence, p(x)
p(x) is a number from 0 to 1.0.
The area under a probability function is
always 1.
n
n
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8. Practice Problem:
n The number of patients seen in the ER in any given hour
is a random variable represented by x. The probability
distribution for x is:
x 10 11 12 13 14
P(x .4 .2 .2 .1 .1)
Find the probability that in a given hour:
a. exactly 14 patients arrive p(x=14)= .1
b. At least 12 patients arrive p(x³12)= (.2 + .1 +.1) = .4
c. At most 11 patients arrive p(x≤11)= (.4 +.2) = .6
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9. Definitions
9
n Probability: the chance that an
uncertain event will occur (always
between 0 and 1)
n Event: Each possible type of
occurrence or outcome
n Simple Event: an event that can be
described by a single characteristic
n Sample Space: the collection of all
possible events
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10. Types of Probability
There are three approaches to assessing the probability of an
uncertain event:
1.a priori classical probability: the probability of an event is based
on prior knowledge of the process involved.
2.empirical classical probability: the probability of an event is
based on observed data.
3.subjective probability: the probability of an event is determined
by an individual, based on that person’s past experience, personal
opinion, and/or analysis of a particular situation.
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11. Calculating Probability
1. a priori classical probability
total number of possibleoutcomes
= number of ways the event can occur
T
Probability of Occurrence = X
2. empirical classical probability
Probability of Occurrence = number of favorableoutcomes observed
total number of outcomesobserved
These equations assume all outcomes are equally likely.
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12. Example of a priori
classical probability
Find the probability of selecting a face card (Jack,
Queen, or King) from a standard deck of 52 cards.
T total number of cards
= number of face cards
Probability of Face Card =
X
X = 12facecards = 3
T 52 total cards 13
12
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13. Example of empirical
classical probability
13
Taking
Stats
Not Taking
Stats
Total
Male 84 145 229
Female 76 134 210
Total 160 279 439
Find the probability of selecting a male taking statistics
from the population described in the following table:
84
Probability of Male Taking Stats =
number of males takingstats
= =0.191
total numberof people 439
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14. Examples of Sample
Space
The Sample Space is the collection of all possible
events
ex. All 6 faces of a die:
ex. All 52 cards in a deck of cards
ex. All possible outcomes when having a child: Boy
or Girl
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15. Events in Sample Space
15
n Simple event
n An outcome from a sample space with one
characteristic
n ex. A red card from a deck of cards
n Complement of an event A (denoted A/)
n All outcomes that are not part of event A
n ex. All cards that are not diamonds
n Joint event
n Involves two or more characteristics
simultaneously
n ex. An ace that is also red from a deck of cards
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16. Visualizing Events in
Sample Space
n Contingency
Tables:
n Tree Diagrams:
Ace Not
Ace
Total
Black 2 24 26
Red 2 24 26
Total 4 48 52
Full Deck
of 52 Cards
Red Card
Black Card
Not an Ace
Ace
Ace
Not anAce
Sample
Space
2
16
24
2
24
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17. Definitions Simple vs.
Joint Probability
n Simple (Marginal) Probability refers to
the probability of a simple event.
17
n ex. P(King)
n Joint Probability refers to the
probability of an occurrence of two or
more events.
n ex. P(King and Spade)
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18. Definitions Mutually
Exclusive Events
n
18
Mutually exclusive events are events that
cannot occur together (simultaneously).
n example:
n
n
A = queen of diamonds; B = queen of clubs
Events A and B are mutually exclusive if only one
card is selected
n example:
n
n
B = having a boy; G = having a girl
Events B and G are mutually exclusive if only one
child is born
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19. Definitions
Collectively Exhaustive
Eventsn
19
Collectively exhaustive events
n
n
One of the events must occur
The set of events covers the entire sample space
n example:
n
n
n
A = aces; B = black cards; C = diamonds; D = hearts
Events A, B, C and D are collectively exhaustive
(but not mutually exclusive – a selected ace may also
be a heart)
Events B, C and D are collectively exhaustive and
also mutually exclusive
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20. Computing Joint and
Marginal Probabilities
n The probability of a joint event, A and B:
n Computing a marginal (or simple)
Pp(Aro)b=aPb(Ailitayn:dB ) + P(Aand B ) + L + P(A and B )
1 2 k
n Where B1, B2, …, Bk are k mutually exclusive
and collectively exhaustive events
P(A and B) = number of outcomes satisfyingA and B
total number of elementary outcomes
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21. Example:
Joint Probability
P(Red and Ace)
=number of cards that are red and ace = 2
total number of cards 52
Ace Not
Ace
Total
Black 2 24 26
Red 2 24 26
Total 4 48 52
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23. Joint Probability Using a
Contingency Table
Event
Event
Total 1
Joint Probabilities Marginal (Simple) Probabilities
A1
A2
B1 B2 Total
P(B1) P(B2)
P(A1 and B1) P(A1 and B2) P(A1)
P(A2 and B1) P(A2 and B2) P(A2)
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24. Probability
Summary So Far
n Probability is the numerical measure
of the likelihood that an event will
occur.
n The probability of any event must be
between 0 and 1, inclusively
n 0 ≤ P(A) ≤ 1 for any event A.
n The sum of the probabilities of all
mutually exclusive and collectively
exhaustive events is 1.
n
n
P(A) + P(B) + P(C) = 1
A, B, and C are mutually exclusive
and collectively exhaustive
Certain
Impossible
.5
1
0
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25. General Addition Rule
25
P(A or B) = P(A) + P(B) - P(Aand B)
General Addition Rule:
If A and B are mutually exclusive,then
P(A and B) = 0, so the rule can besimplified:
P(A or B) = P(A) +P(B)
for mutually exclusive events A andB
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26. General Addition Rule
Example
Find the probability of selecting a male or a statistics student
from the population described in the following table:
P(Male or Stat) = P(M) + P(S) – P(M AND S)
= 229/439 + 160/439 – 84/439 = 305/439
Taking Stats Not Taking Stats Total
Male 84 145 229
Female 76 134 210
Total 160 279 439
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27. Conditional Probability
n A conditional probability is the probability of one
event, given that another event has occurred:
P(B)
P(A| B) =P(A and B)
P(A)
P(A andB)
P(B|A) =
The conditional
probability of A
given that B has
occurred
The conditional
probability of B
given that Ahas
Where P(A and B) = joint probability ofAoacncduBrred
P(A) = marginal probability of A
P(B) = marginal probability of B
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28. Computing Conditional
Probability
n Of the cars on a used car lot, 70%
have air conditioning (AC) and 40%
have a CD player (CD). 20% of the
cars have both.
n What is the probability that a car has
a CD player, given that it has AC ?
n We want to find P(CD | AC).
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29. Computing Conditional
Probability
CD No CD Total
AC 0.2 0.5 0.7
No
AC
0.2 0.1 0.3
Total 0.4 0.6 1.0
=
.7
=.2857P(CD | AC)=P(CDandAC) .2
P(AC)
Given AC, we only consider the top row (70% of the cars). Of
these, 20% have a CD player. 20% of 70% is about 28.57%.
Given
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30. Computing Conditional
Probability: Decision
Trees
Has CD
Doesnot haveCD
HasAC
Does nothaveAC
HasAC
Does nothaveAC
P(CD)= .4
P(CD/
)= .6
P(CD and AC) = .2
P(CD and AC/) = .2
P(CD/ and AC/) =.1
P(CD/ and AC) =.5
.4
.2
.5
.6
.1
.6
30
All
Cars
.2
.4
Given CD or
no CD:
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31. Computing Conditional
Probability: Decision
Trees
HasAC
Doesnot haveAC
Has CD
Does nothave CD
Has CD
Does nothave CD
P(AC)= .7
P(AC/
)= .3
P(AC and CD) = .2
P(AC and CD/) = .5
P(AC/ and CD/) =.1
P(AC/ and CD) =.2
.7
.5
.2
.3
.1
.3
31
All
Cars
.2
.7
Given AC or
no AC:
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32. Statistical
Independence
§ Two events are independent if and only if:
P(A | B)=P(A)
§ Events A and B are independent when the
probability of one event is not affected by the
other event
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33. Multiplication Rules
33
n Multiplication rule for two events A
and B:P(Aand B) =P(A | B) P(B)
P(A| B)=P(A)
If A and B are independent, thenn
and the multiplication rule
simplifiPe(Asatnod:B) =P(A) P(B)
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34. Multiplication Rules
34
n Suppose a city council is composed of 5
democrats, 4 republicans, and 3
independents. Find the probability of
randomly selecting a democrat followed
by an independent.
P(I and D) =P(I | D)P(D) =(3/11)(5/12) =5/44 =.114
n Note that after the democrat is selected
(out of 12 people), there are only 11
people left in the sample space.
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35. Marginal Probability
Using Multiplication
Rules
35
§ Marginal probability for event A:
P(A) =P(A | B1)P(B1) + P(A| B2 )P(B2 ) + L + P(A| Bk )P(Bk )
§ Where B1, B2, …, Bk are k mutually exclusive and
collectively exhaustive events
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36. Bayes’ Theorem
n Bayes’ Theorem is used to revise
previously calculated probabilities
based on new information.
n Developed by Thomas Bayes in the
18th Century.
n It is an extension of conditional
probability.
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37. Bayes’ Theorem
P(A| Bi )P(Bi )
37
iP(B | A) =
P(A| B1)P(B1) + P(A| B2 )P(B2 ) + K + P(A| Bk )P(Bk )
where:
Bi = ith event of k mutually exclusive and
collectively exhaustive events
A = new event that might impact P(Bi)
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38. Bayes’ Theorem
Example
n A drilling company has estimated a 40%
chance of striking oil for their new well.
n A detailed test has been scheduled for more
information. Historically, 60% of successful
wells have had detailed tests, and 20% of
unsuccessful wells have had detailed tests.
n Given that this well has been scheduled for
a detailed test, what is the probability that
the well will be successful?
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39. Bayes’ Theorem
Example
n Let S = successful well
U = unsuccessful well
n P(S) = .4 , P(U) = .6 (prior
probabilities)
n Define the detailed test event as D
n Conditional probabilities:
39
n P(D|S) = .6
n Goal: To find
P(D|U) = .2
P(S|D)
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40. Bayes’ Theorem
Example
= .667
40
.24+.12
.24
(.6)(.4) + (.2)(.6)
P(D |S)P(S)
P(D |S)P(S) + P(D | U)P(U)
(.6)(.4)
P(S| D) =
=
=
Apply Bayes’Theorem:
So, the revised probability of success, given that this
well has been scheduled for a detailed test, is .667
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41. Bayes’ Theorem
Example
n Given the detailed test, the revised probability of
a successful well has risen to .667 from the
original estimate of 0.4.
Event Prior
Prob.
Conditional
Prob.
Joint
Prob.
Revised
Prob.
S (successful) .4 .6 .4*.6 = .24 .24/.36 = .667
U (unsuccessful) .6 .2 .6*.2 = .12
S= .36
.12/.36 = .333
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42. Review Question 1
If you toss a die, what’s the probability that you
roll a 3 or less?
a. 1/6
b. 1/3
c. 1/2
d. 5/6
e. 1.0
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43. Review Question 1
If you toss a die, what’s the probability that you
roll a 3 or less?
a. 1/6
b. 1/3
c. 1/2
d. 5/6
e. 1.0
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44. Review Question 2
Two dice are rolled and the sum of the face
values is six? What is the probability that at
least one of the dice came up a 3?
a. 1/5
b. 2/3
c. 1/2
d. 5/6
e. 1.0
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45. Review Question 2
Two dice are rolled and the sum of the face
values is six. What is the probability that at least
one of the dice came up a 3?
a. 1/5
b. 2/3
c. 1/2
d. 5/6
e. 1.0
How can you get a 6 on
two dice? 1-5, 5-1, 2-4, 4-
2, 3-3
One of these five has a 3.
1/5
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47. Binomial Probability
Distribution
n A fixed number of observations (trials), n
n e.g., 15 tosses of a coin; 20 patients; 1000
people surveyed
A binary outcomen
n e.g., head or tail in each toss of a coin; disease
or no disease
Generally called “success” and “failure”
Probability of success is p, probability of failure
is 1 –p
n
n
n Constant probability for each observation
n e.g., Probability of getting a tail is the same
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48. Binomial distribution
Take the example of 5 coin tosses.
What’s the probability that you flip
exactly 3 heads in 5 coin tosses?
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49. Binomial distribution
Solution:
One way to get exactly 3 heads: HHHTT
What’s the probability of this exact
arrangement?
P(heads)xP(heads) xP(heads)xP(tails)xP(tails)
=(1/2)3 x (1/2)2
Another way to get exactly 3 heads: THHHT
Probability of this exact outcome = (1/2)1 x (1/2)3
x (1/2)1 = (1/2)3 x (1/2)2
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50. Binomial Distribution
In fact, (1/2)3 x (1/2)2 is the probability of
each unique outcome that has exactly 3
heads and 2 tails.
So, the overall probability of 3 heads and 2
tails is:
(1/2)3 x (1/2)2 + (1/2)3 x (1/2)2 + (1/2)3 x (1/2)2
+ ….. for as many unique arrangements as
there are—but how many are there??
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51. Outcome Probability
THHHT
HHHTT
TTHHH
HTTHH
HHTTH
HTHHT
THTHH
HTHTH
HHTHT
THHTH
(1/2)3 x (1/2)2
(1/2)3 x (1/2)2
(1/2)3 x (1/2)2
(1/2)3 x (1/2)2
(1/2)3 x (1/2)2
(1/2)3 x (1/2)2
(1/2)3 x (1/2)2
(1/2)3 x (1/2)2
(1/2)3 x (1/2)2
(1/2)3 x (1/2)2
10 arrangements x (1/2)3 x (1/2)2
The probability
of each unique
outcome (note:
they are all
equal)
ways to
arrange 3
heads in
5 trials
5C3 = 5!/3!2! = 10
Factorial review: n ! = n(n-1)(n-2)…
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52. P(3heads and 2 tails)= x P(heads)3 x P(tails)2 =
10 x (½)5=31.25%
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53. x
4 50 1 32
Binomial distribution
function:
X= the number of heads tossed in
5 coin tosses
p(x)
number of heads
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54. Binomial Distribution,
Generally
1-p = probability
of failure
p =
probability of
success
X = #
successes
out of n
trials
Note the general pattern emerging à if you have only two possible
outcomes (call them 1/0 or yes/no or success/failure) in n independent
trials, then the probability of exactly X “successes”=
n = number of trials
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56. Binomial Distribution:
example
n If I toss a coin 20 times, what’s the
probability of getting of getting 2
or fewer heads?
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57. **All probability distributions are
characterized by an expected value
and a variance:
If X follows a binomial distribution
with parameters n and p: X ~ Bin
(n, p)
Then:
E(X) = np
Var (X) = np(1-p)
SD (X)=
Note: the variance
will always lie
between
0*N-.25 *N
p(1-p) reaches
maximum at p = . 5
P(1-p)=.25
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58. Practice Problem
n 1. You are performing a cohort study. If the
probability of developing disease in the
exposed group is .05 for the study duration,
then if you (randomly) sample 500 exposed
people, how many do you expect to develop
the disease? Give a margin of error (+/- 1
standard deviation) for your estimate.
n 2. What’s the probability that at most 10
exposed people develop the disease?
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59. Answer
1. How many do you expect to develop the disease? Give a
margin of error (+/- 1 standard deviation) for your estimate.
X ~ binomial (500, .05)
E(X) = 500 (.05) = 25
Var(X) = 500 (.05) (.95) = 23.75
StdDev(X) = square root (23.75) = 4.87
25± 4.87
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60. Answer
2. What’s the probability that at most 10
exposed subjects develop the disease?
This is asking for a CUMULATIVE PROBABILITY: the probability of 0 getting the
disease or 1 or 2 or 3 or 4 or up to 10.
P(X≤10) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4)+….+ P(X=10)=
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61. Practice Problem:
You are conducting a case-control study of
smoking and lung cancer. If the probability of
being a smoker among lung cancer cases is .6,
what’s the probability that in a group of 8 cases
you have:
a. Less than 2 smokers?
b. More than 5?
c. What are the expected value and variance of the number
of smokers?
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64. Review Question 4
In your case-control study of smoking and lung-
cancer, 60% of cases are smokers versus only
10% of controls. What is the odds ratio
between smoking and lung cancer?
a. 2.5
b. 13.5
c. 15.0
d. 6.0
e. .05
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65. Review Question 4
In your case-control study of smoking and lung-
cancer, 60% of cases are smokers versus only
10% of controls. What is the odds ratio
between smoking and lung cancer?
a. 2.5
b. 13.5
c. 15.0
d. 6.0
e. .05
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66. Review Question 5
What’s the probability of getting exactly
5 heads in 10 coin tosses?
a.
b.
c.
d.
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67. Review Question 5
What’s the probability of getting exactly
5 heads in 10 coin tosses?
a.
b.
c.
d.
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68. Review Question 6
A coin toss can be thought of as an example of a
binomial distribution with N=1 and p=.5. What are
the expected value and variance of a coin toss?
a. .5, .25
b. 1.0, 1.0
c. 1.5, .5
d. .25, .5
e. .5, .5
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69. Review Question 6
A coin toss can be thought of as an example of a
binomial distribution with N=1 and p=.5. What are
the expected value and variance of a coin toss?
a. .5, .25
b. 1.0, 1.0
c. 1.5, .5
d. .25, .5
e. .5, .5
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70. Review Question 7
If I toss a coin 10 times, what is the expected
value and variance of the number of heads?
a. 5, 5
b. 10, 5
c. 2.5, 5
d. 5, 2.5
e. 2.5, 10
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71. Review Question 7
If I toss a coin 10 times, what is the expected
value and variance of the number of heads?
a. 5, 5
b. 10, 5
c. 2.5, 5
d. 5, 2.5
e. 2.5, 10
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72. Review Question 8
In a randomized trial with n=150, the goal is to
randomize half to treatment and half to control. The
number of people randomized to treatment is a random
variable X. What is the probability distribution of X?
a. X~Normal(µ=75,s=10)
b. X~Exponential(µ=75)
c. X~Uniform
d. X~Binomial(N=150, p=.5)
e. X~Binomial(N=75, p=.5)
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73. Review Question 8
In a randomized trial with n=150, every subject has a
50% chance of being randomized to treatment. The
number of people randomized to treatment is a random
variable X. What is the probability distribution of X?
a. X~Normal(µ=75,s=10)
b. X~Exponential(µ=75)
c. X~Uniform
d. X~Binomial(N=150, p = . 5)
e. X~Binomial(N=75, p=.5)
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74. Review Question 9
More than 2 standard deviationsd.
In the same RCT with n=150, if
69 end up in the treatment group
and 81 in the control group, how
far off is that from expected?
a. Less than 1 standard deviation
b . 1 standard deviation
c . Between 1 and 2 standard deviations
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75. Review Question 9
More than 2 standard deviationsd.
In the same RCT with n=150, if
69 end up in the treatment group
and 81 in the control group, how
far off is that from expe
a. Less than 1 standard deviation
b . 1 standard deviation
c . Between 1 and 2 standard devia
cted?
tions
Expected = 75
81 and 69 are both 6 away
from the expected.
Variance = 150(.25) = 37.5
Std Dev @ 6
Therefore, about 1 SD away
from expected.
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76. Proportions…
n The binomial distribution forms the basis
of statistics for proportions.
A proportion is just a binomial count
divided by n.
n
n For example, if we sample 200 cases and find
60 smokers, X=60 but the observed
proportion=.30.
Statistics for proportions are similar to
binomial counts, but differ by a factor of n.
n
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77. Stats for proportions
For binomial:
For proportion:
P-hat stands for “sample
proportion.”
Differs by
a factor
of n.
Differs
by a
factor
of n.
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78. It all comes back to
normal…
n Statistics for proportions are based
on a normal distribution, because
the binomial can be approximated
as normal if np>5
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79. Multinomial distribution
(beyond the scope of this course)
The multinomial is a generalization of the
binomial. It is used when there are more than
2 possible outcomes (for ordinal or nominal,
rather than binary, random variables).
n Instead of partitioning n trials into 2
outcomes (yes with probability p / no with
probability 1-p), you are partitioning n
trials into 3 or more outcomes (with
probabilities: p1, p2, p3 , ..)
n General formula for 3 outcomes:
pD - pR )zx y
D Rp p (1-
x!y!z!
n!
P(D =x, R =y, G =z) =
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80. Multinomial example
Specific Example: if you are randomly choosing 8
people from an audience that contains 50%
democrats, 30% republicans, and 20% green
party, what’s the probability of choosing exactly 4
democrats, 3 republicans, and 1 green party
member?
(.5)4
(.3)3
(.2)1
4!3!1!
8!
P(D =4, R =3,G =1) =
You can see that it gets hard to calculate very
fast! The multinomial has many uses in
genetics where a person may have 1 of many
possible alleles (that occur with certain
probabilities in a given population) at a gene
locus.
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81. Introduction to the Poisson Distribution
n Poisson distribution is for counts—if events
happen at a constant rate over time, the
Poisson distribution gives the probability of
X number of events occurring in time T.
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82. Poisson Mean and Variance
n Mean
n Variance and Standard
Deviation
µ =l
s 2
=l
s = l
where l = expected number of
hits in a given time period
For a Poisson
random
variable, the
variance and
mean are the
same!
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83. Poisson Distribution, example
The Poisson distribution models counts, such as the number of new
cases of SARS that occur in women in New England next month.
The distribution tells you the probability of all possible numbers of
new cases, from 0 to infinity.
If X= # of new cases next month and X ~ Poisson (l), then the
probability that X=k (a particular count) is:
k!
lk
e-l
p(X =k ) =
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84. Example
n For example, if new cases of West
Nile Virus in New England are
occurring at a rate of about 2 per
month, then these are the
probabilities that: 0,1, 2, 3, 4, 5, 6,
to 1000 to 1 million to… cases will
occur in New England in the next
month:
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85. Poisson Probability table
X P(X)
0 2 0
e - 2
0 !
=.135
1 2 1
e - 2
1 !
=.27
2 2 2 e - 2
2 !
=.27
3 2 3
e - 2
3 !
=.18
4 =.09
5
… …
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86. Example: Poisson
Distribution
Suppose that a rare disease has an incidence of 1 in 1000 person-
years. Assuming that members of the population are affected
independently, find the probability of k cases in a population of
10,000 (followed over 1 year) for k=0,1,2.
The expected value (mean) =l = .001*10,000 = 10
10 new cases expected in this population per yearà
= .00227
2!
=.0000454
0!
= .000454
1!
(10)2
e-(10)
P( X =2) =
(10)1
e-(10)
P( X =1) =
(10)0 e-(10)
P( X =0) =
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87. more on Poisson…
“Poisson Process” (rates)
Note that the Poisson parameter l can be
given as the mean number of events that
occur in a defined time period OR,
equivalently, l can be given as a rate, such as
l=2/month (2 events per 1 month) that must
be multiplied by t=time (called a “Poisson
Process”) à
X ~ Poisson(lt()lk)e-lt
k!
P(X =k ) =
E(X) = lt
Var(X) = lt Visit: Learnbay.co
88. Example
For example, if new cases of West Nile
in New England are occurring at a rate
of about 2 per month, then what’s the
probability that exactly 4 cases will
occur in the next 3 months?
X ~ Poisson (l=2/month)
= 13.4%
4!4!
P(X = 4 in 3 months)= =
(2* 3)4
e-(2*3)
64
e-(6)
Exactly 6 cases?
= 16%
6!6!
P(X =6 in 3 months) =
(2 * 3)6
e- (2*3)
=
66
e-(6)
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89. Practice problems
1a. If calls to your cell phone are a
Poisson process with a constant rate
l=2 calls per hour, what’s the
probability that, if you forget to turn
your phone off in a 1.5 hour movie, your
phone rings during that time?
1b. How many phone calls do you
expect to get during the movie?
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90. Answer
1a. If calls to your cell phone are a Poisson process with
a constant rate l=2 calls per hour, what’s the
probability that, if you forget to turn your phone off in a
1.5 hour movie, your phone rings during that time?
X ~ Poisson (l=2 calls/hour)
P(X≥1)=1 – P(X=0)
= .05
0! 0!
-3(3)0
e-3(2*1.5)0
e-2(1.5)
P(X = 0) = = e
P(X≥1)=1 – .05 = 95%
chance
1b. How many phone calls do you expect to get during the movie?
E(X) = lt = 2(1.5) = 3
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