香港中文大学：《Probability and Statistics for Engineers》课程教学资源（课件讲稿）Sample space and probability

ENGG40Probataistisfor Chapter 1:Sample space and probabilty Instructor:Shengyu Zhang

Instructor: Shengyu Zhang

About the course Website: http://www.cse.cuhk.edu.hk/~syzhang/course /Prob17/ You can find the lecture slides,tutorial slides, info for time,venue,TA,textbook,grading method,etc. No tutorial in the first week. Announcements will be posted on web. The important ones will be sent to your cuhk email as well

About the course ◼ Website: http://www.cse.cuhk.edu.hk/~syzhang/course /Prob17/ ◼ You can find the lecture slides, tutorial slides, info for time, venue, TA, textbook, grading method, etc. ◼ No tutorial in the first week. ◼ Announcements will be posted on web. ❑ The important ones will be sent to your cuhk email as well

Content Sets. Probabilistic models. Conditional probability. Total Probability Theorem and Bayes'Rule. Independence. Counting

Content ◼ Sets. ◼ Probabilistic models. ◼ Conditional probability. ◼ Total Probability Theorem and Bayes’ Rule. ◼ Independence. ◼ Counting

Sets Probability makes extensive use of set operations. A set is a collection of objects,which are the elements of the set. x E S:S is a set and x is an element of S xS:x is not an element of S. ■ 0:A set that has no elements;called empty set

Sets ◼ Probability makes extensive use of set operations. ◼ A set is a collection of objects, which are the elements of the set. ◼ 𝑥 ∈ 𝑆: 𝑆 is a set and 𝑥 is an element of 𝑆 ◼ 𝑥 ∉ 𝑆: 𝑥 is not an element of 𝑆. ◼ ∅: A set that has no elements; called empty set

Sets Subset:S C T ■Equal sets:S=T Countable vs.uncountable Universal set N:The set which contains all objects that could conceivably be of interest in a particular context. Complement:S=Sc=2-S

Sets ◼ Subset: 𝑆 ⊆ 𝑇 ◼ Equal sets: 𝑆 = 𝑇 ◼ Countable vs. uncountable ◼ Universal set Ω: The set which contains all objects that could conceivably be of interest in a particular context. ◼ Complement: 𝑆ҧ= 𝑆 𝑐 = Ω − 𝑆

Sets Union of sets:SUT,USi,Uie Si. Intersection of sets:SnT,nSi,nie Si. Disjoint sets:empty pairwise intersection. Partition of set S:a collection of disjoint sets whose union is S. De Morgan's laws: U:S=∩：S,∩S=U:S

Sets ◼ Union of sets: 𝑆 ∪ 𝑇, ڂ=��1 ∞ 𝑆𝑖 �𝑆� ��∋��ڂ , . ◼ Intersection of sets: 𝑆 ∩ 𝑇, ځ=��1 ∞ 𝑆𝑖 �𝑆� ��∋��ځ , . ◼ Disjoint sets: empty pairwise intersection. ◼ Partition of set 𝑆: a collection of disjoint sets whose union is 𝑆. ◼ De Morgan’s laws: ��ڂ ��ഥ𝑖 𝑆ځ = �𝑆� ��ځ , ��ഥ𝑖 𝑆ڂ = �𝑆�

Content Sets. Probabilistic models. Conditional probability. Total Probability Theorem and Bayes'Rule. Independence. Counting

Content ◼ Sets. ◼ Probabilistic models. ◼ Conditional probability. ◼ Total Probability Theorem and Bayes’ Rule. ◼ Independence. ◼ Counting

Experiment and outcomes A probabilistic model is a mathematical description of an uncertain situation. Every probabilistic model involves an underlying process,called the experiment Example.Flip two coins. The experiment produces exactly one out of several possible outcomes. Example.four outcomes:{HH,HT,TH,TT}

Experiment and outcomes ◼ A probabilistic model is a mathematical description of an uncertain situation. ◼ Every probabilistic model involves an underlying process, called the experiment. ❑ Example. Flip two coins. ◼ The experiment produces exactly one out of several possible outcomes. ❑ Example. four outcomes: 𝐻𝐻, 𝐻𝑇, 𝑇𝐻, 𝑇𝑇

Sample space and events The set of all possible outcomes is the sample space,usually denoted by 0. o i Example.=HH,HT,TH,TT. Event:a subset of sample space. A is a set of possible outcomes Example.A =(HH,T'T,the event that the two coins give the same side

Sample space and events ◼ The set of all possible outcomes is the sample space, usually denoted by Ω. ❑ Example. Ω = 𝐻𝐻, 𝐻𝑇, 𝑇𝐻, 𝑇𝑇 . ◼ Event: a subset of sample space. ❑ 𝐴 ⊆ Ω is a set of possible outcomes ❑ Example. 𝐴 = 𝐻𝐻, 𝑇𝑇 , the event that the two coins give the same side

Infinite sample space The sample space of an experiment may consist of a finite or an infinite number of possible outcomes. Finite sample spaces are conceptually and mathematically simpler. Sample spaces with an infinite number of elements are quite common. 0ts501.co02 As an example,consider throwing Double 2x Single Single a dart on a board and viewing Treble 3x Single Single the point of impact as the Outer Bullseye (25 Points) outcome. Bullseye(50 Points】 The region“Bullseye”is an event: it's a subset of the sample space. 9ts501.c0

Infinite sample space ◼ The sample space of an experiment may consist of a finite or an infinite number of possible outcomes. ❑ Finite sample spaces are conceptually and mathematically simpler. ❑ Sample spaces with an infinite number of elements are quite common. ◼ As an example, consider throwing a dart on a board and viewing the point of impact as the outcome. ◼ The region “Bullseye” is an event: it’s a subset of the sample space