概率论教案(全英)
概率论英文课件：ch5_4 Hypergeometric Distribution
When n is small compared to N, the nature of the N items changes very little in each draw. Thus, the quantity k/N plays the role of the parameter p. As a result, the binomial distribution may be viewed as a large population edition of the hypergeometric distributions.
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Theorem 5.3 Mean and variance of the hypergeometric distribution h(x; N, n, k) are
= nk/N
,
N n k k n (1 ) N 1 N N
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Relationship to the Binomial Distribution
5.4 Hypergeometric Distribution
Hypergeometric Experiment A random sample of size n is selected without replacement from N items. k of the N items may be classified as successes and N – k are classified as failures.
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Example:
A state runs a lottery( 彩 票 ) in which six numbers are randomly selected from 40,without replacement. A player chooses six numbers before the state’s sample is selected. Let X be the numbers selected by the player that match those selected by the state. X has a Hypergeometric Distribution. Question: (a) What is the probability that six numbers chosen by player match all six numbers in state’s sample? N = 40, k = 6, x = 6 (b) What is the probability that five of the six numbers chosen by a player appear in the state’s sample? N = 40, k = 6, x = 5
概率论与数理统计教案(48课时)
《概率论与数理统计》课程教案第一章 随机事件及其概率一．本章的教学目标及基本要求(1) 理解随机试验、样本空间、随机事件的概念;(2) 掌握随机事件之间的关系与运算,；(3) 掌握概率的基本性质以及简单的古典概率计算; 学会几何概率的计算；(4) 理解事件频率的概念，了解随机现象的统计规律性以及概率的统计定义。
了解概率的公理化定义。
(5) 理解条件概率、全概率公式、Bayes 公式及其意义。
理解事件的独立性。
二．本章的教学内容及学时分配第一节 随机事件及事件之间的关系第二节 频率与概率 2学时第三节 等可能概型（古典概型） 2 学时第四节 条件概率第五节 事件的独立性 2 学时三．本章教学内容的重点和难点1） 随机事件及随机事件之间的关系；2） 古典概型及概率计算；3）概率的性质；4）条件概率，全概率公式和Bayes 公式5）独立性、n 重伯努利试验和伯努利定理四．教学过程中应注意的问题1） 使学生能正确地描述随机试验的样本空间和各种随机事件；2） 注意让学生理解事件,,,,,A B A B A B A B AB A ⊂⋃⋂-=Φ…的具体含义，理解事件的互斥关系；3） 让学生掌握事件之间的运算法则和德莫根定律；4） 古典概率计算中，为了计算样本点总数和事件的有利场合数，经常要用到排列和组合，复习排列、组合原理；5） 讲清楚抽样的两种方式——有放回和无放回；五．思考题和习题思考题：1. 集合的并运算⋃和差运算－是否存在消去律？2. 怎样理解互斥事件和逆事件？3. 古典概率的计算与几何概率的计算有哪些不同点？哪些相同点？习题：第二章 随机变量及其分布一．本章的教学目标及基本要求(1) 理解随机变量的概念，理解随机变量分布函数的概念及性质, 理解离散型和连续型随机变量的概率分布及其性质，会运用概率分布计算各种随机事件的概率；(2) 熟记两点分布、二项分布、泊松分布、正态分布、均匀分布和指数分布的分布律或密度函数及性质；二．本章的教学内容及学时分配第一节 随机变量第二节 第二节 离散型随机变量及其分布离散随机变量及分布律、分布律的特征第三节 常用的离散型随机变量常见分布（0-1分布、二项分布、泊松分布） 2学时第四节 随机变量的分布函数分布函数的定义和基本性质，公式第五节 连续型随机变量及其分布连续随机变量及密度函数、密度函数的性质 2学时第六节 常用的连续型随机变量常见分布（均匀分布、指数分布、正态分布）及概率计算 2学时三．本章教学内容的重点和难点a) 随机变量的定义、分布函数及性质；b) 离散型、连续型随机变量及其分布律或密度函数，如何用分布律或密度函数求任何事件的概率；c) 六个常见分布(二项分布、泊松分布、几何分布、均匀分布、指数分布、正态分布)；四．教学过程中应注意的问题a) 注意分布函数(){}F x P X x =<的特殊值及左连续性概念的理解；b) 构成离散随机变量X 的分布律的条件，它与分布函数()F x 之间的关系；c) 构成连续随机变量X 的密度函数的条件，它与分布函数()F x 之间的关系；d) 连续型随机变量的分布函数()F x 关于x 处处连续，且()0P X x ==，其中x 为任意实数，同时说明了()0P A =不能推导A =Φ。
概率论英文课件：ch4_1 Mean of Random Variables
Chapter 4 Mathematical Expectation4.1 Mean of Random Variables4.2 Variance and Covariance4.3 Means and Variances of LinearCombinations of Random variables4.4 Chebyshev’s Theorem13Definition 4.1 Let X be a random variable with probability distribution f(x).The mean or expected value of X isif X is discrete, andif X is continuous.Remark:The mean of a random variable X can be thought of as a measure of the “center of location”in the sense that it indicates where the “center”of the density line. xx xf X E )()( dx x xf X E)()(4Example 4.1, page 89The probability distribution of a random variable X is given byx =0,1,2,3.f(0)=1/35 f(1)=12/35 f(2)=18/35 f(3)=4/3537334)(x x x f 71235435183512351))(3())(2())(1())(0(5ExampleThe probability distribution of a random variable X is given byelsewherex e x f x 00)( 0)(dx e x X E x 00dx e xe xx 1 0 ( )6Example 4.2, page 90In a gambling game a man is paid $5 if he gets all heads or all tails when three coins are tossed, and he will pay out $3 if either one or two heads show. What is his expected gain?Let Y be the amount of gain per bet. The possible values are 5 and –3 dollars.Let X be the number of heads that occur in tossing three coins. The possible values of X are 0, 1, 2, and 3.Solution:P(Y = 5) = P(X = 0 or X = 3) = 1/8 + 1/8 = ¼P(Y = -3) = P(X =1 or X = 2) = 6/8 = ¾= (5)(1/4) + (-3)(3/4) = –1Interpretation : Over the long run, the gambler will, on average, lose $1 per bet. Most likely, the more the gambler plays the games, the more he would lose.7Notice that in the preceding example, there are two random variables, X and Y ; and Y is a function of X , for example if we letE(Y) = E(g(X))= (5)P(Y = 5) + (-3)P(Y = -3)= (5)[P(X = 0) + P(X = 3)] + (-3)[P(X = 1) + P(X = 2)]= (5)P(X = 0) + (5)P(X = 3) + (-3)P(X = 1) + (-3)P(X = 2) = g(0)P(X = 0) + g(3)P(X = 3) + g(1)P(X = 1) + g(2)P(X = 2)= 2,133,05)(X X X g Y x x f x g )()(8Theorem 4.1 Let X be a random variable with probability distribution f(x). The mean or expected value of random variable g(X)isif X is discrete, andif X is continuous.xX g x f x g X g E )()()]([)(dxx f x g X g E X g )()()]([)(9ExampleLet X denote the length in minutes of a long-distance telephone conversation. Assume that the density for X is given byFind E(X) and E(2X+3)Solution:E(X )= = = 10 E(2X+3)= = 2(10) + 3 = 23 .0)(10/101x e x f x dx e x x 10/0101)32( dxx f x )(dx e x x 10/010110Definition 4.2 Let X and Y be random variables with joint probability distribution f(x, y).The mean or expected value of the random variable g(X, Y)isif X and Y are discrete, andif X and Y are continuous. ),(),(),(),()],([y x Y X g y x f y x g Y X g E dydx y x f y x g Y X g E Y X g),(),()],([),( Extension11ExampleExample:Suppose two dice are rolled, one red and one white. Let X be the number on the top face of the red die, and Y be the number on the top face of the white one. Find E(X+Y).E[X + Y]= == ==3.5 + 3.5 = 7 ),(),()(y x y Y x X P y x 6161),()(x y y Y x X P y x 61616161),(),(x y x y y Y x X yP y Y x X xP 61616161)36/1()36/1(x y x y y x12Example 4.7, page 93. Find E[Y/X]for the densitySolution:= = elsewhere ,0 2020 ,16),(33 y , x y x y x f dydx y x f y x g Y X g E Y X g),(),()],([),( dy dx y x x y 20332016dy dx y x 2042201613In generalIf X and Y are two random variables, f(x, y)is thejoint density function, then:E(X)= = (discrete case) E(X)= (continuous case)E(Y)= = (discrete case)E(Y)= (continuous case)g(x)and h(y)are marginal probability distributions of X and Y , respectively. x y y x xf ),( xx xg )( dxx xg dxdy y x xf )(),( y x y x yf ),( y y yh )( dyy yh dxdy y x yf )(),(。
概率论与数理统计教案(48课时)
概率论与数理统计教案(48课时)Chapter 1: XXX1.Learning Objectives and Basic Requirements:1) Understand the concepts of random experiments。
sample space。
and random events;2) Master the nships and ns een random events;3) Master the basic XXX。
learn how to XXX;4) Understand the concept of event frequency。
know the XXX random phenomena。
and the XXX.5) XXX。
the law of total probability。
Bayes' theorem。
and their XXX.2.Teaching Content and Time n:n 1: XXXn 2: XXX (2 hours)n 3: XXX (Classical Probability) (2 hours)n 4: XXXn 5: Independence of Events (2 hours)3.XXX:1) Random events and nships een random events;2) XXX;3) Properties of probability;4) nal probability。
the law of total probability。
and Bayes' theorem;5) XXX。
XXX。
XXX.4.XXX:1) Enable students to correctly describe the sample space of random experiments and us random events;2) Pay n to helping students understand the specific meanings of events such as A∪B。
概率论与数理统计教案(48课时)(最新整理)
( x, y )G
，注意二重积分运算知识点的复习。
d) 二维均匀分布的密度函数的具体表达形式。
五．思考题和习题
思考题：1. 由随机变量 X ,Y 的边缘分布能否决定它们的联合分布？
2. 条件分布是否可以由条件概率公式推导？ 3. 事件的独立性与随机变量的独立性是否一致？ 4．如何利用随机变量之间的独立性去简化概率计算，试举例说明。 习题：
第四章 随机变量的数字特征 一．教学目标及基本要求
(1)理解数学期望和方差的定义并且掌握它们的计算公式；
(2)掌握数学期望和方差的性质与计算，会求随机变量函数的数学期望，特别是利用
期望或方差的性质计算某些随机变量函数的期望和方差。
(3)熟记 0-1 分布、二项分布、泊松分布、正态分布、均匀分布和指数分布的数学期
第四节 二维随机变量的函数的分布
已知(X,Y)的分布率 pij 或密度函数 (x, y) ，求 Z f ( X ,Y ) 的分布律或密度
函数Z (z) 。特别如函数形式： Z X Y , Z max( X ,Y ), Z min( X ,Y ) 。
2 学时
三．本章教学内容的重点和难点
a) 二维随机变量的分布函数及性质，与一维情形比较有哪些不同之处；
5．列举正态分布的应用。
习题：
第三章 多维随机变量及其分布
一．教学目标及基本要求
(1)了解二维随机变量概念及其联合分布函数概念和性质，了解二维离散型和连续 型随机变量定义及其概率分布和性质，了解二维均匀分布和正态分布。
(2)会用联合概率分布计算有关事件的概率，会求边缘分布。 (3)掌握随机变量独立性的概念，掌握运用随机变量的独立性进行概率计算。 (4)会求两个独立随机变量的简单函数(如函数 X+Y, max(X, Y), min(X, Y))的分布。
概率论与数理统计英文版总结电子教案
概率论与数理统计英文版总结Sample Space样本空间The set of all possible outcomes of a statistical experiment is called the sample space. Event 事件An event is a subset of a sample space.certain event（必然事件）：The sample space S itself, is certainly an event, which is called a certain event, means that it always occurs in the experiment.impossible event（不可能事件）：The empty set, denoted by∅, is also an event, called an impossible event, means that it never occurs in the experiment.Probability of events （概率）If the number of successes in n trails is denoted by s, and if the sequence of relative frequencies /s n obtained for larger and larger value of n approaches a limit, then this limit is defined as the probability of success in a single trial.“equally likely to occur”------probability（古典概率）If a sample space S consists of N sample points, each is equally likely to occur. Assume that the event A consists of n sample points, then the probability p that A occurs is()np P AN==Mutually exclusive(互斥事件)Mutually independent 事件的独立性Two events A and B are said to be independent if ()()()P A B P A P B=⋅IOr Two events A and B are independent if and only if(|)()P B A P B=.Conditional Probability 条件概率The multiplication theorem 乘法定理 If 12k ,,,A A A L are events, then12k 121312121()()(|)(|)(|)k k P A A A P A P A A P A A A P A A A A -=⋅⋅I I L I L I I L I If the events 12k ,,,A A A L are independent, then for any subset12{,,,}{1,2,,}m i i i k ⊂L L ,1212()()()()m m P A A A P A P A P A i i i i i i =I I L L(全概率公式 total probability)（贝叶斯公式Bayes ’ formula.）()(|)()i i P B A P B A P A =IUsing the theorem of total probability, we have1()(|)(|)()(|)i i i kjjj P B P A B P B A P B P A B ==∑ 1,2,,i k =L1. random variable definition2. Distribution functionNote The distribution function ()F X is defined on real numbers, not on sample space. 3. PropertiesThe distribution function ()F x of a random variable X has the following properties:3.2 Discrete Random Variables 离散型随机变量geometric distribution （几何分布）Binomial distribution（二项分布）poisson distribution（泊松分布）Expectation (mean) 数学期望2．Variance 方差 standard deviation （标准差）probability density function概率密度函数5. Mean（均值）6. variance 方差4.2 Uniform Distribution 均匀分布The uniform distribution, with the parameters a a nd b , has probability density function1for ,()0 elsewhere,a x b f x b a⎧<<⎪=-⎨⎪⎩4.5 Exponential Distribution 指数分布4.3 Normal Distribution 正态分布4.4 Normal Approximation to the Binomial Distribution （二项分布）4.7 Chebyshev’s Theorem （切比雪夫定理）Joint probability distribution （联合分布）In the study of probability, given at least two random variables X, Y, ..., that are defined on a probability space, the joint probability distribution for X, Y, ... is a probability distribution that gives the probability that each of X, Y, ... falls in any particular range or discrete set of values specified for that variable. 5.2 C onditional distribution 条件分布Consistent with the definition of conditional probability of events when A is the event X =x and B is the event Y =y , the conditional probability distribution of X given Y =y is defined as(,)(|)()X Y p x y p x y p y =for all x provided ()0Y p y ≠.5.3S tatistical independent 随机变量的独立性5.4 Covariance and Correlation 协方差和相关系数We now define two related quantities whose role in characterizing the interdependence of X and Y we want to examine.理We can find the steadily of the frequency of the events in large number of random phenomenon. And the average of large number of random variables are also steadiness. These results are the law of large numbers.population (总体)sample （样本、子样）中位数Sample Distributions 抽样分布1．sampling distribution of the mean 均值的抽样分布It is customary to write )(X E as X μ and )(X D as 2X σ.Here, ()E X μ= is called the expectation of the mean .均值的期望 n X σσ= is called the standard error of the mean. 均值的标准差7.1 Point Estimate 点估计Unbiased estimator(无偏估计量)minimum variance unbiased estimator（最小方差无偏估计量）3. Method of Moments 矩估计的方法confidence interval----- 置信区间lower confidence limits-----置信下限upper confidence limits----- 置信上限degree of confidence----置信度2．极大似然函数likelihood functionmaximum likelihood estimate（最大似然估计）8.1 Statistical Hypotheses（统计假设）显著性水平Two Types of Errors。
概率论英文课件：ch5_3 Binomial Distribution
3
1 2
4 3
4 1 P(X = 3) = 3 2
3
1 2
43
2
Binomial Distribution A Bernoulli trial can result in a “success” with probability p and a failure with probability q = 1 – p. Then the probability distribution of the binomial random variable X, the number of “successes” in n independent trials, is
8
= b( x;15,0.4)
x 3
5 4
b( x;15,0.4) b( x;15,0.4) P(X = 5) = P(X 5) – P( X 4)= x 0 x 0
Example
8
Example
It is known that the incidence of a disease is 13% in a city. We randomly selected 150 people from this city, estimate the probability of 10 person of them being infection.
of expanding of [p+(1-p)]n, So the distribution of X is called as the Binominal distribution. Obviously, ∑P(X=k)=1.
概率论教案——精选推荐
概率论教案概率论教案 Hessen was revised in January 2021第⼀章随机事件与概率第⼀节随机事件教学⽬的：了解概率的主要任务及其研究对象；掌握随机试验、随机事件等基本概念；掌握随机事件间的关系与运算，了解其运算规律。
教学重点：随机试验，随机事件，事件间的关系与运算。
教学难点：事件（关系、运算）与集合的对应，⽤运算表⽰复杂事件。
教学内容：1、随机现象与概率统计的研究对象随机现象：在⼀定的条件下，出现不确定结果的现象。
研究现象：概率论与数理统计研究随机现象的统计规律性。
2、随机试验（E）对随机现象的观察。
特点①试验可在相同条件下重复；②试验的所有可能结果不只⼀个，但事先已知；③每次试验出现⼀个且出现⼀个，哪个出现事先不知。
3、基本事件与样本空间（1）基本事件：E中的结果（能直接观察到，不可再分），也称为样本点，⽤表⽰。
（2）样本空间：E中所有基本事件的集合称为这个随机试验E的样本空间，⽤Ω表⽰。
4、随机事件（1）随机事件：随机试验中可能发⽣也可能不发⽣的时间。
⽤A、B、C等表⽰。
（2）随机事件的集合表⽰（3）随机事件的图形表⽰必然事件（Ω）和不可能事件（E）5、事件间的关系与运算（1）包含（⼦事件）与相等（2）和事件（加法运算）（2）积事件（乘法运算）（3）互斥关系（4）对⽴关系（逆事件）（5）差事件(减法运算)6、事件间的运算规律（1）交换律；（2）结合律；（3）分配律；（4）对偶律教学时数：2学时作业：习题⼀ 1、2第⼆节概率的定义教学⽬的：掌握概率的古典定义，⼏何定义，统计定义及这三种概率的计算⽅法；了解概率的基本性质。
教学难点：古典概率的计算，频率性质与统计概率。
教学内容：1、概率⽤于表⽰事件A 发⽣可能性⼤⼩的数称为事件A 的概率，⽤P(A)表⽰。
2、古典型试验与古典概率（1）古典型试验：特点①基本事件只有有限个；②所有基本事件的发⽣是等可能的。
（2）古典概率，在古典型试验中规定P(A)=nkA =Ω中基本事件总数中含的基本事件数3、⼏何型试验与⼏何概率（1）⼏何型试验向区域G 内投点，点落在G 内每⼀点处是等可能的，落在⼦区域1G 内(称事件A 发⽣)的概率与1G 的度量成正⽐，⽽与1G 的位置和形状⽆关。
概率论基础教程英文版第九版课程设计
Probability Theory Basic Tutorial Course Design - 9thEdition (English Version)IntroductionThis document outlines the course design for the Probability Theory Basic Tutorial, 9th edition in English. The course is designed for students who have a basic understanding of mathematics and areinterested in probability theory. The m of the course is to provide a strong foundation in probability theory, including basic concepts, principles, and applications.Course ObjectiveThe course ms to:•Provide a basic understanding of probability theory•Introduce students to the fundamental concepts and principles of probability theory•Develop skills in using probability theory to solve real-world problems•Build a solid foundation for further study in probability theoryCourse ContentThe course will cover the following topics:Week 1: Introduction to Probability•Probability and its importance•Sample space and events•Probability axiomsWeek 2: Combinatorics•Permutations and combinations•Binomial theoremWeek 3: Conditional Probability and Independence•Conditional probability•Bayes’ theorem•IndependenceWeek 4: Probability Distributions•Discrete random variables•Probability mass function•Cumulative distribution functionWeek 5: Continuous Random Variables•Probability density function•Cumulative distribution function•Uniform, Exponential, and Normal distributionsWeek 6: Joint and Conditional Distributions•Joint distributions•Marginal distributions•Conditional distributionsWeek 7: Moment Generating Functions•Definition and properties of moment generating functions •Calculation of momentsWeek 8: Central Limit Theorem•Statement of the central limit theorem•Applications of the central limit theoremWeek 9: Markov Chns•Markov chns•Transition probability matrices•Steady-state probabilitiesWeek 10: Introduction to Stochastic Processes•Stochastic processes and random walks•Poisson processes•MartingalesCourse MaterialsThe following materials will be used for the course:•Textbook: Probability Theory: A Comprehensive Course by Achim Klenke, 2nd edition (English version)•Lecture notes and exercises will be provided weekly•Online resources including interactive simulations, quizzes, and videos will be provided.Course AssessmentAssessment will be based on the following components:•Weekly assignments (40%)•Mid-term Exam (30%)•Final Exam (30%)ConclusionThis course design ms to provide students with a strong foundation in probability theory and an ability to apply the principles learned to real-world problems. The course materials provide a comprehensive coverage of the fundamental concepts, principles, and applications of probability theory necessary for further study in the field.。
《概率论与数理统计》教案
《概率论与数理统计》教案第一章：概率的基本概念1.1 概率的定义与性质介绍概率的定义，理解概率是衡量随机事件发生可能性大小的数。
掌握概率的基本性质，如additivity（可加性）和symmetry（对称性）。
1.2 条件概率与独立性引入条件概率的概念，理解在给定一些信息的情况下，事件发生的概率。
学习独立事件的定义，掌握独立性原理，了解如何通过乘法规则计算联合概率。
第二章：随机变量及其分布2.1 随机变量的概念介绍随机变量的定义，理解随机变量是随机现象的数值化描述。
学习离散随机变量和连续随机变量的区别，以及如何列出随机变量的可能取值。
2.2 概率分布学习概率分布的概念，掌握如何计算随机变量取某个值的概率。
掌握期望值和方差的计算方法，了解它们在描述随机变量集中趋势和离散程度方面的作用。
第三章：多维随机变量及其分布3.1 联合随机变量引入多维随机变量的概念，理解多个随机变量共同作用的概率分布。
学习如何列出联合随机变量的可能取值，以及如何计算联合概率。
3.2 独立随机变量掌握独立多维随机变量的概念，了解独立性在概率论中的重要性。
学习如何计算两个独立随机变量的联合分布，以及如何推导条件概率。
第四章：大数定律与中心极限定理4.1 大数定律介绍大数定律的概念，理解在足够多次试验中，随机变量的样本平均将趋近于其期望值。
学习弱大数定律和强大数定律的表述，以及它们在实际应用中的意义。
4.2 中心极限定理掌握中心极限定理的内容，了解当样本量足够大时，样本均值的分布将趋近于正态分布。
学习如何应用中心极限定理进行近似计算，以及其在统计学中的重要性。
第五章：数理统计的基本概念5.1 统计量与样本介绍统计量的概念，理解统计量是用来描述样本特征的函数。
学习如何计算样本均值、样本方差等基本统计量。
5.2 抽样分布与估计掌握抽样分布的概念，了解不同统计量的抽样分布特性。
学习点估计和区间估计的定义，了解如何根据样本数据估计总体参数。
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• A discrete variable takes a specific value
– E.g. P(count=3)
• One of a finite number of outcomes occurs
– E.g. P(fault=‘gearbox failure’)
• Irrespective, P is still a real value
Types of Events
• Mutually exclusive events (e.g. E1=‘Guilty’ E2=‘Not Guilty’):
• If two events, E1 and E2, are mutually exclusive, they have no common sample points in X
• And P( X E1 E2 ) P( X E1) P( X E2 )
E1
E2
• Non-mutually exclusive events (e.g. E1=‘EEE’ E2=‘MAE’):
Байду номын сангаас
• Two events, E1 and E2, are not mutually exclusive when they have one or more common sample points
interpreted as relative frequency • A probability is usually expressed in terms of a random variable.
• In this example, X denotes the part length and the probability statement can be written as :
frequency interpretation is still appropriate
• E.g. outcome X from rolling a dice
• 6 outcomes; P(X=1)=1/6 ; P( X [1,6]) 1
Types of Events
• A continuous variable takes a specific value:
P(X [10.8,11.2]) 0.25 or P(10.8 X 11.2) 0.25
Events
• A measured value is not always obtained from an experiment. Sometimes, the result is only classified into one of several categories.
probability…
*Whaaat???!! Yes. If E2 and E1 have no influence on each other it will
be this simple.
Basic Probability Rules
• Complement P(X E1) 1 P(E1)
Probability of happening is the complement of it not happening…
• The following statement:
“The chance that X, the lifetime of a power transformer, is between 10.8 and 11.2yrs is 25%”
is a statement that quantifies the possibility of plant lifetime • The value of a probability is always restricted to the interval [0,1] and can be
• For example, a manufactured electronic component might only be classified as defective or not – 2 categories.
• For example, a circuit breaker with fail if just one of its 15 subassemblies fails – the cause will be one of 15 categories.
• Previous example demonstrates events form part of a set
• Set theory is applicable then • What was set theory again?
– You may remember it, but in case you don’t… – …quick review and how it relates to
• The possible categories are referred to as outcomes • A set of one or more outcomes is called an event. • The concept of probability can be applied to events and the relative
• And
P( X E1 E2 ) P( X E1) P( X E2 ) P( X E1 E2 )
• And E1 E2 is when E1 and E2 both occur
E1 E2
E1 E2
COMBINING EVENTS
Combinations of Events
EE313/EM301Engineering Analysis Lesson 2a: Basic Probability (Events)
Probability
• Probability is used to quantify the likelihood that a measurement falls within some set of values