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2013 | OriginalPaper | Chapter

1. Elements of Probability Theory

Author : Ron C. Mittelhammer

Published in: Mathematical Statistics for Economics and Business

Publisher: Springer New York

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Abstract

The objective of this chapter is to define a quantitative measure of the propensity for an uncertain outcome to occur, or the degree of belief that a proposition or conjecture is true. This quantitative measure will be called probability and is relevant for quantifying such things as how likely it is that a shipment of smart phones contains less than 5 percent defectives, that a gambler will win a crap game, that next year’s corn yields will exceed 80 bushels per acre, or that electricity demand in Los Angeles will exceed generating capacity on a given day. This probability concept will also be relevant for quantifying the degree of belief in such propositions as it will rain in Seattle tomorrow, Congress will raise taxes next year, and the United States will suffer another recession in the coming year.

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Footnotes
1
A countably infinite set is one that has an infinite number of elements that can be “counted” in the sense of being able to place the elements in a one-to-one correspondence with the positive integers. An uncountable infinite set has an infinite number of elements that cannot be counted, i.e., the elements of the set cannot be placed in a one-to-one correspondence with the positive integers.
 
2
These experiments were actually performed by the author, except the author did not actually physically flip the coins to obtain the results listed here. Rather, the coin flips were simulated by the computer. In the coming chapters the reader will come to understand exactly how the computer might be used to simulate the coin-flipping experiment, and how to simulate other experiments as well.
 
3
See G. Debreu (1959), Theory of Value: An Axiomatic Analysis of Economic Equilibrium. Cowles Monograph 17. New York: John Wiley, pp. 60–63. Note that additional axioms are generally included that are not needed for the existence of a utility function per se but that lead to a simplification of the consumer maximization problem. See L. Phlips (1983), Applied Consumption Analysis. New York: North Holland, pp. 8–11.
 
4
By definition, since S contains all possible outcomes of the experiment, the event S is then certain to occur.
 
5
A partition of a set \( B \) is a collection of disjoint subsets of B, say \( \left\{ {{B_i},i \in I} \right\} \) such that \( B = { \cup_{{i \in I}}}{B_i} \).
 
6
See A.N. Kolmogorov (1956) Foundations of the Theory of Probability, 2nd ed. New York: Chelsea.
 
7
Unlike the previous example which used a countable sample space, when the sample space is uncountable, not all subsets of S can technically be considered events, i.e., there may be subsets of S to which probability cannot be assigned. The collection of subsets defined here are the Borel sets contained in S, all of which can be considered events in S. We discuss this technical question further in the Appendix to the Chapter.
 
8
We will tacitly assume, unless explicitly stated otherwise, that the orientation of integral ranges is from lowest to highest values in defining the integral over any set A.
 
9
Named after the English mathematician and logician George Boole.
 
10
Named for the Italian mathematician, C.E. Bonferroni.
 
11
See R.C. Buck (1978) Advanced Calculus, 3rd edition, McGraw-Hill, p. 44. We will discuss the concept of limits in more detail in Chapter 5. For now, a more intuitive understanding of limits is sufficient.
 
12
Note that, in a sense, all probabilities could be viewed as conditional, where P(A|S) could be used to denote probabilities in previous sections. We will continue to use “unconditional” to refer to the case where the original sample space, S, has been left “unconditioned.”
 
Metadata
Title
Elements of Probability Theory
Author
Ron C. Mittelhammer
Copyright Year
2013
Publisher
Springer New York
DOI
https://doi.org/10.1007/978-1-4614-5022-1_1