MTH 108 Business Math I Lecture 28 Chapter
MTH 108 Business Math I Lecture 28
Chapter 13 Introduction to Probability Theory
Review Sets; elements Representation; two ways Set notation; elements; number of elements Special sets: Universal set; Complement; Null set; Subset • Venn diagram • Set operations: equality; union; intersection • Properties of set operations • •
Review • • Number of elements Counting method; tree diagram Fundamental principles of counting Permutation counting rule Factorial formula Combination Relationship between permutation and combination
Today’s Topics • Sample spaces and events
Introduction • The notion of probability is associated with random process, or random experiments. Probability deals with uncertainty. E. g. consider the following statements with an element of uncertainty: i. I will probably go shopping after work ii. Chances are it will rain this afternoon iii. The probability of heads is ½ iv. The odds on Pacemaker to place are 3 to 2 v. 3 out of 5 smokers develop lung cancer
Basic concepts We hear and read these kinds of "probabilistic" statements every day can they be analyzed mathematically? A random experiment is a process which results in one of a number of possible outcomes. The possible outcomes are known prior to the performance of an experiment, but one can not predict with certainty which particular outcome will result. Random experiment---a process whose outcome cannot be predicted.
Examples Flipping a coin Rolling a die Drawing a card from a well-shuffled deck 3 out of 5 smokers develop lung cancer Defective products produced during the manufacturing in a random manner • The times between the arrivals of telephone calls at a telephone exchange. • • •
Sample Space • Each repetition of an experiment can be thought of as a trial. Each trial has an observable outcome. • The set of all possible outcomes for an experiment is called the sample space S such that each outcome corresponds to exactly one element in S. • The elements of S are called simple outcome or sample points. • n(S) denotes the size of a sample space. If there are finite number of sample points, then S is said to be a finite sample space, otherwise, S is called an infinite sample space.
Examples • Flipping a coin twice Do not mix the sample points with the possible outcomes of permutations!!!
Examples • Time between arrivals of telephone calls The sample space of an experiment which measures the time between the arrivals of telephone calls at a telephone exchange will be defined as: • Rolling a die
Remarks • An experiment may have more than one sample space. • When an experiment has more than one sample space, it will be our practice to consider only a sample space that gives sufficient details to answer all pertinent questions relative to an experiment. This sample space is refer to as the usual sample space.
Basic concepts • Given an experiment, outcomes (sample points) are frequently classified into events. An event E for an experiment is a subset of a sample space S. • event has a specific technical meaning in probability theory. The way in which events are defined depends upon the set of outcomes for which probabilities are to be computed.
Examples Rolling a coin twice: Rolling a coin once:
Examples Time between telephone calls: Suppose that an experiment consists of selecting three manufactured parts from a production process and observing whether they are acceptable or defective a) Determine the sample space S b) The outcomes in event ‘exactly 2 acceptable parts’ c) The outcomes in event ‘at least one defective part’
Examples
Number of events Number: If a sample space has n elements, then the number of possible events is
Observations • If an event consists of only 1 outcome, it is called a simple event. E. g. flipping a coin once. • If an event consists of more than 1 outcome, it is called a compound event. E. g. flipping a coin twice
Observations • If the outcome of an experiment is a sample point in E, then event E is said to occur. e. g. rolling a die • A sample space is a subset of itself, so it is too an event, called a certain event, it must occur no matter what the outcome is. • If an event contains no sample points, it is a null set called impossible event.
Observations • Events can sometimes be described in words e. g. , for the flipping-a-coin-twice experiment: In words Set notation • no heads {TT} • one tail {TH, HT} • at least one head {TH, HT, HH} • more than one head {HH}
Some examples of sample space 1) An urn contains four marbles, 1 red, 1 pink, 1 black, and 1 white. a) A marble is drawn at random, its colour is noted, and is placed back into the urn. Then a marble is again randomly drawn and its colour is noted. Describe a sample space and determine the number of sample points. This is an example of marbles drawn with replacement.
b) Determine the number of sample points in the sample space if two marbles are selected in succession without replacement and the colours are noted.
2) A pair of dice is rolled once and for each die the number that turns up is observed. Describe a sample space.
Set operations: Events Since events and sample space can also be sometimes represented by a Venn diagram. With the help of Venn diagram, we can also show other set operations, i. e. of complement, union and intersection. 1) E is a subset of S 2) Complement of E
3) Union of events 4) Intersection of events a) For the event to occur, at least one of the events E and F must occur. b) If event occurs, both E and F must occur. c) If event occurs, then E does not occur.
Properties of events Since events are sets and subsets of the sample space S, they satisfy the following properties along with the other properties of sets discussed in lecture 26.
Mutually exclusive events A set of events is called mutually exclusive if the occurrence of any one of the events, stops the existence of the other events. Specifically, the events are mutually exclusive if. Events are mutually exclusive if they do not occur simultaneously. E. g. 1) 2) Flipping a coin once
Mutually exclusive events 3) Selecting three manufactured parts from a production process. a) The outcomes in event ‘exactly 2 acceptable parts’ b) The outcomes in event ‘at least one defective part’
Example A red and a green die are thrown and the numbers on each are noted. Which pairs of the following events are mutually exclusive?
Collectively exhaustive events A set of events is said to be collectively exhaustive if their union accounts for all possible outcomes of an experiment (i. e. their union is the sample space). e. g. 1) flipping a coin once 2) Rolling a pair of dice
Example Consider a sample space of college admissions with sample points And the events Then,
Example
Summary Random experiment; outcomes Sample space: trial; sample point, finite, infinite Event; examples Number of events Simple event; compound event; certain event; impossible event • Size of a sample space • Set operations and properties of events • Mutually exclusive and collectively exhaustive events • • •
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