Probability and Statistics Dr. Saeid Moloudzadeh Sample Space and Events 1 Contents Descriptive Statistics Axioms of Probability Combinatorial Methods Conditional Probability and Independence Distribution Functions and Discrete Random Variables Special Discrete Distributions Continuous Random Variables Special Continuous Distributions Bivariate Distributions

Probability and Statistics Contents Descriptive Statistics Axioms of Probability Combinatorial Methods Conditional Probability and Independence Distribution Functions and Discrete Random Variables Special Discrete Distributions Continuous Random Variables Special Continuous Distributions Bivariate Distributions 2

Chapter 1: Axioms of Probability Context Sample Space and Events Axioms of Probability Basic Theorems 3

Chapter 1: Axioms of Probability Context Sample Space and Events Axioms of Probability Basic Theorems 4

Section 2: Sample Space and Events Definition (Experiment): An experiment is an act or process of observation that leads to a single outcome that cannot be predicted with certainty. A sample point is the most basic outcome of an experiment. Definition (Sample space): The set of all possible outcomes to a random experiment (all the sample points) is called the sample space and is denoted by S. 5

Section 2: Sample Space and Events Example 2.1: Some examples of experiments and their sample spaces are as follows: (a) If the outcome of the experiment is the gender of a child, then S = {g, b} where outcome g means that the child is a girl and b that it is a boy. (b) If the experiment consists of flipping two coins and noting whether they land heads or tails, then : S = {(H,H), (H, T), (T,H), (T, T)} The outcome is (H, H) if both coins land heads, (H, T) if the first coin lands heads and the second tails, (T, H) if the first is tails and the second is heads, and (T, T) if both coins land tails. 6

Section 2: Sample Space and Events (c) If the experiment consists of tossing two dice, then the sample space consists of the 36 points S = {(i, j): i, j = 1, 2, 3, 4, 5, 6} where the outcome (i, j) is said to occur if i appears on the leftmost die and j on the other die. Definition (Sample events): Any set of outcomes of the experiment is called an event. That is, an event is a subset of the sample space. Events will be denoted by the capital letters A, B, C and so on. We say that the event A occurs whenever the outcome is contained in A. 7

Section 2: Sample Space and Events Example 2.2: In Example 2.1 (a), if A = {g}, then A is the event that the child is a girl. Similarly, if B = {b}, then B is the event that the child is a boy. In Example 2.1 (b), if A = {(H, H), (H, T)}, then A is the event that the first coin lands on heads. In Example 2.1 (c), if A = {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)} then A is the event that the sum of the dice is 7. 8

Section 2: Sample Space and Events In the study of probability theory the relations between different events of an experiment play a central role. In the remainder of this section we study these relations. In all of the following definitions the events belong to a fixed sample space S. Definition: Subset: An event A is said to be a subset of the event B if, whenever A occurs, B also occurs. This means that all of the sample points of A are contained in B. Hence considering A and B solely as two sets, A is a subset of B in the usual set-theoretic sense: that is, 9

Section 2: Sample Space and Events Equality: Events A and B are said to be equal if the occurrence of A implies the occurrence of B, and vice versa; that is, if and hence A=B. Intersection: An event is called the intersection of two events A and B if it occurs only whenever A and B occur simultaneously. In the language of sets this event is denoted by AB or because it is the set containing exactly the common points of A and B. 10

Section 2: Sample Space and Events Union: An event is called the union of two events A and B if it occurs whenever at least one of them occurs. This event is since all of its points are in A or B or both. Complement: An event is called the complement of the event A if it only occurs whenever A does not occur. The complement of A is denoted by Difference: An event is called the difference of two events A and B if it occurs whenever A occurs but B does not. The difference of the events A and B is denoted by A-B. 11

Section 2: Sample Space and Events Certainty: An event is called certain if its occurrence is inevitable. Thus the sample space is a certain event. Impossibility: An event is called impossible if there is certainty in its nonoccurrence. Therefore, the empty set is an impossible event. Mutually Exclusiveness : If the joint occurrence of two events A and B is impossible, we say that A and B are mutually exclusive or disjoint. Thus A and B are mutually exclusive if the occurrence of A precludes the occurrence of B, and vice versa. Since the event representing the joint occurrence of A and B is AB, their intersection, A and B, are mutually exclusive if AB = 12

Section 2: Sample Space and Events Shaded region in the following figure presents Venn diagrams for AB, A and B are disjoint events. 13

Section 2: Sample Space and Events Example 2.3: At a busy international airport, arriving planes land on a first-come, first-served basis. Let E = there are at least five planes waiting to land, F = there are at most three planes waiting to land, H = there are exactly two planes waiting to land. Then 14

Section 2: Sample Space and Events 15

Section 2: Sample Space and Events 16

Section 2: Sample Space and Events 17