Chapter 1 Random events and the probability §1. Random event

1 Random Experiments The basic notion in probability is that of a random experiment: an experiment whose outcome cannot be determined in advance, but is nevertheless still subject to analysis.

Examples of random experiments are: E1. tossing a die E2. measuring the amount of rainfall in ZhengZhou in February, E3. counting the number of calls arriving at a telephone exchange during a fixed time period, E4. selecting a random sample of fifty people and observing the number of left-handers, E5. choosing at random ten people and measuring their height. 1, 2, 3, 4, 5 or 6.

2Sample Space and Event 2 Sample Space and Event

Definition The set of all possible outcomes of a random experiment is called the sample space of the experiment and is denoted by S or Ω. The possible outcomes themselves are called sample points or elements and are denoted by e 1,e 2 etc.

Examples of random experiments with their sample spaces are:

Discrete and continuous sample spaces Definition: A sample space is finite if it has a finite number of elements. Definition: A sample space is discrete if there are “gaps” between the di ff erent elements, or if the elements can be “listed”, even if an infinite list (eg. 1, 2, 3,...). In mathematical language, a sample space is discrete if it is countable. Definition: A sample space is continuous if there are no gaps between the elements, so the elements cannot be listed (eg. the interval [0, 1]).

Definition of events Definition 1.2: An event is a subset of the sample space. That is, any collection of outcomes forms an event. Events will be denoted by capital letters A,B,C,.... Note: We say that event A occurs if the outcome of the experiment is one of the elements in A. Note: Ω is a subset of itself, so Ω is an event, it is called certain event. The empty set, ∅ = {}, is also a subset of Ω. This is called the null event, or the event with no outcomes.

Examples of events are: 1. The event that the sum of two dice is 10 or more, 2. The event that a machine lives less than 1000 days, 3. The event that out of fifty selected people, five are left-handed,

Example: Suppose our random experiment is to pick a person in the class and see what forms of transport they used to get to campus yesterday.

Opposites: the complement or ‘not’ operator Definition The complementof 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 of. S B A

3.the intersection ‘and’ operator Definition The intersection of two events A and B, denoted by the symbol A∩B or A, is the event containing all elements that are common to A and B.

Definition Two events A and B are said to be disjointif A∩B =Ø. More generally the events A 1,A 2,A 3,…… are said to be pairwise disjoint or mutually exclusive if A i ∩ A j =Ø whenever i≠j. S A B

Note: S S A BA B A 、 B Opposite A 、 B disjoint disjoint Opposite A ∪ B=S and AB=Ø

the union ‘or’ operator Definition The union of the two events A and B, denoted by the symbol A ∪ B, is the event containing all the elements that belong to A or B or both.

Definition An event A is said to imply an event B if A ⊂ B. This means that if A occurs then B necessarily occurs since the outcomes of the experiment is also an element of B. S B A

Definition An event A is equal to event B if and only if (iff) A ⊂ B and B ⊂ A (denoted by A=B).

Examples: Experiment: Pick a person in this class at random. Sample space: Ω = {all people in class}. Let event A =“person is male” and event B = “person travelled by bike today”. Suppose I pick a male who did not travel by bike. Say whether the following events have occurred:

A =“person is male” B = “person travelled by bike today”. pick a male who did not travel by bike

Properties of union, intersection, and complement

Distributive laws

Exercise 1. Consider the set S={1,2,3,4,5,6,7,8,9} with subsets A={1,3,5,7,9},B={2,4,6,8},C={1,2,3,4}, D={7,8} Find the following sets: (1) (2) A∩D;(3)A ∪ B (4) (5)

2. If the sample space is S=A ∪ B and if P(A)=0.8 and P(B)=0.5, find P(A ∩ B)