Chapter 1: Discrete and Continuous Probability Distributions
TEM1116 Probability and Statistics
Tri1 2013/14
Chapter 1
Chapter 1: Discrete and Continuous Probability Distributions
Section 1: Probability
Contents: 1.1 1.2 1.3 1.4 1.5 Some basics of probability theory Axioms, Interpretations, and Properties of Probability Counting Techniques and Probability Conditional Probability Independence
TEM1116
1
TEM1116 Probability and Statistics
Tri1 2013/14
Chapter 1
1.1
Basics of Probability Theory
Probability refers to the study of randomness and uncertainty. The word “probability” as used in “probability of an event” is a numerical measure of the chance for the occurrence of an event. Experiment: a repeatable procedure with a well-defined set of possible outcomes. (Devore: Any action or process whose outcome is subject to uncertainty.) Sample Space and Events Sample space of an experiment is the set of all possible outcomes. An event is a set of outcomes (it is a subset of the sample space). Example: Consider an experiment of rolling a 6-sided die.
Sample Space, S :
{1, 2, 3, 4, 5, 6}
S
Events, Ek: E1: composite number is rolled. → Equivalently, {4, 6}. E2: number less than four is rolled. → Equivalently, {1, 2, 3}.
E1
E2
Example 1.1 : An experiment consists of tossing three coins. Find the sample space if (i) We are interested in the observed face of each coin, (ii) We are interested in the total number of heads obtained. Solution: (i) S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT} (ii) S = {0, 1, 2, 3}
TEM1116
2
TEM1116 Probability and Statistics
Tri1 2013/14
Chapter 1
Example 1.2 : How many sample points are in the sample space when a pair of dice is thrown once? [The answer would depend on what is observed.] Solution: Suppose we observe the numbers that appear face-up. S = {(1,1), (1,2), …, (6,6)} Number of sample points 6x6=36 Relations from Set Theory Intersection: The intersection of two events A and B, denoted by A∩ B, is the event containing all
Tri1 2013/14
Chapter 1
Chapter 1: Discrete and Continuous Probability Distributions
Section 1: Probability
Contents: 1.1 1.2 1.3 1.4 1.5 Some basics of probability theory Axioms, Interpretations, and Properties of Probability Counting Techniques and Probability Conditional Probability Independence
TEM1116
1
TEM1116 Probability and Statistics
Tri1 2013/14
Chapter 1
1.1
Basics of Probability Theory
Probability refers to the study of randomness and uncertainty. The word “probability” as used in “probability of an event” is a numerical measure of the chance for the occurrence of an event. Experiment: a repeatable procedure with a well-defined set of possible outcomes. (Devore: Any action or process whose outcome is subject to uncertainty.) Sample Space and Events Sample space of an experiment is the set of all possible outcomes. An event is a set of outcomes (it is a subset of the sample space). Example: Consider an experiment of rolling a 6-sided die.
Sample Space, S :
{1, 2, 3, 4, 5, 6}
S
Events, Ek: E1: composite number is rolled. → Equivalently, {4, 6}. E2: number less than four is rolled. → Equivalently, {1, 2, 3}.
E1
E2
Example 1.1 : An experiment consists of tossing three coins. Find the sample space if (i) We are interested in the observed face of each coin, (ii) We are interested in the total number of heads obtained. Solution: (i) S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT} (ii) S = {0, 1, 2, 3}
TEM1116
2
TEM1116 Probability and Statistics
Tri1 2013/14
Chapter 1
Example 1.2 : How many sample points are in the sample space when a pair of dice is thrown once? [The answer would depend on what is observed.] Solution: Suppose we observe the numbers that appear face-up. S = {(1,1), (1,2), …, (6,6)} Number of sample points 6x6=36 Relations from Set Theory Intersection: The intersection of two events A and B, denoted by A∩ B, is the event containing all