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Appendix A

Probability Theory

STEVEN N. TANI

A reasonable probability is the only certainty.

—E.W. Howe

A.1 Introduction
A.2 Distinctions and the Clarity Test
A.3 Possibility Tree Representation of a Distinction
A.4 Probability as an Expression of Degree of Belief
A.5 Inferential Notation
A.6 Multiple Distinctions
A.7 Joint, Conditional, and Marginal Probabilities
A.7.1 Joint Probability
A.7.2 Marginal Probability
A.7.3 Conditional Probability
A.8 Calculating Joint Probabilities
A.9 Dependent and Independent Probabilities
A.10 Reversing Conditional Probabilities: Bayes’ Rule
A.11 Probability Distributions
A.11.1 Summary Statistics for a Probability Distribution
A.12 Combining Uncertain Quantities
References

# A.1 Introduction

This appendix is not intended to be a comprehensive discourse on probability theory. For that, please refer to any good textbook on probability, such as those by William Feller (1968) and by K.L. Chung and Farid AitSahlia (2003). Rather, we here present a number of key ideas from probability theory that every decision practitioner should know.

# A.2 Distinctions and the Clarity Test

A fundamental requirement for addressing the important uncertainties in a decision is to create distinctions that are both clear and useful. A distinction is a separation of the universe of possibilities (outcomes) into two or more subsets. By definition, these subsets will be mutually exclusive (i.e., a possibility cannot be in more than one subset) and collectively exhaustive (i.e., ...

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