# Chapter 1: Discrete and Continuous Probability Distributions

**Topics:**Random variable, Probability theory, Probability distribution

**Pages:**22 (5123 words)

**Published:**August 31, 2013

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

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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}

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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 elements that are common to A and B. A∩ B is the event that both A and B occur. Union: The union of the events A and B, denoted by A∪ B, is the event containing all the elements that belong to A or B or both. A∪ B is the event that A or B occurs Complement: The complement of an event A with respect to S is the set of all elements of S that are not in A. We denote the complement of A by the c symbol A', ̅ or A . A' is the event that A does not occur. Mutually Exclusive: When two events A and B have no outcomes in common, they are said to be mutually exclusive, or disjoint events. In other words, A I B = ∅ If A is an event and A’ is its complement, then P(A) = 1 − P(A’)

Sample space S with events A, B and C.

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Chapter 1

Example 1.3 : Let A be an event of obtaining ‘at least three head’ when 4 coins are tossed. Find A and A’. Solutions: S={HHHH, HHHT, HHTH, HTHH, THHH, HHTT, THTH, THHT, HTTH, HTHT, HHTT, TTTH, TTHT, THTT, HTTT, TTTT}; A = {HHHH, HHHT, HHTH, HTHH, THHH}; # S(A U A’) = 16

# S(A) = 5

A = {HHTT, THTH, THHT, HTTH, HTHT, HHTT, TTTH, TTHT, THTT, HTTT, TTTT}; # S(A’) = 11

A follow-up question for you: If B is the event that the first toss and the last have the same result, find B', A∩ B, A∩ B' and A'∩B . Example 1.4 : For the experiment in which the number of pumps in use at a single six-pump gas station is observed. Let A = {0, 1, 2, 3, 4}, B = {3, 4, 5, 6}, and C = {1, 3, 5}. Then find (b) A U B , (c) A U C , (d) ( A U C ) ' (a) A I B Solutions: (a) AIB = {0, 1, 2, 3, 4} I {3, 4, 5, 6} = {3, 4} (b) AUB = {0,1,2,3,4} U {3, 4, 5, 6} = {0, 1, 2, 3, 4, 5, 6} = S (c) AUC = {0, 1, 2, 3, 4}U {1, 3, 5}= {0, 1, 2, 3, 4, 5} (d) (AUC)’= S – (AUC) = {6} [Given sets E and F , the difference E −F is the set { x | x∈ E but x∉ F}.]

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Tri1 2013/14

Chapter 1

1.2

Axioms, Interpretations, and Properties of Probability

Axioms...

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