Basic ideas 1.2 Sample space Event

Definition1.1 The of a random experiment The set of all possible outcomes of a random experiment is called the of the experiment and is is called the sample space of the experiment and is denoted by S. The possible outcomes themselves are called The possible outcomes themselves are called sample and are denoted by etc. points or elements and are denoted by etc. Running our random experiment should result in exactly one of these outcomes. E1:If we denote the five people as A, B, C, D, E and

E3: Let S 3 =the event that the outcome n is an odd disregard the order in which they were chosen. Let = the event that person E is chosen. S 1 = the event that person E is chosen. E2: Let S 2 =the event that two heads are obtained. number. er of elements or an infinite number which can be listed If the sample space S contains either a finite numb- in Then Then

(E1) and (E2) are finite whereas(E3) is succession (countably infinite) then we call S a discrete sample space. Examples (E1),(E2) and (E3) each have a discrete sample sample space; Countably infinite. If S consists of an uncountable set of outcomes, example all possible values in the range [a, b] where for a<b, Example (E4) has a continuous sample space. then we call S a continuous sample space.

Give out the sample spaces of the following random 1.Record the sum of all numbers for a roll of three products which meet the standard. Answer Class Exercise experiments: dice ; 2.Record the sum of the products until there are 10

in A then we say that A has occurred. If the actual outcome of the experiment is contained Definition 1.2 An event is a subset A, of the sample space. i.e. set of some of the possible outcomes of the experiment. is a it is a life in years of a certain electronic component, then the event A that the component fails before the end of the fifth year is the subset Example 1.1 where t is the Given the sample space

the null set and denoted by the symbol Ø, It is conceivable that an event may a subset that inc- ludes the entire sample space S, or a subset of S called Which conta- ins no elements at all. of 7 are the odd numbers 1 and 7. For instance,if then Bmust be the null set, the only possible Factors

Definition 1.3 The complement of an event A with respect to S is the subset of all elements of S that are not in A.We denote the complement of A by the symbol A C A B S

the deck is not a red but a black card. Example 1.2 Let R be the event that a red card is selected from an ordinary deck of 52 playing cards, and let S be the entire deck. Example 1.3 Consider the sample space S={book, catalyst, cigarette, precipitate, engineer, river} Let A ={catalyst, river, book, cigarette}. Then is the event that the card selected from Then ={precipitate, engineer}.

Definition 1.4 The intersection of two events A and B,denoted by the symbol are common to A and B. AB A B S is the event containing all elements that Example 1.4 Let P be the event that a person selected at random while dining at a popular cafeteria is a taxpayer, and let Q be the event that the person is over 65 years of age.

cafeteria who are over 65 years of age. Then the event is the set of all taxpayers in the Example 1.5 Let That is, M and N have no elements in common and, therefore, cannot both occur simultaneously. then For certain statistical experiments it is by no means unusual to define two events,A and B, that cannot both occur simultaneously. The events A and B are then said to be mutually exclusive. Stated more formally,we have the following definition:

Definition 1.5 Two events A and B are said to be disjoint or mutuall yexclusive if More generally the events A 1, A 2, A 3,…are said to be S A B pairwise disjoint or mutually exclusive if whenever

Example 1.6 A cable television company offers programs of eight Different channels, three of which are affiliated with ABC,two with NBC, The other two are an educational channel and the ESPN sports channel. and one with CBS. Solution: Suppose that a person subscribing to this Service turns on a television set without first selecting to the NBC network and B the event that it belongs to the CBS network. the channel. Let A be the event that the program belongs Since a television program cannot belong to more than one network, the events A and B

have no programs in common. ally exclusive. Therefore, grams, the intersection contains no pro- and consequently the events A and B are mutu- Definition 1.6 The union of the two events A and B,denoted by the that belong to A or B or both. symbol is the event containing all the elements Example 1.7 S B A Let then

Example 1.8 Let P be event that an employee selected at random from an oil drilling company smokes cigarettes. Let Q be the event that the employee selected drinks alcoholic beverages. either drink or smoke,or do both. Example 1.9 Then the event is the set of all employees who then If

Definition 1.7 An event A is said to imply an event B if S B A This means that if A occurs then B necessarily occurs since the outcomes of the experiment is also an element of B. We denoted by A=B. Definition 1.8 An event A is said equal to event B if

disjointcomplement S A B S A B A 、 B disjoint A 、 B complement The difference of disjoint and complement

(ii) Associativity: (iii) Distributivity: (vi) de Morgan’s laws: (i) commutatively: Let A, B and C be subsets of some universal set S:

Class Exercise B and C, where A, B and C are three random events: (1) A happens, but B and C do not; (5) Three events do not happen at all; (2) A and B happen but C does not; (3) All three events happen; (4) At least one of the three events happens; (6) Not more than one event happens; e.g. 1 Try to represent the following events with A, Solution:

(7) Not more than two of them happen; (8) At least two of them happen ; (9) At least one of A and B happen but C does not; (10) Just two of them happen. Solution:

Conclusion ple space and random event. Random experimentSample space subset Random event Basic/Simple event Determined event Impossible event Complicated event Random event 1.The relationship among random experiment, sam-

theory and Set theory: Sample space, determined eventWhole set Impossible eventEmpty set Basic or simple event Element Event or random eventSubset The opposite event of A Complement set of A Event A implies event BA is a subset of B Event A equals event B A is B SignProbability theorySet theory 2. The signs & its relationship between Probability

The difference of event a And event B The difference of set A and B Event A and event B are Mutually exclusive (disjointed) Set a and B are disjointed The union of event A and event B The union of set A and B The intersection of event A and event B The intersection of set A and B

(1) A happens, but B and C do not: (5) Three events do not happen at all: (2) A and B happen but C does not: (3) All three events happen: (4) At least one of the three events happens: (6) Not more than one event happens: Solution:

(7) Not more than two of them happen: (8) At least two of them happen : (9) At least one of A and B happen but C does not: (10) Just two of them happen :