EC255 Chapter Notes - Chapter 4: Elementary Event, Statistical Inference, Sample Space
4.1 Introduction to Probability
• Probability is the basis for inferential statistics
• Recall: Inferential statistics involves taking a sample from a population, computing a statistic on
the sample, and inferring from it the value of the corresponding parameter of the population
4.2 Methods of Assigning Probabilities
• Three general methods: (1) the classical method, (2) the relative frequency of occurrence
method, (3) subjective probabilities
Classical Method of Assigning Probabilities
• Classical method: when probabilities are assigned based on laws and rules
• Involves an experiment (a process that produces outcome) and an event (an outcome of an
experiment)
• Probability of an individual event occurring is determined as the ratio of the number of items in
a population containing the event (ne) to the total number of items in the population (N)
o P(E) = ne/N
▪ N = total possible number of outcomes of an experiment
▪ ne = the number of outcomes in which the event occurs out of N outcomes
o E.g. If a company has 200 workers and 70 are female, probability of randomly selecting a
female from company is 70/200 = 0.35
• Probabilities can be determined a priori—they can be determined prior to the experiment
• Bc ne can never be greater that N, the highest value of any probability is 1 (certain to occur)
• Smallest possible probability is 0 (event is certain to not occur
• Range of possile proailities: ≤ PE) ≤
Relative Frequency of Occurrence
• Relative frequency of occurrence method: based on cumulated historical data
• The probability of an event occurring is equal to the number of times the event has occurred in
that past divided by the total number of opportunities for the event to have occurred
o # of times the event has occurred
Total number of opportunities for the event to occur
• Not based on rules or laws but on what has occurred in past
• E.g. supplier sent company 90 batches in the past, and inspectors rejected 10 of them. Using this
method, the probability that the inspectors will reject next batch is 10/90, or 0.11
o If next batch is rejected, relative frequency of occurrence probability for the subsequent
shipment would change to 11/91 = 0.12
Subjective Probability
• Subjective method: based on the feeling or insights of the person determining the probability
• Comes from the person's intuition or reasoning
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• Can be used to capitalize on background of experienced workers and managers in decision
making
4.3 Structure of Probability
Experiment
• Experiment: a process that produces outcomes
• Examples of business-oriented experiments:
o Interviewing 20 randomly selected consumers and asking which brand of appliance they
prefer
o Sampling every 200th bottle of ketchup from an assembly line and weighing the contents
o Auditing every 10th account to detect any errors
Event
• Event: an outcome of an experiment
• Experiment defines the possibilities of the event
• If the experiment is to sample five bottles coming off production line, an event could be to get
one defective and 4 good bottles
• Denoted by uppercase letters
o Italic capital letters (e.g. A, and E1, E2,… represent the general or astrat ase
o Roman capital letters (e.g. H and T for heads and tails) denote specific things/people
Elementary Events
• Elementary events: events that cannot be decomposed or broken down into other events
• Denoted by lowercase letters (e.g. e1, e2,…
• Suppose experiment is to roll a die
o Elementary events may be to roll a 1 or roll a 2 or 3, etc.
o Rolling an even number is an event but not an elementary event bc it can be broken
down further (2, 4, 6)
o There are six elementary events {1, 2, 3, 4, 5, 6}
o Rolling a pair of dice => 36 possible elementary events
Sample Space
• Sample space: a complete roster or listing of all elementary events for an experiment
• E.g. sample space for the roll of a single die is {1, 2, 3, 4, 5, 6}
• Can help in finding probabilities but can be unwieldy when sample space is large
Table 4.1 All Possible Elementary Events in the Roll of a Pair of Dice (Sample Space)
(1,1)
(2,1)
(3,1)
(4,1)
(5,1)
(6,1)
(1,2)
(2,2)
(3,2)
(4,2)
(5,2)
(6,2)
(1,3)
(2,3)
(3,3)
(4,3)
(5,3)
(6,3)
(1,4)
(2,4)
(3,4)
(4,4)
(5,4)
(6,4)
(1,5)
(2,5)
(3,5)
(4,5)
(5,5)
(6,5)
(1,6)
(2,6)
(3,6)
(4,6)
(5,6)
(6,6)
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Unions and Intersections
• Set notation: the use of braces to group numbers
o Used as a symbolic tool for unions and intersections
• The union of X, Y, is formed by combining elements from both sets and is denoted "X U Y"
o An element qualifies for the union of X, Y, if it is in either X or Y or both X and Y
o The union expression X U Y can be translated to "X or Y"
o e.g. if
(not 8)
o Note: all the values of X and all the values of Y qualify for the union but none of the
values is listed more than once in the union
• An intersection is denoted X ᴖ Y
o To qualify for an intersection, an element must be in both X and Y
o The intersection contains the elements common to both sets
o Intersection of X, Y is referred to as X and Y
o e.g. if
o More exclusive to qualify
Mutually Exclusive Events
• Two or more events are mutually exclusive events if the occurrence of one event precludes the
occurrence of the other event(s)
o Means that they cannot occur simultaneously and therefore can have no intersection
• E.g. a manufactured part is either defective or acceptable; cannot be both
• Probability of two mutually exclusive events occurring at the same time is zero
• Mutually exclusive events X and Y:
P (X ᴖ Y) = 0
Independent Events
• Two or more events are independent events if the occurrence of one of the events does not
affect the occurrence or nonoccurrence of the other event(s)
• E.g. Rolling dice yield independent events
• E.g. Event of getting a head on the first coin toss is independent of getting a head on the second
toss
• If X and Y are independent:
P (X | Y) = P (X) and P (Y | X) = P (Y)
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Document Summary
4. 2 methods of assigning probabilities: three general methods: (1) the classical method, (2) the relative frequency of occurrence method, (3) subjective probabilities. Classical method of assigning probabilities: classical method: when probabilities are assigned based on laws and rules. Total number of opportunities for the event to occur: not based on rules or laws but on what has occurred in past, e. g. supplier sent company 90 batches in the past, and inspectors rejected 10 of them. Using this method, the probability that the inspectors will reject next batch is 10/90, or 0. 11. If next batch is rejected, relative frequency of occurrence probability for the subsequent shipment would change to 11/91 = 0. 12. Subjective probability: subjective method: based on the feeling or insights of the person determining the probability, comes from the person"s intuition or reasoning, can be used to capitalize on background of experienced workers and managers in decision making. Experiment: experiment: a process that produces outcomes, examples of business-oriented experiments:
