Probability Theory Review
An Intuitive Approach
Probability is the branch of mathematics concerned with the assessment and analysis of uncertainty. The theory of probability provides the means to rationally model, analyze and solve problems where future events cannot be foreseen with certitude. In numerous managerial decision problems, especially those with strategic implications, it is not possible to ascertain the results to be obtained when choosing a course of action. In such cases, if one wants to act rationally —to maximize the chances of attaining one’s goal— it becomes necessary to explicitly deal with the uncertainty posed by the problem. Thus, probability theory is indispensable for rational decision making.
Probability is not statistics. Statistics is a distinct field of applied mathematics dedicated to the collection, analysis, interpretation, and presentation of quantitative and qualitative data. Statistics employs probability theory to make inferences about contingent events based on sample information (statistical data) pertaining to those events or related events deemed of relevance. Consequently, statistical models can be used only if the required data is obtainable. This precludes their use in managerial situations where the problem under consideration is, as often happens, unique and little or no empirical data is available or procurable. Probability, however, can always be invoked when facing uncertainty, even in the absence of statistical data.
Sets, Outcomes and Events
The fundamental concepts of probability are best explained by means of the elementary theory of sets. (Actually, modern probability theory is thoroughly grounded on set theory.) First we need to define some basic terms.
• Set – a clearly defined collection of elements.
Sets are usually represented by listing their elements within braces { } or by stating some condition the elements must satisfy for membership in the set.
• Subset – a set where each element is also a member of some referred set.
• Outcome – the actual result obtained from engaging in a specific activity.
• Event – a set of possible outcomes. Two types of events can be distinguished:
? Compound event – a set consisting of two or more possible outcomes.
? Elementary event – a set consisting of only one possible outcome.
Thus, an elementary event limits the range of possible outcomes to a single result. In practice, elementary events are generally synonymous with outcomes. There is, however, a subtle distinction of import in decision modeling: outcomes refer to concrete results; events are abstractions defined by the problem analyst. Outcomes comprise the low-level domain of objective results associated with a given activity. Events reflect the high-level perspective adopted by the analyst to refer to a range of possible outcomes. Hence, events are subjective categorizations considered by the analyst to be germane to the problem. The caveat here is that different perspectives and categorizations can lead to different models and, conceivably, even different conclusions regarding the very same problem. This is something the analyst should always keep in mind.
Moreover, outcomes are always unique whereas more than one event can refer to a given outcome. Again, just what gets defined as an event depends on the analyst’s conception of the problem. Defining an elementary event implies that the analyst’s view of possible outcomes in a given problem situation coincides with a single actual possibility. The analyst’s subjectivity in envisioning and modeling the problem, however, is still very much operant. It is important to be mindful of the nature of the distinction between outcomes and events when building and interpreting models.
Outcomes, then, are the actual elemental (atomistic or non-decomposable) results that arise from a given activity. Events are subjectively defined categorizations (groupings) consisting of one or more possible outcomes. An event is said to occur if any one outcome in the event set happens to transpire.
Example 1: Consider a die (a cube with six sides each marked with one to six pips (dots)). A roll of the die can result in six possible outcomes denoted by the set of numbers {1, 2, … , 6}. Each of the six outcomes can also be viewed as an elementary event. The event “Even” (face comes up an even number) is a compound event that can be defined as E = {2, 4, 6}.
Colorful dice from India come in a variety of shapes and sizes.
Many are rectangular parallelepipeds with only four marked sides.
The principle, however, is the same.
Example 2: Consider the introduction of a new product into the market. Define the event S to be “Introduction is a success” if first-month sales reach or exceed 500 units. S is then a compound event equal to the set of outcomes {x | x = 500}, where x is the amount of units sold during the first month. (In words, S is the set of all x such that x is greater than or equal to five hundred.)
Example 3: Consider the toss of a fair coin. “Heads” is a possible outcome that can also be regarded as an elementary event: H = {heads}.
“Fair” in relation to tossing coins and rolling dice means that the devices are symmetrically balanced and that the tossing and rolling are honest.
We can think of events as subsets of the universe of possible outcomes that are deemed significant in the context of a given problem.
Example 4: Consider two tosses of a fair coin. The universe of possible outcomes is the set U = {HH, HT, TH, TT}, where the paired letters inside the braces stand for the outcomes “heads” or “tails” in the first and second tosses, respectively. Let A be the event “At least one head comes up.” Then A = {HH, HT, TH}. Note that A is a subset of U.
In general, all events are subsets of U. Thus, we obtain a second definition for the term:
• Event – a subset of the universe of possible outcomes U.
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Online Resources
Events
Probability
Probability Theory
Set
Set Theory
Axiomatic Set Theory
Naïve Set Theory
Probability Theory
Probability
Set
Set Theory
Set Theory
Basic Set Theory
The Beginnings of Modern Probability Theory
A Brief History of Probability
Figures from the History of Probability & Statistics
A Short History of Probability & Statistics
A History of Set Theory
Sets: An Introduction
Set Theory
